# Subdivision Schemes Based Collocation Algorithms for Solution of Fourth Order Boundary Value Problems

Authors: Syeda Tehmina Ejaz; Ghulam Mustafa; Faheem Khan
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

We present two collocation algorithms based on interpolating and approximating subdivision schemes for the solution of fourth order boundary value problems arising in the mathematical modeling of viscoelastic, and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems. Numerical examples are given to illustrate the algorithms. We conclude that approximating schemes based collocation algorithms give better solution than interpolating schemes based collocation algorithms. Main purpose of this paper is to explore and seek the applications of interpolating and approximating subdivision schemes in the field of boundary value problems along with intrinsic comparison of the results obtained by algorithms based on these schemes. A comparison with other approaches of this type of boundary value problems in order to see the advantages of the proposed methods is also given.


---

## Body

## 1. Introduction

Subdivision schemes propose consistent and efficient iterative algorithms to produce smooth curves and surfaces from a discrete set of control points by subdividing them according to some refining rules, recursively. In recent years, subdivision techniques have become an integral part of computer graphics due to their wide range of applications in many areas such as engineering, medical science, graphic visualization, and image processing. The idea of subdivision has been initiated by de Rham [1]. Later on, Dyn et al. [2] studied a family of schemes with mask of size four, indexed by a tension parameter. Subdivision schemes can be classified into two important branches, approximating and interpolating ones. Approximating scheme means that the limit curve approximates the initial polygon and that, after subdivision, only the newly generated control points are in the limit curve, while interpolating scheme means that, after subdivision, the control points of the original control polygon and the newly generated control points both lie on the limit curve. Mustafa and Rehman [3] unified existing even-point interpolating and approximating schemes by offering general formula for the mask of (
2
m
+
4
)-point even-ary subdivision scheme. Aslam et al. [4] presented an explicit formula which unified the mask of (
2
m
-
1
)-point interpolating as well as approximating schemes. Mustafa et al. [5, 6] presented an explicit formula for the mask of odd-points b-ary interpolating subdivision schemes. Following are the advantages and disadvantages of interpolating and approximating subdivision schemes in the field of geometric modeling:(i)
Interpolating schemes are more useful for engineering applications, especially the schemes with the shape control but approximating schemes do not satisfy the shape control property.
(ii)
Interpolating subdivision schemes have the drawback that, in order to create smoother curves, it is necessary to enlarge the support of the mask. The designers in geometric modeling require subdivision schemes to have their masks with a possibly smaller support and to create good smooth curves.
(iii)
Approximating schemes yield smoother curves with smaller support as compared to the interpolating schemes.In this paper, we want to find the answers of the following questions. Are there attractive characteristics of these schemes in the context of solution of boundary value problems? Do approximating (interpolating) schemes play better role than interpolating (approximation) schemes in this case? To seek answers of these questions, we consider the following interpolating [7–9] and approximating [10] subdivision schemes:(1)
p
2
i
k
+
1
=
p
i
k
,
p
2
i
+
1
k
+
1
=
35
65536
p
i
-
4
k
+
p
i
+
5
k
-
405
65536
p
i
-
3
k
+
p
i
+
4
k
+
567
16384
p
i
-
2
k
+
p
i
+
3
k
-
2205
16384
p
i
-
3
k
+
p
i
+
4
k
+
19845
32768
p
i
k
+
p
i
+
1
k
,
(2)
p
2
i
k
+
1
=
23
16384
p
i
-
3
k
+
p
i
+
3
k
-
285
8192
p
i
-
2
k
+
p
i
+
2
k
+
2073
16384
p
i
-
1
k
+
p
i
+
1
k
+
3333
4096
p
i
k
,
p
2
i
+
1
k
+
1
=
-
3
32768
p
i
-
3
k
+
p
i
+
4
k
-
33
32768
p
i
-
2
k
+
p
i
+
3
k
-
1931
32768
p
i
-
1
k
+
p
i
+
2
k
+
18351
32768
p
i
k
+
p
i
+
1
kwith order of continuity C
4. Schemes (1) and (2) reproduce polynomial curves of degree nine and three by [11] and [10], respectively. Cardinal supports of these schemes are [
-
8,8
] and [
-
6,6
], respectively.We construct collocation algorithms by using the basis functions of the above interpolating and approximating subdivision schemes for the numerical solution of linear fourth order boundary value problems arising in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems, where elastic members serve as key elements for shedding or transmitting loads and in plate deflection theory and many other areas of engineering and applied mathematics. The mathematical form of these types of problems is given by(3)
y
i
v
x
=
a
x
y
x
+
b
x
,
0
⩽
x
⩽
1
,subject to the boundary condition(4)
y
0
=
α
,
y
′
0
=
β
,
y
1
=
γ
,
y
′
1
=
ω
,where a
(
x
) and b
(
x
) are continuous and a
(
x
)
⩾
0 on [
0,1
]. Analytic solution of such type of boundary value problem is possible only in very rare cases. Qu and Agarwal [12, 13] solved this type of problems by interpolatory subdivision scheme based collocation algorithm. But they have computed the solution of second order boundary value problems using 6-point interpolating scheme based collocation algorithm. Mustafa and Ejaz [14] solved third order boundary value problems by using 8-point interpolatory subdivision scheme based collocation algorithm. Until now fourth order boundary value problems have not been solved by subdivision based collocation algorithms. This motivates us to find numerical solution of fourth order boundary value problems by interpolating and approximating subdivision schemes based collocation algorithms.The outline of the paper is as follows. In Section2, we construct subdivision matrices of subdivision schemes (1) and (2) for the computation of eigenvalues and their corresponding (right and left) eigenvectors. Basis functions and their derivatives have been also discussed in this section. In Section 3, subdivision based collocation algorithms for solution of (3) are formulated. Approximation properties of these algorithms are also given in Section 3. In Section 4, numerical examples are presented. Comparison of approximate solutions by interpolating and approximating schemes based collocation algorithms is also given. Conclusion is given in Section 5.


## 2. Basic Properties of the Schemes

In this section, we construct subdivision matrices of the schemes defined in (1) and (2) for the computation of eigenvalues and their corresponding eigenvectors. Basis functions of these schemes and their derivatives have also been discussed in this section.


### 2.1. Subdivision Matrices

IfS
1 and S
2 are subdivision matrices of schemes (1) and (2), then these matrices are defined as(5)
S
1
=
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
,where L
1
=
19845
/
32768, L
2
=
-
2205
/
32768, L
3
=
567
/
16384, L
4
=
-
405
/
65536, L
5
=
35
/
65536, and(6)
S
2
=
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
,where s
1
=
23
/
16384, s
2
=
-
285
/
8192, s
3
=
2073
/
16384, s
4
=
3333
/
4096, s
1
′
=
-
3
/
32768, s
2
′
=
-
33
/
32768, s
3
′
=
-
1931
/
32768, and s
4
′
=
18351
/
32768.The first ten real eigenvalues of matricesS
1 and S
2 are the same which are given as follows: (7)
λ
i
=
1
,
1
2
,
1
4
,
1
8
,
1
16
,
1
32
,
1
64
,
1
128
,
1
256
,
1
512
,
i
=
0,1
,
2
,
…
,
9
.The remaining eigenvalues are complex which are not required. For above eigenvalues λ
i, the eigenvectors υ
R
i and υ
L
i that satisfies S
1
υ
R
i
=
λ
i
υ
R
i and υ
L
i
S
1
T
=
υ
L
i
λ
i are called right and left eigenvectors of the matrix S
1, respectively. We can also define the right ν
R
i and left ν
L
i eigenvectors of S
2 in a similar way. The normalized left and right eigenvectors corresponding to first five eigenvalues of S
1 and S
2 are given in Tables 1 and 2, respectively.Table 1
Eigenvalues and eigenvectors of the matrixS
1.
Eigenvaluesλ
i
Corresponding right and left eigenvectors
1
υ
R
0
=
(
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1
)
T
υ
L
0
=
(
0,0
,
0,0
,
0,0
,
0,0
,
1,0
,
0,0
,
0,0
,
0,0
,
0
)
T
1
2
υ
R
1
=
(
-
8
,
-
7
,
-
6
,
-
5
,
-
4
,
-
3
,
-
2
,
-
1,0
,
1,2
,
3,4
,
5,6
,
7,8
)
T
υ
L
1
=
-
1575
,
-
1474560
,
-
315738080
,
-
1397587968,43588613880
,
-
4311679549440,1336741045920
,
-
4824847319040
,
υ
L
1
=
(
0
,
4824847319040
,
-
1336741045920,4311679549440
,
-
43588613880,1397587968,315738080
,
υ
L
1
=
(
1474560,1575
T
/
5841884245680
1
4
υ
R
2
=
(
64,49,36,25,16,9
,
4,1
,
0,1
,
4,9
,
16,25,36,49,64
)
T
υ
L
2
=
(
459375,215040000,47660030080,103151616000
,
-
3882427261296,23490017902592
,
-
84313449846912
,
υ
L
2
=
(
313469774708736
,
-
497829885297150,313469774708736
,
-
84313449846912,23490017902592
,
-
3882427261296
,
υ
L
2
=
(
103151616000,47660030080,215040000,459375
)
T
/
259624836213120
1
8
υ
R
3
=
(
-
512
,
-
343
,
-
216
,
-
125
,
-
64
,
-
27
,
-
8
,
-
1,0
,
1,8
,
27,64,125,216,343,512
)
T
υ
L
3
=
(
-
104125
,
-
24371200,1177382520
,
-
5986263040
,
-
10571778214,207884427264
,
-
972244098856
,
-
1386160480256
,
υ
L
3
=
(
0
,
1386160480256,972244098856
,
-
207884427264,10571778214
,
5986263040
,
-
1177382520
,
24371200
,
υ
L
3
=
(
104125
)
T
/
9460416859904
1
16
υ
R
4
=
(
4096,2401,1296,625,256,81,16,1
,
0,1
,
16,81,256,625,1296,2401,4096
)
T
υ
L
4
=
(
392875,45977600
,
-
1296269280,5912719360,1180083476
,
-
86261280768,332951715808
,
-
67767008256
,
υ
L
4
=
(
850467338370
,
-
67767008256,332951715808
,
-
86261280768
,
1180083476,5912719360
,
-
1296269280
,
υ
L
4
=
(
45977600,392875
)
T
/
183768238080Table 2
Eigenvalues and eigenvectors of the matrixS
2.
Eigenvaluesλ
i
Corresponding right and left eigenvectors
1
ν
R
0
=
(
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1
)
T
ν
L
0
=
(
0,0
,
0,0
,
0,0
,
1,0
,
0,0
,
0,0
,
0
)
T
1
2
ν
R
1
=
(
-
6
,
-
5
,
-
4
,
-
3
,
-
2
,
-
1,0
,
1,2
,
3,4
,
5,6
)
T
ν
L
1
=
-
271,1475866,286711948
,
-
3680077858,73546185237
,
-
507541369116,0
,
507541369116
,
ν
L
1
=
(
-
73546185237,3680077858
,
-
286711948
,
-
1475866,271
)
T
/
740670013440
1
4
ν
R
2
=
(
36,25,16,9
,
4,1
,
0,1
,
4,9
,
16,25,36
)
T
ν
L
2
=
(
-
3927,10663114
,
-
582891346
,
-
38170879262,32176934487,2719571096148
,
-
5426009838428
,
ν
L
2
=
(
2719571096148,32176934487
,
-
38170879262
,
-
582891346,10663114
,
-
3927
)
T
/
4991362191360
1
8
ν
R
3
=
(
-
216
,
-
125
,
-
64
,
-
27
,
-
8
,
-
1,0
,
1,8
,
27,64,125,216
)
T
ν
L
3
=
23
,
-
31050
,
-
1351836,21998258
,
-
507201588,193168476,0
,
ν
L
3
=
(
-
193168476,507201588
,
-
21998258,1351836,31050
,
-
23
)
T
/
635351040
1
16
ν
R
4
=
(
1
/
27
)
(
51867,25027,10267,3267,667,67,27,67,667,3267,10267,25027,51867
)
T
ν
L
4
=
-
6693,4466462,117537274
,
-
1293454266,6372836901
,
-
15842276196,21281793036
,
ν
L
4
=
(
-
15842276196,6372836901
,
-
1293454266,117537274,4466462
,
-
6693
)
T
/
42150297600


### 2.2. Basis Functions

The basis functions for the convergent subdivision schemes (1) and (2) are the limit curves ϕ
(
x
) and Φ
(
x
) generated from the cardinal data {
p
i
=
(
i
,
δ
0
)
T
}. ϕ
(
x
) and Φ
(
x
) are also known as fundamental solutions of the subdivision schemes, so(8)
ϕ
i
=
Φ
i
=
1
,
i
=
0
,
0
,
i
≠
0
.Furthermore, ϕ
(
x
)
=
Φ
(
x
) satisfies the following two-scale equation:(9)
ϕ
x
=
∑
j
=
-
p
p
a
i
ϕ
2
x
-
i
,
S
u
p
ϕ
=
-
p
-
1
,
p
+
1
.The lth derivative of basis function at integers satisfies the relation(10)
ϕ
l
x
=
∑
j
=
-
p
p
a
i
ϕ
l
2
x
-
i
,
S
u
p
ϕ
=
-
p
,
p
.Since υ
R
i
T
υ
L
j
=
1 and ν
R
i
T
ν
L
j
=
1 for i
=
j and 0 otherwise, then, by [16], we get the following result.Lemma 1.
The fundamental solution (cardinal basis)ϕ
(
x
) of the subdivision scheme (1) is four times continuously differentiable, supported on [
-
8,8
], and its derivatives at integers are given by(11)
ϕ
′
t
=
2
sgn
⁡
t
e
t
T
υ
L
1
,
ϕ
′
′
t
=
2
2
e
t
T
υ
L
2
,
ϕ
′
′
′
t
=
2
3
sgn
⁡
t
e
t
T
υ
L
3
,
ϕ
i
v
t
=
2
4
e
t
T
υ
L
4
,
-
8
⩽
t
⩽
8
,where υ
L
i, 0
⩽
i
⩽
4, are defined in Table 1; the sgn
⁡ function of a real number t is defined as(12)
sgn
⁡
t
=
-
1
,
t
<
0
,
0
,
t
=
0
,
1
,
t
>
0
.
e
t’s are the column matrices defined as (13)
e
t
=
a
8
t
,
a
7
t
,
a
6
t
,
a
5
t
,
a
4
t
,
a
3
t
,
a
2
t
,
a
1
t
,
a
0
t
,
a
-
1
t
,
a
-
2
t
,
a
-
3
t
,
a
-
4
t
,
a
-
5
t
,
a
-
6
t
,
a
-
7
t
,
a
-
8
t
T
,where 0
≤
t
≤
8 and(14)
a
i
t
=
1
,
i
=
t
,
0
,
i
≠
t
.Lemma 2.
The fundamental solutionΦ
(
x
) of subdivision scheme (2) defined in (9) is four times continuously differentiable, supported on [
-
6,6
], and its derivatives at integers are defined as(15)
Φ
′
t
=
2
sgn
⁡
t
e
t
T
ν
L
1
,
Φ
′
′
t
=
2
2
e
t
T
ν
L
2
,
Φ
′
′
′
t
=
2
3
sgn
⁡
t
e
t
T
ν
L
3
,
Φ
i
v
t
=
2
4
e
t
T
ν
L
4
-
6
⩽
t
⩽
6
,where ν
L
i for 0
⩽
i
⩽
4 are defined in Table 2; the sgn
⁡ function of a real number t is defined by (12); e
t’s are the column matrices defined as(16)
e
t
=
a
6
t
,
a
5
t
,
a
4
t
,
a
3
t
,
a
2
t
,
a
1
t
,
a
0
t
,
a
-
1
t
,
a
-
2
t
,
a
-
3
t
,
a
-
4
t
,
a
-
5
t
,
a
-
6
t
T
,
0
⩽
t
⩽
6
,where a
i
t are defined by (14).From (11) and (15), we get values of derivatives at the integers given in Tables 3 and 4, respectively.Table 3
Derivatives ofϕ at cardinal data by (11).
At∖Derivatives
ϕ
′
ϕ
′
′
ϕ
′
′
′
ϕ
iv
0
0
-
2370618501415
309077185968
0
33869667
457408
±
1
∓
1914621952
1159104017
3265310153216
676106344305
±
43317515008
15295995855
-
5295054752
89730585
±
2
±
530452796
1159104017
-
878265102572
676106344305
∓
121530512357
61183983420
10404741119
358922340
±
3
∓
1470464
13780629
734063059456
2028319032915
±
240606976
566518365
-
74879584
9970065
±
4
±
17297069
1159104017
-
80883901277
1352212688610
∓
5285889107
244735933680
295020869
2871378720
±
5
∓
2772992
5795520085
214899200
135221268861
∓
37414144
3059199171
9238624
17946117
±
6
∓
1127636
10431936153
297875188
405663806583
±
1090169
453214692
-
900187
7976052
±
7
∓
4096
8113728119
64000
19317324123
∓
21760
437028453
71840
17946117
±
8
∓
5
9272832136
4375
618154371936
∓
2975
13984910496
11225
328157568Table 4
Derivatives ofΦ at cardinal data by (15).
At∖Derivatives
Φ
′
Φ
′
′
Φ
′
′
′
Φ
iv
0
0
-
1356502459607
311960136960
0
1773482753
219532800
±
1
∓
42295114093
30861250560
226630924679
103986712320
±
596199
245120
-
1320189683
219532800
±
2
±
3502199297
17635000320
3575214943
138648949760
∓
1565437
980480
708092989
292710400
±
3
∓
1840038929
185167503360
-
19085439631
623920273920
±
10999129
39709440
-
215575711
439065600
±
4
±
71677987
92583751680
-
291445673
623920273920
∓
12517
735360
58768637
1317196800
±
5
±
105419
26452500480
5331557
623920273920
∓
115
294144
2233231
1317196800
±
6
∓
271
370335006720
-
1309
415946849280
±
23
79418880
-
2231
878131200


## 2.1. Subdivision Matrices

IfS
1 and S
2 are subdivision matrices of schemes (1) and (2), then these matrices are defined as(5)
S
1
=
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
L
5
L
4
L
3
L
2
L
1
L
1
L
2
L
3
L
4
L
5
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
,where L
1
=
19845
/
32768, L
2
=
-
2205
/
32768, L
3
=
567
/
16384, L
4
=
-
405
/
65536, L
5
=
35
/
65536, and(6)
S
2
=
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
0
0
0
0
0
0
s
1
′
s
2
′
s
3
′
s
4
′
s
4
′
s
3
′
s
2
′
s
1
′
0
0
0
0
0
0
s
1
s
2
s
3
s
4
s
3
s
2
s
1
,where s
1
=
23
/
16384, s
2
=
-
285
/
8192, s
3
=
2073
/
16384, s
4
=
3333
/
4096, s
1
′
=
-
3
/
32768, s
2
′
=
-
33
/
32768, s
3
′
=
-
1931
/
32768, and s
4
′
=
18351
/
32768.The first ten real eigenvalues of matricesS
1 and S
2 are the same which are given as follows: (7)
λ
i
=
1
,
1
2
,
1
4
,
1
8
,
1
16
,
1
32
,
1
64
,
1
128
,
1
256
,
1
512
,
i
=
0,1
,
2
,
…
,
9
.The remaining eigenvalues are complex which are not required. For above eigenvalues λ
i, the eigenvectors υ
R
i and υ
L
i that satisfies S
1
υ
R
i
=
λ
i
υ
R
i and υ
L
i
S
1
T
=
υ
L
i
λ
i are called right and left eigenvectors of the matrix S
1, respectively. We can also define the right ν
R
i and left ν
L
i eigenvectors of S
2 in a similar way. The normalized left and right eigenvectors corresponding to first five eigenvalues of S
1 and S
2 are given in Tables 1 and 2, respectively.Table 1
Eigenvalues and eigenvectors of the matrixS
1.
Eigenvaluesλ
i
Corresponding right and left eigenvectors
1
υ
R
0
=
(
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1
)
T
υ
L
0
=
(
0,0
,
0,0
,
0,0
,
0,0
,
1,0
,
0,0
,
0,0
,
0,0
,
0
)
T
1
2
υ
R
1
=
(
-
8
,
-
7
,
-
6
,
-
5
,
-
4
,
-
3
,
-
2
,
-
1,0
,
1,2
,
3,4
,
5,6
,
7,8
)
T
υ
L
1
=
-
1575
,
-
1474560
,
-
315738080
,
-
1397587968,43588613880
,
-
4311679549440,1336741045920
,
-
4824847319040
,
υ
L
1
=
(
0
,
4824847319040
,
-
1336741045920,4311679549440
,
-
43588613880,1397587968,315738080
,
υ
L
1
=
(
1474560,1575
T
/
5841884245680
1
4
υ
R
2
=
(
64,49,36,25,16,9
,
4,1
,
0,1
,
4,9
,
16,25,36,49,64
)
T
υ
L
2
=
(
459375,215040000,47660030080,103151616000
,
-
3882427261296,23490017902592
,
-
84313449846912
,
υ
L
2
=
(
313469774708736
,
-
497829885297150,313469774708736
,
-
84313449846912,23490017902592
,
-
3882427261296
,
υ
L
2
=
(
103151616000,47660030080,215040000,459375
)
T
/
259624836213120
1
8
υ
R
3
=
(
-
512
,
-
343
,
-
216
,
-
125
,
-
64
,
-
27
,
-
8
,
-
1,0
,
1,8
,
27,64,125,216,343,512
)
T
υ
L
3
=
(
-
104125
,
-
24371200,1177382520
,
-
5986263040
,
-
10571778214,207884427264
,
-
972244098856
,
-
1386160480256
,
υ
L
3
=
(
0
,
1386160480256,972244098856
,
-
207884427264,10571778214
,
5986263040
,
-
1177382520
,
24371200
,
υ
L
3
=
(
104125
)
T
/
9460416859904
1
16
υ
R
4
=
(
4096,2401,1296,625,256,81,16,1
,
0,1
,
16,81,256,625,1296,2401,4096
)
T
υ
L
4
=
(
392875,45977600
,
-
1296269280,5912719360,1180083476
,
-
86261280768,332951715808
,
-
67767008256
,
υ
L
4
=
(
850467338370
,
-
67767008256,332951715808
,
-
86261280768
,
1180083476,5912719360
,
-
1296269280
,
υ
L
4
=
(
45977600,392875
)
T
/
183768238080Table 2
Eigenvalues and eigenvectors of the matrixS
2.
Eigenvaluesλ
i
Corresponding right and left eigenvectors
1
ν
R
0
=
(
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1,1
,
1
)
T
ν
L
0
=
(
0,0
,
0,0
,
0,0
,
1,0
,
0,0
,
0,0
,
0
)
T
1
2
ν
R
1
=
(
-
6
,
-
5
,
-
4
,
-
3
,
-
2
,
-
1,0
,
1,2
,
3,4
,
5,6
)
T
ν
L
1
=
-
271,1475866,286711948
,
-
3680077858,73546185237
,
-
507541369116,0
,
507541369116
,
ν
L
1
=
(
-
73546185237,3680077858
,
-
286711948
,
-
1475866,271
)
T
/
740670013440
1
4
ν
R
2
=
(
36,25,16,9
,
4,1
,
0,1
,
4,9
,
16,25,36
)
T
ν
L
2
=
(
-
3927,10663114
,
-
582891346
,
-
38170879262,32176934487,2719571096148
,
-
5426009838428
,
ν
L
2
=
(
2719571096148,32176934487
,
-
38170879262
,
-
582891346,10663114
,
-
3927
)
T
/
4991362191360
1
8
ν
R
3
=
(
-
216
,
-
125
,
-
64
,
-
27
,
-
8
,
-
1,0
,
1,8
,
27,64,125,216
)
T
ν
L
3
=
23
,
-
31050
,
-
1351836,21998258
,
-
507201588,193168476,0
,
ν
L
3
=
(
-
193168476,507201588
,
-
21998258,1351836,31050
,
-
23
)
T
/
635351040
1
16
ν
R
4
=
(
1
/
27
)
(
51867,25027,10267,3267,667,67,27,67,667,3267,10267,25027,51867
)
T
ν
L
4
=
-
6693,4466462,117537274
,
-
1293454266,6372836901
,
-
15842276196,21281793036
,
ν
L
4
=
(
-
15842276196,6372836901
,
-
1293454266,117537274,4466462
,
-
6693
)
T
/
42150297600


## 2.2. Basis Functions

The basis functions for the convergent subdivision schemes (1) and (2) are the limit curves ϕ
(
x
) and Φ
(
x
) generated from the cardinal data {
p
i
=
(
i
,
δ
0
)
T
}. ϕ
(
x
) and Φ
(
x
) are also known as fundamental solutions of the subdivision schemes, so(8)
ϕ
i
=
Φ
i
=
1
,
i
=
0
,
0
,
i
≠
0
.Furthermore, ϕ
(
x
)
=
Φ
(
x
) satisfies the following two-scale equation:(9)
ϕ
x
=
∑
j
=
-
p
p
a
i
ϕ
2
x
-
i
,
S
u
p
ϕ
=
-
p
-
1
,
p
+
1
.The lth derivative of basis function at integers satisfies the relation(10)
ϕ
l
x
=
∑
j
=
-
p
p
a
i
ϕ
l
2
x
-
i
,
S
u
p
ϕ
=
-
p
,
p
.Since υ
R
i
T
υ
L
j
=
1 and ν
R
i
T
ν
L
j
=
1 for i
=
j and 0 otherwise, then, by [16], we get the following result.Lemma 1.
The fundamental solution (cardinal basis)ϕ
(
x
) of the subdivision scheme (1) is four times continuously differentiable, supported on [
-
8,8
], and its derivatives at integers are given by(11)
ϕ
′
t
=
2
sgn
⁡
t
e
t
T
υ
L
1
,
ϕ
′
′
t
=
2
2
e
t
T
υ
L
2
,
ϕ
′
′
′
t
=
2
3
sgn
⁡
t
e
t
T
υ
L
3
,
ϕ
i
v
t
=
2
4
e
t
T
υ
L
4
,
-
8
⩽
t
⩽
8
,where υ
L
i, 0
⩽
i
⩽
4, are defined in Table 1; the sgn
⁡ function of a real number t is defined as(12)
sgn
⁡
t
=
-
1
,
t
<
0
,
0
,
t
=
0
,
1
,
t
>
0
.
e
t’s are the column matrices defined as (13)
e
t
=
a
8
t
,
a
7
t
,
a
6
t
,
a
5
t
,
a
4
t
,
a
3
t
,
a
2
t
,
a
1
t
,
a
0
t
,
a
-
1
t
,
a
-
2
t
,
a
-
3
t
,
a
-
4
t
,
a
-
5
t
,
a
-
6
t
,
a
-
7
t
,
a
-
8
t
T
,where 0
≤
t
≤
8 and(14)
a
i
t
=
1
,
i
=
t
,
0
,
i
≠
t
.Lemma 2.
The fundamental solutionΦ
(
x
) of subdivision scheme (2) defined in (9) is four times continuously differentiable, supported on [
-
6,6
], and its derivatives at integers are defined as(15)
Φ
′
t
=
2
sgn
⁡
t
e
t
T
ν
L
1
,
Φ
′
′
t
=
2
2
e
t
T
ν
L
2
,
Φ
′
′
′
t
=
2
3
sgn
⁡
t
e
t
T
ν
L
3
,
Φ
i
v
t
=
2
4
e
t
T
ν
L
4
-
6
⩽
t
⩽
6
,where ν
L
i for 0
⩽
i
⩽
4 are defined in Table 2; the sgn
⁡ function of a real number t is defined by (12); e
t’s are the column matrices defined as(16)
e
t
=
a
6
t
,
a
5
t
,
a
4
t
,
a
3
t
,
a
2
t
,
a
1
t
,
a
0
t
,
a
-
1
t
,
a
-
2
t
,
a
-
3
t
,
a
-
4
t
,
a
-
5
t
,
a
-
6
t
T
,
0
⩽
t
⩽
6
,where a
i
t are defined by (14).From (11) and (15), we get values of derivatives at the integers given in Tables 3 and 4, respectively.Table 3
Derivatives ofϕ at cardinal data by (11).
At∖Derivatives
ϕ
′
ϕ
′
′
ϕ
′
′
′
ϕ
iv
0
0
-
2370618501415
309077185968
0
33869667
457408
±
1
∓
1914621952
1159104017
3265310153216
676106344305
±
43317515008
15295995855
-
5295054752
89730585
±
2
±
530452796
1159104017
-
878265102572
676106344305
∓
121530512357
61183983420
10404741119
358922340
±
3
∓
1470464
13780629
734063059456
2028319032915
±
240606976
566518365
-
74879584
9970065
±
4
±
17297069
1159104017
-
80883901277
1352212688610
∓
5285889107
244735933680
295020869
2871378720
±
5
∓
2772992
5795520085
214899200
135221268861
∓
37414144
3059199171
9238624
17946117
±
6
∓
1127636
10431936153
297875188
405663806583
±
1090169
453214692
-
900187
7976052
±
7
∓
4096
8113728119
64000
19317324123
∓
21760
437028453
71840
17946117
±
8
∓
5
9272832136
4375
618154371936
∓
2975
13984910496
11225
328157568Table 4
Derivatives ofΦ at cardinal data by (15).
At∖Derivatives
Φ
′
Φ
′
′
Φ
′
′
′
Φ
iv
0
0
-
1356502459607
311960136960
0
1773482753
219532800
±
1
∓
42295114093
30861250560
226630924679
103986712320
±
596199
245120
-
1320189683
219532800
±
2
±
3502199297
17635000320
3575214943
138648949760
∓
1565437
980480
708092989
292710400
±
3
∓
1840038929
185167503360
-
19085439631
623920273920
±
10999129
39709440
-
215575711
439065600
±
4
±
71677987
92583751680
-
291445673
623920273920
∓
12517
735360
58768637
1317196800
±
5
±
105419
26452500480
5331557
623920273920
∓
115
294144
2233231
1317196800
±
6
∓
271
370335006720
-
1309
415946849280
±
23
79418880
-
2231
878131200


## 3. Description of Numerical Algorithms

In this section, first we formulate two collocation algorithms which are based on interpolating (1) and approximating (2) subdivision schemes for the solution of (3). Then we settle down the boundary conditions to get unique solution.


### 3.1. Collocation Algorithms

Here we formulate two collocation algorithms based on two subdivision schemes. These collocation algorithms are defined in coming subsections.


#### 3.1.1. Interpolating Collocation Algorithm

The collocation algorithm based on interpolating scheme (1), say, interpolating collocating algorithm, is given below. In this algorithm, we assume approximate solution Z
1
(
x
) of (3) as(17)
Z
1
x
=
∑
i
=
-
8
N
+
8
z
i
ϕ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
8, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(18)
Z
1
i
v
x
j
=
a
x
j
Z
1
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(19)
Z
1
0
=
α
,
Z
1
′
0
=
β
,
Z
1
N
=
γ
,
Z
1
′
N
=
ω
,where α, β, γ, and ω are constants. Let a
j
=
a
(
x
j
), b
j
=
b
(
x
j
); then (18) can be written as(20)
Z
1
i
v
x
j
=
a
j
Z
1
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(21)
Z
1
i
v
x
j
=
1
h
4
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
.Using (17) and (21) in (20), we get the following N
+
1 system of equations:(22)
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
8
N
+
8
z
i
ϕ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.


#### 3.1.2. Approximating Collocation Algorithm

In approximating collocating algorithm (i.e., algorithm based on approximating scheme (2)), we assume approximate solution Z
2
(
x
) of (3) as(23)
Z
2
x
=
∑
i
=
-
6
N
+
6
z
i
Φ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
6, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(24)
Z
2
i
v
x
j
=
a
x
j
Z
2
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(25)
Z
2
0
=
α
,
Z
2
′
0
=
β
,
Z
2
N
=
γ
,
Z
2
′
N
=
ω
.Equation (24) can be written as(26)
Z
2
i
v
x
j
=
a
j
Z
2
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(27)
Z
2
i
v
x
j
=
1
h
4
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
.Using (23) and (27) in (26), we get the following N
+
1 system of equations:(28)
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
6
N
+
6
z
i
Φ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.Now we simplify systems (22) and (28) in the following theorems.Theorem 3 (interpolating collocation algorithm).
Forj
=
0 by (22), one gets(29)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
q
0
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
0
=
ϕ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (22), we get(30)
z
-
8
ϕ
i
v
x
0
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
0
-
x
N
+
8
h
-
a
0
h
4
z
-
8
ϕ
x
0
-
x
-
8
h
+
z
-
7
ϕ
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
0
-
x
N
+
8
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(31)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
⋯
+
z
N
+
7
ϕ
i
v
-
N
-
7
+
z
N
+
8
ϕ
i
v
-
N
-
8
-
a
0
h
4
z
-
8
ϕ
8
+
z
-
7
ϕ
7
+
⋯
+
z
N
+
7
ϕ
-
N
-
7
+
z
N
+
8
ϕ
-
N
-
8
=
h
4
b
0
.Since the cardinal support of basis function ϕ
(
x
) is [
-
8,8
], so ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
]; also, by Table 3, we get(32)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
z
-
6
ϕ
i
v
6
+
z
-
5
ϕ
i
v
5
+
z
-
4
ϕ
i
v
4
+
z
-
3
ϕ
i
v
3
+
z
-
2
ϕ
i
v
2
+
z
-
1
ϕ
i
v
1
+
z
0
ϕ
i
v
0
+
z
1
ϕ
i
v
-
1
+
z
2
ϕ
i
v
-
2
+
z
3
ϕ
i
v
-
3
+
z
4
ϕ
i
v
-
4
+
z
5
ϕ
i
v
-
5
+
z
6
ϕ
i
v
-
6
+
z
7
ϕ
i
v
-
7
+
z
8
ϕ
i
v
-
8
-
a
0
h
4
z
0
ϕ
0
=
h
4
b
0
.If ϕ
i
i
v
=
ϕ
i
v
(
i
), then(33)
z
-
8
ϕ
8
i
v
+
z
-
7
ϕ
7
i
v
+
z
-
6
ϕ
6
i
v
+
z
-
5
ϕ
5
i
v
+
z
-
4
ϕ
4
i
v
+
z
-
3
ϕ
3
i
v
+
z
-
2
ϕ
2
i
v
+
z
-
1
ϕ
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
-
1
i
v
+
z
2
ϕ
-
2
i
v
+
z
3
ϕ
-
3
i
v
+
z
4
ϕ
-
4
i
v
+
z
5
ϕ
-
5
i
v
+
z
6
ϕ
-
6
i
v
+
z
7
ϕ
-
7
i
v
+
z
8
ϕ
-
8
i
v
=
h
4
b
0
.As we observe from Table 3, ϕ
-
i
i
v
=
ϕ
i
i
v; we have (34)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
.
Forq
0
=
ϕ
0
i
v
-
a
0
h
4, we get (29). This completes the proof.Theorem 4 (interpolating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (22) is equivalent to(35)
z
-
8
ϕ
-
j
-
8
i
v
+
z
-
7
ϕ
-
j
-
7
i
v
+
⋯
+
z
0
ϕ
-
j
i
v
+
z
1
ϕ
1
-
j
i
v
-
a
j
h
4
ϕ
1
-
j
+
z
2
ϕ
2
-
j
i
v
-
a
j
h
4
ϕ
2
-
j
+
⋯
+
z
N
ϕ
N
-
j
i
v
-
a
j
h
4
ϕ
N
-
j
+
z
N
+
1
ϕ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (22), we get(36)
z
-
8
ϕ
i
v
x
j
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
j
-
x
N
+
8
h
-
a
j
h
4
z
-
8
ϕ
x
j
-
x
-
8
h
+
z
-
7
ϕ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
j
-
x
N
+
8
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(37)
z
-
8
ϕ
i
v
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
z
-
8
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
=
h
4
b
j
.This implies (38)
z
-
8
ϕ
i
v
j
+
8
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.If ϕ
j
i
v
=
ϕ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(39)
z
-
8
ϕ
j
+
8
i
v
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
i
v
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
i
v
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
i
v
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.As we observe from Table 3, ϕ
-
j
i
v
=
ϕ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (40)
z
-
8
ϕ
-
j
-
8
i
v
-
a
j
h
4
ϕ
-
j
-
8
+
z
-
7
ϕ
-
j
-
7
i
v
-
a
j
h
4
ϕ
-
j
-
7
+
⋯
+
z
N
+
7
ϕ
N
+
7
-
j
i
v
-
a
j
h
4
ϕ
N
+
7
-
j
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
-
a
j
h
4
ϕ
N
+
8
-
j
=
h
4
b
j
.Since ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
], then by Table 3, we get (35).Theorem 5 (approximating collocation algorithm).
Forj
=
0, by (28), one gets(41)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
q
0
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
=
h
4
b
0
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
0
=
Φ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (28), we get(42)
z
-
6
Φ
i
v
x
0
-
x
-
6
h
+
z
-
5
Φ
i
v
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
0
-
x
N
+
6
h
-
a
0
h
4
z
-
6
Φ
x
0
-
x
-
6
h
+
z
-
5
Φ
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
x
0
-
x
N
+
6
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(43)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
⋯
+
z
N
+
5
Φ
i
v
-
N
-
5
+
z
N
+
6
Φ
i
v
-
N
-
6
-
a
0
h
4
z
-
6
Φ
6
+
z
-
5
Φ
5
+
⋯
+
z
N
+
5
Φ
-
N
-
5
+
z
N
+
6
Φ
-
N
-
6
=
h
4
b
0
.Since the cardinal support of basis function Φ
(
x
) is [
-
6,6
], so Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
]; also, by Table 4, we get(44)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
z
-
4
Φ
i
v
4
+
z
-
3
Φ
i
v
3
+
z
-
2
Φ
i
v
2
+
z
-
1
Φ
i
v
1
+
z
0
Φ
i
v
0
+
z
1
Φ
i
v
-
1
+
z
2
Φ
i
v
-
2
+
z
3
Φ
i
v
-
3
+
z
4
Φ
i
v
-
4
+
z
5
Φ
i
v
-
5
+
z
6
Φ
i
v
-
6
-
a
0
h
4
z
0
Φ
0
=
h
4
b
0
.If Φ
i
i
v
=
Φ
i
v
(
i
), then(45)
z
-
6
Φ
6
i
v
+
z
-
5
Φ
5
i
v
+
z
-
4
Φ
4
i
v
+
z
-
3
Φ
3
i
v
+
z
-
2
Φ
2
i
v
+
z
-
1
Φ
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
-
1
i
v
+
z
2
Φ
-
2
i
v
+
z
3
Φ
-
3
i
v
+
z
4
Φ
-
4
i
v
+
z
5
Φ
-
5
i
v
+
z
6
Φ
-
6
i
v
=
h
4
b
0
.As we observe from Table 4, Φ
-
i
i
v
=
Φ
i
i
v; we have(46)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
+
z
7
Φ
7
i
v
+
z
8
Φ
8
i
v
=
h
4
b
0
.
ForΥ
0
=
Φ
0
i
v
-
a
0
h
4, we get (41). This completes the proof.Theorem 6 (approximating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (28) is equivalent to(47)
z
-
6
Φ
-
j
-
6
i
v
+
z
-
5
Φ
-
j
-
5
i
v
+
⋯
+
z
0
Φ
-
j
i
v
+
z
1
Φ
1
-
j
i
v
-
a
j
h
4
Φ
1
-
j
+
z
2
Φ
2
-
j
i
v
-
a
j
h
4
Φ
2
-
j
+
⋯
+
z
N
Φ
N
-
j
i
v
-
a
j
h
4
Φ
N
-
j
+
z
N
+
1
Φ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
6
Φ
N
+
6
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (28), we get(48)
z
-
6
Φ
i
v
x
j
-
x
-
6
h
+
z
-
5
Φ
i
v
x
j
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
j
-
x
N
+
6
h
-
a
j
h
4
z
-
8
Φ
x
j
-
x
-
8
h
+
z
-
7
Φ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
5
Φ
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
x
j
-
x
N
+
6
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(49)
z
-
6
Φ
i
v
j
+
6
+
z
-
5
Φ
i
v
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
z
-
6
Φ
j
+
6
+
z
-
5
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
=
h
4
b
j
.This implies (50)
z
-
6
Φ
i
v
j
+
6
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
i
v
j
+
5
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.If Φ
j
i
v
=
Φ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(51)
z
-
6
Φ
j
+
6
i
v
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
j
+
5
i
v
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
i
v
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
i
v
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.As we observe from Table 4, Φ
-
j
i
v
=
Φ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (52)
z
-
6
Φ
-
j
-
6
i
v
-
a
j
h
4
Φ
-
j
-
6
+
z
-
5
Φ
-
j
-
5
i
v
-
a
j
h
4
Φ
-
j
-
5
+
⋯
+
z
N
+
5
Φ
N
+
5
-
j
i
v
-
a
j
h
4
Φ
N
+
5
-
j
+
z
N
+
6
Φ
N
+
6
-
j
i
v
-
a
j
h
4
Φ
N
+
6
-
j
=
h
4
b
j
.Since Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
], then, by Table 4, we get (47).


### 3.2. Boundary Conditions at End Points

We have two different systems of(
N
+
1
) equations defined by (22) and (28). In order to get unique solution of these systems, we need sixteen more conditions for system (22) and twelve more conditions for system (28). Four conditions can be achieved from boundary conditions given in (4) for both systems of linear equations in which first order derivatives are involved and remaining conditions are achieved by some extrapolation method at the end points. First we find the approximation of the first derivative by difference operators and after that we define the extrapolation method at end points for both systems of linear equations.


#### 3.2.1. Approximation of Derivative Boundary Conditions

In this section, we approximate the derivative boundary conditions by difference operators. Since approximation order of interpolating scheme (1) and approximating scheme (2) is ten and four, respectively, so we approximate derivative boundary conditions at end points with approximation orders ten and four for interpolating and approximating collocation algorithms.If we use interpolating collocation algorithm for the solution of (3), then approximation of derivative conditions at ends point is defined as(53)
Z
1
′
0
=
N
2520
-
7381
z
0
+
25200
z
1
-
56700
z
2
+
100800
z
3
-
132300
z
4
+
127008
z
5
-
88200
z
6
+
43200
z
7
-
14175
z
8
+
28800
z
9
-
252
z
10
+
O
h
10
,
(54)
Z
1
′
N
=
N
2520
7381
z
N
-
25200
z
N
-
1
+
56700
z
N
-
2
-
100800
z
N
-
3
+
132300
z
N
-
4
-
127008
z
N
-
5
+
88200
z
N
-
6
-
43200
z
N
-
7
+
14175
z
N
-
8
-
28800
z
N
-
9
+
252
z
N
-
10
+
O
h
10and if we use approximating collocation algorithm for the solution of (3), then approximation of derivative conditions at end points is defined as(55)
Z
2
′
0
=
N
12
-
25
z
0
+
48
z
1
-
36
z
2
+
16
z
3
-
3
z
4
+
O
h
4
,
(56)
Z
2
′
N
=
N
12
·
25
z
N
-
48
z
N
-
1
+
36
z
N
-
2
-
16
z
N
-
3
+
3
z
N
-
4
+
O
h
4
.


#### 3.2.2. Adjustment of Boundary Conditions

Still we need twelve and eight more conditions for systems (22) and (28), respectively, to get stable systems for the solution of (3). For this we made some adjustment of boundary conditions for systems (22) and (28), which are defined below.Case 1.
If we use interpolating collocation algorithm for the approximate solution of (3), then we define six conditions at left end points and six conditions at the right end points. Since subdivision scheme (1) reproduces nine-degree (i.e., tenth order) polynomials, so we define boundary conditions of order ten for solution of (22). For simplicity only left end points z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 are discussed and the values of right end points z
N
+
2
,
z
N
+
3
,
z
N
+
4
,
z
N
+
5
,
z
N
+
6
,
z
N
+
7 can be treated similarly.
The valuesz
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 can be determined by the nonic polynomial q
(
x
) interpolating (
x
i
,
z
i
), 2
≤
i
≤
7. Precisely, we have(57)
z
-
i
=
q
-
x
i
,
i
=
2,3
,
4,5
,
6,7
,where (58)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
Z
1
x
i
-
j
.Since by (17) Z
1
(
x
i
)
=
z
i for i
=
2,3
,
4,5
,
6,7, then, by replacing x
i by -
x
i, we have(59)
q
-
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
-
i
+
j
.Hence the following boundary conditions can be employed at the left end:(60)
∑
j
=
0
10
10
j
-
1
j
z
-
i
+
j
=
0
,
i
=
7,6
,
5,4
,
3,2
.Similarly, for the right end, we can define z
i
=
q
(
-
x
i
), i
=
N
+
2, N
+
3, N
+
4, N
+
5, N
+
6, N
+
7, and(61)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
i
-
j
.So we have the following boundary conditions at the right end:(62)
∑
j
=
0
10
10
j
-
1
j
z
i
-
j
=
0
,
i
=
N
+
2
,
N
+
3
,
N
+
4
,
N
+
5
,
N
+
6
,
N
+
7
.Finally, we get the following new system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (29) and (35), four equations from boundary conditions (19), and twelve from boundary conditions (60) and (62).Case 2.
If we use approximating collocation algorithm (28) for the solution of (3) then we need eight more conditions. So in this case, we define four extra conditions at the left end points and four conditions at the right end points by some extrapolation method. Since the subdivision scheme reproduces cubic (i.e., fourth order) polynomial, so we define quartic polynomial for the adjustment of boundary treatment. The values z
-
4
,
z
-
3
,
z
-
2
,
z
-
1 are determined by the quartic polynomial p
(
x
) interpolating (
x
i
,
z
i
). This polynomial is defined as(63)
z
-
i
+
1
=
p
-
x
i
+
1
,
i
=
1,2
,
3,4
,where (64)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
Z
2
x
i
-
j
+
1
.Since by (23) Z
2
(
x
i
)
=
z
i for i
=
1,2
,
3,4, then, by replacing x
i by -
x
i, we have (65)
p
-
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
-
i
+
j
+
1
.Hence the following boundary conditions can be employed at the left end:(66)
∑
j
=
0
4
4
j
-
1
j
z
-
i
+
j
+
1
=
0
,
i
=
1,2
,
3,4
.Similarly, for the right end, we can define z
i
+
1
=
p
(
x
i
+
1
), i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4, and (67)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
i
-
j
+
1
.So we have the following boundary conditions at the right end:(68)
∑
j
=
0
4
4
j
-
1
j
z
i
-
j
+
1
=
0
,
i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4
.Finally, we get a following new system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (41) and (47), four equations from boundary conditions (25), and eight from boundary conditions (66) and (68).


### 3.3. Stable Systems of Linear Equations

In this section, we present stable systems of linear equations for both interpolating and approximating collocation algorithms.


#### 3.3.1. Stable System for Interpolating Collocation Algorithm

From (29) and (35), we get the following undetermined system of (
N
+
1
) equations with (
N
+
17
) unknowns {
z
i
}:(69)
A
1
Z
1
=
G
1
,where the matrices A
1, Z
1, and G
1 of orders (
N
+
1
)
×
(
N
+
17
), N
+
17, and N
+
1, respectively, are given by(70)
A
1
=
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
0
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
ϕ
8
i
v
⋯
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
1
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
⋯
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
2
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
⋯
0
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
3
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
7
′
′
′
ϕ
8
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
6
′
′
′
ϕ
7
′
′
′
ϕ
8
′
′
′
,
(71)
Z
1
=
z
-
8
,
z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
…
,
z
N
+
8
T
,
(72)
G
1
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
j
=
ϕ
0
i
v
-
a
j
h
4.For obtaining the unique solution of (69), we made some adjustment of boundary conditions in previous section which is defined in (53), (54), (60), and (62). By using this adjustment, we get the following system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, defined as(73)
D
1
Z
1
=
R
1
,where the coefficient matrix(74)
D
1
=
B
0
T
,
A
1
T
,
B
1
T
T
.
A
1 is defined by (70); B
0 and B
1 are defined as(75)
B
0
=
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
⋯
0
0
0
0
0
0
0
0
0
0
7381
N
2520
25200
N
2520
-
56700
N
2520
100800
N
2520
-
132300
N
2520
127008
N
2520
-
88200
N
2520
43200
N
2520
-
14175
N
2520
2800
N
2520
-
252
N
2520
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
.First six rows of B
0 are obtained from (60), second last row is obtained from (53), and last row is taken from given boundary conditions Z
1
(
0
) which is defined in (19) and(76)
B
1
=
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
N
10
-
10
N
9
45
N
8
-
120
N
7
35
N
-
252
N
5
105
N
2
-
40
N
45
N
2
-
10
N
7381
N
2520
0
0
0
0
0
0
0
0
0
0
⋯
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
0
⋯
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
.First row of B
1 is obtained from Z
1
(
N
) which is defined in (19), second row is obtained from (54), and the last six rows are obtained from (62) and Z
1 which is defined in (71) and R
1 is defined as (77)
R
1
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
1
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
1 is defined by (72).


#### 3.3.2. Stable System for Approximating Collocation Algorithm

From (41) and (47), we get the following undetermined system of (
N
+
1
) equations with (
N
+
13
) unknowns {
z
i
}:(78)
A
2
Z
2
=
G
2
,where the matrices A
2, Z
2, and G
2 of orders (
N
+
1
)
×
(
N
+
13
), N
+
13, and N
+
1, respectively, are given by(79)
A
2
=
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
0
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
⋯
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
1
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
⋯
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
2
Φ
1
i
v
Φ
3
i
v
Φ
4
i
v
⋯
0
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
3
Φ
2
i
v
Φ
3
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
5
′
′
′
Φ
6
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
4
′
′
′
Φ
5
′
′
′
Φ
6
′
′
′
,
(80)
Z
2
=
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2
,
…
,
z
N
+
6
T
,
(81)
G
2
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
j
=
Φ
0
i
v
-
a
j
h
4.In order to get the unique solution of system (78), we have defined some extra conditions in (55), (56), (66), and (68). By using these extra conditions we get the following system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}:(82)
D
2
Z
2
=
R
2
,where the coefficient matrix(83)
D
2
=
B
0
T
,
A
2
T
,
B
1
T
T
.
A
2 is defined by (79); B
0 and B
1 are defined as(84)
B
0
=
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
4
1
0
0
0
0
⋯
0
0
0
0
0
0
0
0
-
251
N
12
48
N
12
-
36
N
12
16
N
12
-
3
N
12
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
⋯
0
0
.First four rows of B
0 are obtained from (66), second last row is obtained from (55), and the last row is taken from the given boundary conditions Z
2
(
0
) which is defined in (25) and(85)
B
1
=
0
0
0
0
⋯
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
3
N
12
-
16
N
12
36
N
12
-
48
N
12
25
N
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
.First row of B
1 is obtained from Z
2
(
N
) which is defined in (25), second row is obtained from (56), and the last four rows are obtained from (68); Z
2 is defined in (80) and R
2 is defined as (86)
R
2
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
2
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
2 is defined by (81). Hence to obtain the approximate solution of the fourth order boundary value problem (3) by interpolating and approximating collocation algorithms we need to solve systems (73) and (82), respectively.


### 3.4.  Existence of the Solution

In this section, we discuss the nonsingularity of the coefficient matricesD
1 and D
2 defined in (74) and (83), respectively. We observe that the coefficient matrices D
1 and D
2 are neither symmetric nor diagonally dominant. However it can be shown that D
1 and D
2 are nonsingular. Since D
1 and D
2 are band matrices with half bandwidth 9 and 7, numerical complexities for solving the linear systems using Gaussian elimination are about 81
(
N
+
17
) and 49
(
N
+
13
) multiplications, respectively. For large N, the matrices are almost symmetric except the first and last eight rows and columns of D
1 and first and last four rows and columns of D
2 due to the boundary conditions. Therefore we first consider their symmetric part, that is, square symmetric matrices E
1 and E
2 of orders N
+
1 defined as(87)
E
1
=
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
ϕ
8
i
v
⋯
0
0
0
0
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
⋯
0
0
0
0
ϕ
-
2
i
v
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
⋯
0
0
0
0
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
ϕ
4
i
v
ϕ
3
i
v
ϕ
2
i
v
ϕ
1
i
v
0
0
0
0
0
0
0
0
0
⋯
ϕ
3
i
v
ϕ
2
i
v
ϕ
1
i
v
ϕ
0
i
v
,
E
2
=
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
0
0
⋯
0
0
0
0
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
0
⋯
0
0
0
0
Φ
-
2
i
v
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
⋯
0
0
0
0
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
Φ
4
i
v
Φ
3
i
v
Φ
2
i
v
Φ
1
i
v
0
0
0
0
0
0
0
0
0
⋯
Φ
3
i
v
Φ
2
i
v
Φ
1
i
v
Φ
0
i
v
.So E
1 and E
2 are symmetric matrices obtained from D
1 and D
2, respectively. It can be shown that E
1 and E
2 are nonsingular when N increase. The nonsingularity of the matrices E
1 and E
2 is shown in Table 5 by finding their determinants. The matrices E
1 and E
2 remain nonsingular for N
⩽
500 and N
⩽
100, respectively. For large N the determinants of the matrices may or may not be equal to zero. The nonsingularity of the matrices D
1 and D
2 has been checked by finding their eigenvalues for N
⩽
500 and for N
⩽
100, respectively. Since the eigenvalues for both the matrices are nonzero, then by [17] matrices D
1 and D
2 are nonsingular. However the matrices D
1 for N
⩾
500 and D
2 for N
⩾
100 may or may not be nonsingular. Therefore we claim that systems of (73) and (82) are stable.Table 5
Determinants of the matrices.
N
E
1
E
2
10
-
8667
/
56
1.11401292
×
10
3
50
-
177183
3.2517495
×
10
2
100
-
552709050
0.753776508473953
200
-
5.033491472
×
1
0
36
⋯
300
-
4.477989536
×
1
0
71
⋯
400
3987757210720454
⋯
500
3987757210720454
⋯


### 3.5. Error Estimation

In this section, we discuss the approximation properties of the interpolating and approximating collocation algorithms. Since schemes (1) and (2) reproduce polynomial curves of degree nine and three, so by [10] schemes have approximation orders ten and four, respectively. Here we present our main results for error estimation. Proof of these results is similar to the proof of Proposition [12, 14].Proposition 7.
Suppose the exact solutiony
(
x
)
∈
C
4
[
0,1
] and {
z
i
} are obtained by (73); then absolute error by interpolating collocation algorithm is (88)
e
r
r
1
x
∞
=
Z
1
l
x
-
y
l
x
∞
=
O
h
3
-
l
,
l
=
0,1
,
2,3
,where l denotes the order of derivative.Proposition 8.
Suppose the exact solutiony
(
x
)
∈
C
4
[
0,1
] and {
z
i
} are obtained by (82); then absolute error by approximating collocation algorithm is (89)
e
r
r
2
x
∞
=
Z
2
l
x
-
y
l
x
∞
=
O
h
3
-
l
,
l
=
0,1
,
2,3
.


## 3.1. Collocation Algorithms

Here we formulate two collocation algorithms based on two subdivision schemes. These collocation algorithms are defined in coming subsections.


### 3.1.1. Interpolating Collocation Algorithm

The collocation algorithm based on interpolating scheme (1), say, interpolating collocating algorithm, is given below. In this algorithm, we assume approximate solution Z
1
(
x
) of (3) as(17)
Z
1
x
=
∑
i
=
-
8
N
+
8
z
i
ϕ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
8, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(18)
Z
1
i
v
x
j
=
a
x
j
Z
1
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(19)
Z
1
0
=
α
,
Z
1
′
0
=
β
,
Z
1
N
=
γ
,
Z
1
′
N
=
ω
,where α, β, γ, and ω are constants. Let a
j
=
a
(
x
j
), b
j
=
b
(
x
j
); then (18) can be written as(20)
Z
1
i
v
x
j
=
a
j
Z
1
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(21)
Z
1
i
v
x
j
=
1
h
4
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
.Using (17) and (21) in (20), we get the following N
+
1 system of equations:(22)
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
8
N
+
8
z
i
ϕ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.


### 3.1.2. Approximating Collocation Algorithm

In approximating collocating algorithm (i.e., algorithm based on approximating scheme (2)), we assume approximate solution Z
2
(
x
) of (3) as(23)
Z
2
x
=
∑
i
=
-
6
N
+
6
z
i
Φ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
6, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(24)
Z
2
i
v
x
j
=
a
x
j
Z
2
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(25)
Z
2
0
=
α
,
Z
2
′
0
=
β
,
Z
2
N
=
γ
,
Z
2
′
N
=
ω
.Equation (24) can be written as(26)
Z
2
i
v
x
j
=
a
j
Z
2
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(27)
Z
2
i
v
x
j
=
1
h
4
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
.Using (23) and (27) in (26), we get the following N
+
1 system of equations:(28)
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
6
N
+
6
z
i
Φ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.Now we simplify systems (22) and (28) in the following theorems.Theorem 3 (interpolating collocation algorithm).
Forj
=
0 by (22), one gets(29)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
q
0
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
0
=
ϕ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (22), we get(30)
z
-
8
ϕ
i
v
x
0
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
0
-
x
N
+
8
h
-
a
0
h
4
z
-
8
ϕ
x
0
-
x
-
8
h
+
z
-
7
ϕ
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
0
-
x
N
+
8
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(31)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
⋯
+
z
N
+
7
ϕ
i
v
-
N
-
7
+
z
N
+
8
ϕ
i
v
-
N
-
8
-
a
0
h
4
z
-
8
ϕ
8
+
z
-
7
ϕ
7
+
⋯
+
z
N
+
7
ϕ
-
N
-
7
+
z
N
+
8
ϕ
-
N
-
8
=
h
4
b
0
.Since the cardinal support of basis function ϕ
(
x
) is [
-
8,8
], so ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
]; also, by Table 3, we get(32)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
z
-
6
ϕ
i
v
6
+
z
-
5
ϕ
i
v
5
+
z
-
4
ϕ
i
v
4
+
z
-
3
ϕ
i
v
3
+
z
-
2
ϕ
i
v
2
+
z
-
1
ϕ
i
v
1
+
z
0
ϕ
i
v
0
+
z
1
ϕ
i
v
-
1
+
z
2
ϕ
i
v
-
2
+
z
3
ϕ
i
v
-
3
+
z
4
ϕ
i
v
-
4
+
z
5
ϕ
i
v
-
5
+
z
6
ϕ
i
v
-
6
+
z
7
ϕ
i
v
-
7
+
z
8
ϕ
i
v
-
8
-
a
0
h
4
z
0
ϕ
0
=
h
4
b
0
.If ϕ
i
i
v
=
ϕ
i
v
(
i
), then(33)
z
-
8
ϕ
8
i
v
+
z
-
7
ϕ
7
i
v
+
z
-
6
ϕ
6
i
v
+
z
-
5
ϕ
5
i
v
+
z
-
4
ϕ
4
i
v
+
z
-
3
ϕ
3
i
v
+
z
-
2
ϕ
2
i
v
+
z
-
1
ϕ
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
-
1
i
v
+
z
2
ϕ
-
2
i
v
+
z
3
ϕ
-
3
i
v
+
z
4
ϕ
-
4
i
v
+
z
5
ϕ
-
5
i
v
+
z
6
ϕ
-
6
i
v
+
z
7
ϕ
-
7
i
v
+
z
8
ϕ
-
8
i
v
=
h
4
b
0
.As we observe from Table 3, ϕ
-
i
i
v
=
ϕ
i
i
v; we have (34)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
.
Forq
0
=
ϕ
0
i
v
-
a
0
h
4, we get (29). This completes the proof.Theorem 4 (interpolating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (22) is equivalent to(35)
z
-
8
ϕ
-
j
-
8
i
v
+
z
-
7
ϕ
-
j
-
7
i
v
+
⋯
+
z
0
ϕ
-
j
i
v
+
z
1
ϕ
1
-
j
i
v
-
a
j
h
4
ϕ
1
-
j
+
z
2
ϕ
2
-
j
i
v
-
a
j
h
4
ϕ
2
-
j
+
⋯
+
z
N
ϕ
N
-
j
i
v
-
a
j
h
4
ϕ
N
-
j
+
z
N
+
1
ϕ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (22), we get(36)
z
-
8
ϕ
i
v
x
j
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
j
-
x
N
+
8
h
-
a
j
h
4
z
-
8
ϕ
x
j
-
x
-
8
h
+
z
-
7
ϕ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
j
-
x
N
+
8
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(37)
z
-
8
ϕ
i
v
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
z
-
8
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
=
h
4
b
j
.This implies (38)
z
-
8
ϕ
i
v
j
+
8
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.If ϕ
j
i
v
=
ϕ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(39)
z
-
8
ϕ
j
+
8
i
v
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
i
v
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
i
v
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
i
v
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.As we observe from Table 3, ϕ
-
j
i
v
=
ϕ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (40)
z
-
8
ϕ
-
j
-
8
i
v
-
a
j
h
4
ϕ
-
j
-
8
+
z
-
7
ϕ
-
j
-
7
i
v
-
a
j
h
4
ϕ
-
j
-
7
+
⋯
+
z
N
+
7
ϕ
N
+
7
-
j
i
v
-
a
j
h
4
ϕ
N
+
7
-
j
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
-
a
j
h
4
ϕ
N
+
8
-
j
=
h
4
b
j
.Since ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
], then by Table 3, we get (35).Theorem 5 (approximating collocation algorithm).
Forj
=
0, by (28), one gets(41)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
q
0
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
=
h
4
b
0
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
0
=
Φ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (28), we get(42)
z
-
6
Φ
i
v
x
0
-
x
-
6
h
+
z
-
5
Φ
i
v
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
0
-
x
N
+
6
h
-
a
0
h
4
z
-
6
Φ
x
0
-
x
-
6
h
+
z
-
5
Φ
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
x
0
-
x
N
+
6
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(43)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
⋯
+
z
N
+
5
Φ
i
v
-
N
-
5
+
z
N
+
6
Φ
i
v
-
N
-
6
-
a
0
h
4
z
-
6
Φ
6
+
z
-
5
Φ
5
+
⋯
+
z
N
+
5
Φ
-
N
-
5
+
z
N
+
6
Φ
-
N
-
6
=
h
4
b
0
.Since the cardinal support of basis function Φ
(
x
) is [
-
6,6
], so Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
]; also, by Table 4, we get(44)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
z
-
4
Φ
i
v
4
+
z
-
3
Φ
i
v
3
+
z
-
2
Φ
i
v
2
+
z
-
1
Φ
i
v
1
+
z
0
Φ
i
v
0
+
z
1
Φ
i
v
-
1
+
z
2
Φ
i
v
-
2
+
z
3
Φ
i
v
-
3
+
z
4
Φ
i
v
-
4
+
z
5
Φ
i
v
-
5
+
z
6
Φ
i
v
-
6
-
a
0
h
4
z
0
Φ
0
=
h
4
b
0
.If Φ
i
i
v
=
Φ
i
v
(
i
), then(45)
z
-
6
Φ
6
i
v
+
z
-
5
Φ
5
i
v
+
z
-
4
Φ
4
i
v
+
z
-
3
Φ
3
i
v
+
z
-
2
Φ
2
i
v
+
z
-
1
Φ
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
-
1
i
v
+
z
2
Φ
-
2
i
v
+
z
3
Φ
-
3
i
v
+
z
4
Φ
-
4
i
v
+
z
5
Φ
-
5
i
v
+
z
6
Φ
-
6
i
v
=
h
4
b
0
.As we observe from Table 4, Φ
-
i
i
v
=
Φ
i
i
v; we have(46)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
+
z
7
Φ
7
i
v
+
z
8
Φ
8
i
v
=
h
4
b
0
.
ForΥ
0
=
Φ
0
i
v
-
a
0
h
4, we get (41). This completes the proof.Theorem 6 (approximating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (28) is equivalent to(47)
z
-
6
Φ
-
j
-
6
i
v
+
z
-
5
Φ
-
j
-
5
i
v
+
⋯
+
z
0
Φ
-
j
i
v
+
z
1
Φ
1
-
j
i
v
-
a
j
h
4
Φ
1
-
j
+
z
2
Φ
2
-
j
i
v
-
a
j
h
4
Φ
2
-
j
+
⋯
+
z
N
Φ
N
-
j
i
v
-
a
j
h
4
Φ
N
-
j
+
z
N
+
1
Φ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
6
Φ
N
+
6
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (28), we get(48)
z
-
6
Φ
i
v
x
j
-
x
-
6
h
+
z
-
5
Φ
i
v
x
j
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
j
-
x
N
+
6
h
-
a
j
h
4
z
-
8
Φ
x
j
-
x
-
8
h
+
z
-
7
Φ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
5
Φ
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
x
j
-
x
N
+
6
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(49)
z
-
6
Φ
i
v
j
+
6
+
z
-
5
Φ
i
v
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
z
-
6
Φ
j
+
6
+
z
-
5
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
=
h
4
b
j
.This implies (50)
z
-
6
Φ
i
v
j
+
6
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
i
v
j
+
5
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.If Φ
j
i
v
=
Φ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(51)
z
-
6
Φ
j
+
6
i
v
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
j
+
5
i
v
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
i
v
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
i
v
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.As we observe from Table 4, Φ
-
j
i
v
=
Φ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (52)
z
-
6
Φ
-
j
-
6
i
v
-
a
j
h
4
Φ
-
j
-
6
+
z
-
5
Φ
-
j
-
5
i
v
-
a
j
h
4
Φ
-
j
-
5
+
⋯
+
z
N
+
5
Φ
N
+
5
-
j
i
v
-
a
j
h
4
Φ
N
+
5
-
j
+
z
N
+
6
Φ
N
+
6
-
j
i
v
-
a
j
h
4
Φ
N
+
6
-
j
=
h
4
b
j
.Since Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
], then, by Table 4, we get (47).


## 3.1.1. Interpolating Collocation Algorithm

The collocation algorithm based on interpolating scheme (1), say, interpolating collocating algorithm, is given below. In this algorithm, we assume approximate solution Z
1
(
x
) of (3) as(17)
Z
1
x
=
∑
i
=
-
8
N
+
8
z
i
ϕ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
8, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(18)
Z
1
i
v
x
j
=
a
x
j
Z
1
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(19)
Z
1
0
=
α
,
Z
1
′
0
=
β
,
Z
1
N
=
γ
,
Z
1
′
N
=
ω
,where α, β, γ, and ω are constants. Let a
j
=
a
(
x
j
), b
j
=
b
(
x
j
); then (18) can be written as(20)
Z
1
i
v
x
j
=
a
j
Z
1
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(21)
Z
1
i
v
x
j
=
1
h
4
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
.Using (17) and (21) in (20), we get the following N
+
1 system of equations:(22)
∑
i
=
-
8
N
+
8
z
i
ϕ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
8
N
+
8
z
i
ϕ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.


## 3.1.2. Approximating Collocation Algorithm

In approximating collocating algorithm (i.e., algorithm based on approximating scheme (2)), we assume approximate solution Z
2
(
x
) of (3) as(23)
Z
2
x
=
∑
i
=
-
6
N
+
6
z
i
Φ
x
-
x
i
h
,
0
⩽
x
⩽
1
,where N is the positive integer N
⩾
6, h
=
1
/
N and x
i
=
i
/
N
=
i
h, and {
z
i
} are the unknown to be determined for the solution of (3). The collocation algorithm, together with the boundary conditions to be discussed, is given by(24)
Z
2
i
v
x
j
=
a
x
j
Z
2
x
j
+
b
x
j
,
j
=
0,1
,
2
,
…
,
N
,with the following type of boundary conditions:(25)
Z
2
0
=
α
,
Z
2
′
0
=
β
,
Z
2
N
=
γ
,
Z
2
′
N
=
ω
.Equation (24) can be written as(26)
Z
2
i
v
x
j
=
a
j
Z
2
x
j
+
b
j
,
j
=
0,1
,
2
,
…
,
N
,where(27)
Z
2
i
v
x
j
=
1
h
4
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
.Using (23) and (27) in (26), we get the following N
+
1 system of equations:(28)
∑
i
=
-
6
N
+
6
z
i
Φ
i
v
x
j
-
x
i
h
-
h
4
a
j
∑
i
=
-
6
N
+
6
z
i
Φ
x
j
-
x
i
h
=
h
4
b
j
,
j
=
0,1
,
2
,
…
,
N
.Now we simplify systems (22) and (28) in the following theorems.Theorem 3 (interpolating collocation algorithm).
Forj
=
0 by (22), one gets(29)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
q
0
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
0
=
ϕ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (22), we get(30)
z
-
8
ϕ
i
v
x
0
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
0
-
x
N
+
8
h
-
a
0
h
4
z
-
8
ϕ
x
0
-
x
-
8
h
+
z
-
7
ϕ
x
0
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
0
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
0
-
x
N
+
8
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(31)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
⋯
+
z
N
+
7
ϕ
i
v
-
N
-
7
+
z
N
+
8
ϕ
i
v
-
N
-
8
-
a
0
h
4
z
-
8
ϕ
8
+
z
-
7
ϕ
7
+
⋯
+
z
N
+
7
ϕ
-
N
-
7
+
z
N
+
8
ϕ
-
N
-
8
=
h
4
b
0
.Since the cardinal support of basis function ϕ
(
x
) is [
-
8,8
], so ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
]; also, by Table 3, we get(32)
z
-
8
ϕ
i
v
8
+
z
-
7
ϕ
i
v
7
+
z
-
6
ϕ
i
v
6
+
z
-
5
ϕ
i
v
5
+
z
-
4
ϕ
i
v
4
+
z
-
3
ϕ
i
v
3
+
z
-
2
ϕ
i
v
2
+
z
-
1
ϕ
i
v
1
+
z
0
ϕ
i
v
0
+
z
1
ϕ
i
v
-
1
+
z
2
ϕ
i
v
-
2
+
z
3
ϕ
i
v
-
3
+
z
4
ϕ
i
v
-
4
+
z
5
ϕ
i
v
-
5
+
z
6
ϕ
i
v
-
6
+
z
7
ϕ
i
v
-
7
+
z
8
ϕ
i
v
-
8
-
a
0
h
4
z
0
ϕ
0
=
h
4
b
0
.If ϕ
i
i
v
=
ϕ
i
v
(
i
), then(33)
z
-
8
ϕ
8
i
v
+
z
-
7
ϕ
7
i
v
+
z
-
6
ϕ
6
i
v
+
z
-
5
ϕ
5
i
v
+
z
-
4
ϕ
4
i
v
+
z
-
3
ϕ
3
i
v
+
z
-
2
ϕ
2
i
v
+
z
-
1
ϕ
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
-
1
i
v
+
z
2
ϕ
-
2
i
v
+
z
3
ϕ
-
3
i
v
+
z
4
ϕ
-
4
i
v
+
z
5
ϕ
-
5
i
v
+
z
6
ϕ
-
6
i
v
+
z
7
ϕ
-
7
i
v
+
z
8
ϕ
-
8
i
v
=
h
4
b
0
.As we observe from Table 3, ϕ
-
i
i
v
=
ϕ
i
i
v; we have (34)
z
-
8
ϕ
-
8
i
v
+
z
-
7
ϕ
-
7
i
v
+
z
-
6
ϕ
-
6
i
v
+
z
-
5
ϕ
-
5
i
v
+
z
-
4
ϕ
-
4
i
v
+
z
-
3
ϕ
-
3
i
v
+
z
-
2
ϕ
-
2
i
v
+
z
-
1
ϕ
-
1
i
v
+
z
0
ϕ
0
i
v
-
a
0
h
4
+
z
1
ϕ
1
i
v
+
z
2
ϕ
2
i
v
+
z
3
ϕ
3
i
v
+
z
4
ϕ
4
i
v
+
z
5
ϕ
5
i
v
+
z
6
ϕ
6
i
v
+
z
7
ϕ
7
i
v
+
z
8
ϕ
8
i
v
=
h
4
b
0
.
Forq
0
=
ϕ
0
i
v
-
a
0
h
4, we get (29). This completes the proof.Theorem 4 (interpolating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (22) is equivalent to(35)
z
-
8
ϕ
-
j
-
8
i
v
+
z
-
7
ϕ
-
j
-
7
i
v
+
⋯
+
z
0
ϕ
-
j
i
v
+
z
1
ϕ
1
-
j
i
v
-
a
j
h
4
ϕ
1
-
j
+
z
2
ϕ
2
-
j
i
v
-
a
j
h
4
ϕ
2
-
j
+
⋯
+
z
N
ϕ
N
-
j
i
v
-
a
j
h
4
ϕ
N
-
j
+
z
N
+
1
ϕ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (22), we get(36)
z
-
8
ϕ
i
v
x
j
-
x
-
8
h
+
z
-
7
ϕ
i
v
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
i
v
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
i
v
x
j
-
x
N
+
8
h
-
a
j
h
4
z
-
8
ϕ
x
j
-
x
-
8
h
+
z
-
7
ϕ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
7
ϕ
x
j
-
x
N
+
7
h
+
z
N
+
8
ϕ
x
j
-
x
N
+
8
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(37)
z
-
8
ϕ
i
v
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
z
-
8
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
=
h
4
b
j
.This implies (38)
z
-
8
ϕ
i
v
j
+
8
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
i
v
j
+
7
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
i
v
j
-
N
-
7
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
i
v
j
-
N
-
8
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.If ϕ
j
i
v
=
ϕ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(39)
z
-
8
ϕ
j
+
8
i
v
-
a
j
h
4
ϕ
j
+
8
+
z
-
7
ϕ
j
+
7
i
v
-
a
j
h
4
ϕ
j
+
7
+
⋯
+
z
N
+
7
ϕ
j
-
N
-
7
i
v
-
a
j
h
4
ϕ
j
-
N
-
7
+
z
N
+
8
ϕ
j
-
N
-
8
i
v
-
a
j
h
4
ϕ
j
-
N
-
8
=
h
4
b
j
.As we observe from Table 3, ϕ
-
j
i
v
=
ϕ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (40)
z
-
8
ϕ
-
j
-
8
i
v
-
a
j
h
4
ϕ
-
j
-
8
+
z
-
7
ϕ
-
j
-
7
i
v
-
a
j
h
4
ϕ
-
j
-
7
+
⋯
+
z
N
+
7
ϕ
N
+
7
-
j
i
v
-
a
j
h
4
ϕ
N
+
7
-
j
+
z
N
+
8
ϕ
N
+
8
-
j
i
v
-
a
j
h
4
ϕ
N
+
8
-
j
=
h
4
b
j
.Since ϕ
′
(
x
), ϕ
′
′
(
x
), ϕ
′
′
′
(
x
), and ϕ
i
v
(
x
) are zero outside the interval [
-
8,8
], then by Table 3, we get (35).Theorem 5 (approximating collocation algorithm).
Forj
=
0, by (28), one gets(41)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
q
0
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
=
h
4
b
0
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
0
=
Φ
0
i
v
-
a
0
h
4.Proof.
Substitutingj
=
0 in (28), we get(42)
z
-
6
Φ
i
v
x
0
-
x
-
6
h
+
z
-
5
Φ
i
v
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
0
-
x
N
+
6
h
-
a
0
h
4
z
-
6
Φ
x
0
-
x
-
6
h
+
z
-
5
Φ
x
0
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
x
0
-
x
N
+
5
h
+
z
N
+
6
Φ
x
0
-
x
N
+
6
h
=
h
4
b
0
.For x
i
=
i
h, i
=
0,1
,
2
,
…
,
N, this implies(43)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
⋯
+
z
N
+
5
Φ
i
v
-
N
-
5
+
z
N
+
6
Φ
i
v
-
N
-
6
-
a
0
h
4
z
-
6
Φ
6
+
z
-
5
Φ
5
+
⋯
+
z
N
+
5
Φ
-
N
-
5
+
z
N
+
6
Φ
-
N
-
6
=
h
4
b
0
.Since the cardinal support of basis function Φ
(
x
) is [
-
6,6
], so Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
]; also, by Table 4, we get(44)
z
-
6
Φ
i
v
6
+
z
-
5
Φ
i
v
5
+
z
-
4
Φ
i
v
4
+
z
-
3
Φ
i
v
3
+
z
-
2
Φ
i
v
2
+
z
-
1
Φ
i
v
1
+
z
0
Φ
i
v
0
+
z
1
Φ
i
v
-
1
+
z
2
Φ
i
v
-
2
+
z
3
Φ
i
v
-
3
+
z
4
Φ
i
v
-
4
+
z
5
Φ
i
v
-
5
+
z
6
Φ
i
v
-
6
-
a
0
h
4
z
0
Φ
0
=
h
4
b
0
.If Φ
i
i
v
=
Φ
i
v
(
i
), then(45)
z
-
6
Φ
6
i
v
+
z
-
5
Φ
5
i
v
+
z
-
4
Φ
4
i
v
+
z
-
3
Φ
3
i
v
+
z
-
2
Φ
2
i
v
+
z
-
1
Φ
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
-
1
i
v
+
z
2
Φ
-
2
i
v
+
z
3
Φ
-
3
i
v
+
z
4
Φ
-
4
i
v
+
z
5
Φ
-
5
i
v
+
z
6
Φ
-
6
i
v
=
h
4
b
0
.As we observe from Table 4, Φ
-
i
i
v
=
Φ
i
i
v; we have(46)
z
-
6
Φ
-
6
i
v
+
z
-
5
Φ
-
5
i
v
+
z
-
4
Φ
-
4
i
v
+
z
-
3
Φ
-
3
i
v
+
z
-
2
Φ
-
2
i
v
+
z
-
1
Φ
-
1
i
v
+
z
0
Φ
0
i
v
-
a
0
h
4
+
z
1
Φ
1
i
v
+
z
2
Φ
2
i
v
+
z
3
Φ
3
i
v
+
z
4
Φ
4
i
v
+
z
5
Φ
5
i
v
+
z
6
Φ
6
i
v
+
z
7
Φ
7
i
v
+
z
8
Φ
8
i
v
=
h
4
b
0
.
ForΥ
0
=
Φ
0
i
v
-
a
0
h
4, we get (41). This completes the proof.Theorem 6 (approximating collocation algorithm).
Forj
=
1,2
,
3
,
…
,
N, system (28) is equivalent to(47)
z
-
6
Φ
-
j
-
6
i
v
+
z
-
5
Φ
-
j
-
5
i
v
+
⋯
+
z
0
Φ
-
j
i
v
+
z
1
Φ
1
-
j
i
v
-
a
j
h
4
Φ
1
-
j
+
z
2
Φ
2
-
j
i
v
-
a
j
h
4
Φ
2
-
j
+
⋯
+
z
N
Φ
N
-
j
i
v
-
a
j
h
4
Φ
N
-
j
+
z
N
+
1
Φ
N
+
1
-
j
i
v
+
⋯
+
z
N
+
6
Φ
N
+
6
-
j
i
v
=
h
4
b
j
.Proof.
By expanding (28), we get(48)
z
-
6
Φ
i
v
x
j
-
x
-
6
h
+
z
-
5
Φ
i
v
x
j
-
x
-
5
h
+
⋯
+
z
N
+
5
Φ
i
v
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
i
v
x
j
-
x
N
+
6
h
-
a
j
h
4
z
-
8
Φ
x
j
-
x
-
8
h
+
z
-
7
Φ
x
j
-
x
-
7
h
+
⋯
+
z
N
+
5
Φ
x
j
-
x
N
+
5
h
+
z
N
+
6
Φ
x
j
-
x
N
+
6
h
=
h
4
b
j
.For x
j
=
j
h, j
=
1,2
,
…
,
N, we get(49)
z
-
6
Φ
i
v
j
+
6
+
z
-
5
Φ
i
v
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
z
-
6
Φ
j
+
6
+
z
-
5
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
=
h
4
b
j
.This implies (50)
z
-
6
Φ
i
v
j
+
6
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
i
v
j
+
5
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
i
v
j
-
N
-
5
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
i
v
j
-
N
-
6
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.If Φ
j
i
v
=
Φ
i
v
(
j
), for j
=
1,2
,
…
,
N, then(51)
z
-
6
Φ
j
+
6
i
v
-
a
j
h
4
Φ
j
+
6
+
z
-
5
Φ
j
+
5
i
v
-
a
j
h
4
Φ
j
+
5
+
⋯
+
z
N
+
5
Φ
j
-
N
-
5
i
v
-
a
j
h
4
Φ
j
-
N
-
5
+
z
N
+
6
Φ
j
-
N
-
6
i
v
-
a
j
h
4
Φ
j
-
N
-
6
=
h
4
b
j
.As we observe from Table 4, Φ
-
j
i
v
=
Φ
j
i
v, j
=
1,2
,
…
,
N; then the above equation can be written as (52)
z
-
6
Φ
-
j
-
6
i
v
-
a
j
h
4
Φ
-
j
-
6
+
z
-
5
Φ
-
j
-
5
i
v
-
a
j
h
4
Φ
-
j
-
5
+
⋯
+
z
N
+
5
Φ
N
+
5
-
j
i
v
-
a
j
h
4
Φ
N
+
5
-
j
+
z
N
+
6
Φ
N
+
6
-
j
i
v
-
a
j
h
4
Φ
N
+
6
-
j
=
h
4
b
j
.Since Φ
′
(
x
), Φ
′
′
(
x
), Φ
′
′
′
(
x
), and Φ
i
v
(
x
) are zero outside the interval [
-
6,6
], then, by Table 4, we get (47).


## 3.2. Boundary Conditions at End Points

We have two different systems of(
N
+
1
) equations defined by (22) and (28). In order to get unique solution of these systems, we need sixteen more conditions for system (22) and twelve more conditions for system (28). Four conditions can be achieved from boundary conditions given in (4) for both systems of linear equations in which first order derivatives are involved and remaining conditions are achieved by some extrapolation method at the end points. First we find the approximation of the first derivative by difference operators and after that we define the extrapolation method at end points for both systems of linear equations.


### 3.2.1. Approximation of Derivative Boundary Conditions

In this section, we approximate the derivative boundary conditions by difference operators. Since approximation order of interpolating scheme (1) and approximating scheme (2) is ten and four, respectively, so we approximate derivative boundary conditions at end points with approximation orders ten and four for interpolating and approximating collocation algorithms.If we use interpolating collocation algorithm for the solution of (3), then approximation of derivative conditions at ends point is defined as(53)
Z
1
′
0
=
N
2520
-
7381
z
0
+
25200
z
1
-
56700
z
2
+
100800
z
3
-
132300
z
4
+
127008
z
5
-
88200
z
6
+
43200
z
7
-
14175
z
8
+
28800
z
9
-
252
z
10
+
O
h
10
,
(54)
Z
1
′
N
=
N
2520
7381
z
N
-
25200
z
N
-
1
+
56700
z
N
-
2
-
100800
z
N
-
3
+
132300
z
N
-
4
-
127008
z
N
-
5
+
88200
z
N
-
6
-
43200
z
N
-
7
+
14175
z
N
-
8
-
28800
z
N
-
9
+
252
z
N
-
10
+
O
h
10and if we use approximating collocation algorithm for the solution of (3), then approximation of derivative conditions at end points is defined as(55)
Z
2
′
0
=
N
12
-
25
z
0
+
48
z
1
-
36
z
2
+
16
z
3
-
3
z
4
+
O
h
4
,
(56)
Z
2
′
N
=
N
12
·
25
z
N
-
48
z
N
-
1
+
36
z
N
-
2
-
16
z
N
-
3
+
3
z
N
-
4
+
O
h
4
.


### 3.2.2. Adjustment of Boundary Conditions

Still we need twelve and eight more conditions for systems (22) and (28), respectively, to get stable systems for the solution of (3). For this we made some adjustment of boundary conditions for systems (22) and (28), which are defined below.Case 1.
If we use interpolating collocation algorithm for the approximate solution of (3), then we define six conditions at left end points and six conditions at the right end points. Since subdivision scheme (1) reproduces nine-degree (i.e., tenth order) polynomials, so we define boundary conditions of order ten for solution of (22). For simplicity only left end points z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 are discussed and the values of right end points z
N
+
2
,
z
N
+
3
,
z
N
+
4
,
z
N
+
5
,
z
N
+
6
,
z
N
+
7 can be treated similarly.
The valuesz
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 can be determined by the nonic polynomial q
(
x
) interpolating (
x
i
,
z
i
), 2
≤
i
≤
7. Precisely, we have(57)
z
-
i
=
q
-
x
i
,
i
=
2,3
,
4,5
,
6,7
,where (58)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
Z
1
x
i
-
j
.Since by (17) Z
1
(
x
i
)
=
z
i for i
=
2,3
,
4,5
,
6,7, then, by replacing x
i by -
x
i, we have(59)
q
-
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
-
i
+
j
.Hence the following boundary conditions can be employed at the left end:(60)
∑
j
=
0
10
10
j
-
1
j
z
-
i
+
j
=
0
,
i
=
7,6
,
5,4
,
3,2
.Similarly, for the right end, we can define z
i
=
q
(
-
x
i
), i
=
N
+
2, N
+
3, N
+
4, N
+
5, N
+
6, N
+
7, and(61)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
i
-
j
.So we have the following boundary conditions at the right end:(62)
∑
j
=
0
10
10
j
-
1
j
z
i
-
j
=
0
,
i
=
N
+
2
,
N
+
3
,
N
+
4
,
N
+
5
,
N
+
6
,
N
+
7
.Finally, we get the following new system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (29) and (35), four equations from boundary conditions (19), and twelve from boundary conditions (60) and (62).Case 2.
If we use approximating collocation algorithm (28) for the solution of (3) then we need eight more conditions. So in this case, we define four extra conditions at the left end points and four conditions at the right end points by some extrapolation method. Since the subdivision scheme reproduces cubic (i.e., fourth order) polynomial, so we define quartic polynomial for the adjustment of boundary treatment. The values z
-
4
,
z
-
3
,
z
-
2
,
z
-
1 are determined by the quartic polynomial p
(
x
) interpolating (
x
i
,
z
i
). This polynomial is defined as(63)
z
-
i
+
1
=
p
-
x
i
+
1
,
i
=
1,2
,
3,4
,where (64)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
Z
2
x
i
-
j
+
1
.Since by (23) Z
2
(
x
i
)
=
z
i for i
=
1,2
,
3,4, then, by replacing x
i by -
x
i, we have (65)
p
-
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
-
i
+
j
+
1
.Hence the following boundary conditions can be employed at the left end:(66)
∑
j
=
0
4
4
j
-
1
j
z
-
i
+
j
+
1
=
0
,
i
=
1,2
,
3,4
.Similarly, for the right end, we can define z
i
+
1
=
p
(
x
i
+
1
), i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4, and (67)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
i
-
j
+
1
.So we have the following boundary conditions at the right end:(68)
∑
j
=
0
4
4
j
-
1
j
z
i
-
j
+
1
=
0
,
i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4
.Finally, we get a following new system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (41) and (47), four equations from boundary conditions (25), and eight from boundary conditions (66) and (68).


## 3.2.1. Approximation of Derivative Boundary Conditions

In this section, we approximate the derivative boundary conditions by difference operators. Since approximation order of interpolating scheme (1) and approximating scheme (2) is ten and four, respectively, so we approximate derivative boundary conditions at end points with approximation orders ten and four for interpolating and approximating collocation algorithms.If we use interpolating collocation algorithm for the solution of (3), then approximation of derivative conditions at ends point is defined as(53)
Z
1
′
0
=
N
2520
-
7381
z
0
+
25200
z
1
-
56700
z
2
+
100800
z
3
-
132300
z
4
+
127008
z
5
-
88200
z
6
+
43200
z
7
-
14175
z
8
+
28800
z
9
-
252
z
10
+
O
h
10
,
(54)
Z
1
′
N
=
N
2520
7381
z
N
-
25200
z
N
-
1
+
56700
z
N
-
2
-
100800
z
N
-
3
+
132300
z
N
-
4
-
127008
z
N
-
5
+
88200
z
N
-
6
-
43200
z
N
-
7
+
14175
z
N
-
8
-
28800
z
N
-
9
+
252
z
N
-
10
+
O
h
10and if we use approximating collocation algorithm for the solution of (3), then approximation of derivative conditions at end points is defined as(55)
Z
2
′
0
=
N
12
-
25
z
0
+
48
z
1
-
36
z
2
+
16
z
3
-
3
z
4
+
O
h
4
,
(56)
Z
2
′
N
=
N
12
·
25
z
N
-
48
z
N
-
1
+
36
z
N
-
2
-
16
z
N
-
3
+
3
z
N
-
4
+
O
h
4
.


## 3.2.2. Adjustment of Boundary Conditions

Still we need twelve and eight more conditions for systems (22) and (28), respectively, to get stable systems for the solution of (3). For this we made some adjustment of boundary conditions for systems (22) and (28), which are defined below.Case 1.
If we use interpolating collocation algorithm for the approximate solution of (3), then we define six conditions at left end points and six conditions at the right end points. Since subdivision scheme (1) reproduces nine-degree (i.e., tenth order) polynomials, so we define boundary conditions of order ten for solution of (22). For simplicity only left end points z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 are discussed and the values of right end points z
N
+
2
,
z
N
+
3
,
z
N
+
4
,
z
N
+
5
,
z
N
+
6
,
z
N
+
7 can be treated similarly.
The valuesz
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2 can be determined by the nonic polynomial q
(
x
) interpolating (
x
i
,
z
i
), 2
≤
i
≤
7. Precisely, we have(57)
z
-
i
=
q
-
x
i
,
i
=
2,3
,
4,5
,
6,7
,where (58)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
Z
1
x
i
-
j
.Since by (17) Z
1
(
x
i
)
=
z
i for i
=
2,3
,
4,5
,
6,7, then, by replacing x
i by -
x
i, we have(59)
q
-
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
-
i
+
j
.Hence the following boundary conditions can be employed at the left end:(60)
∑
j
=
0
10
10
j
-
1
j
z
-
i
+
j
=
0
,
i
=
7,6
,
5,4
,
3,2
.Similarly, for the right end, we can define z
i
=
q
(
-
x
i
), i
=
N
+
2, N
+
3, N
+
4, N
+
5, N
+
6, N
+
7, and(61)
q
x
i
=
∑
j
=
1
10
10
j
-
1
j
+
1
z
i
-
j
.So we have the following boundary conditions at the right end:(62)
∑
j
=
0
10
10
j
-
1
j
z
i
-
j
=
0
,
i
=
N
+
2
,
N
+
3
,
N
+
4
,
N
+
5
,
N
+
6
,
N
+
7
.Finally, we get the following new system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (29) and (35), four equations from boundary conditions (19), and twelve from boundary conditions (60) and (62).Case 2.
If we use approximating collocation algorithm (28) for the solution of (3) then we need eight more conditions. So in this case, we define four extra conditions at the left end points and four conditions at the right end points by some extrapolation method. Since the subdivision scheme reproduces cubic (i.e., fourth order) polynomial, so we define quartic polynomial for the adjustment of boundary treatment. The values z
-
4
,
z
-
3
,
z
-
2
,
z
-
1 are determined by the quartic polynomial p
(
x
) interpolating (
x
i
,
z
i
). This polynomial is defined as(63)
z
-
i
+
1
=
p
-
x
i
+
1
,
i
=
1,2
,
3,4
,where (64)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
Z
2
x
i
-
j
+
1
.Since by (23) Z
2
(
x
i
)
=
z
i for i
=
1,2
,
3,4, then, by replacing x
i by -
x
i, we have (65)
p
-
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
-
i
+
j
+
1
.Hence the following boundary conditions can be employed at the left end:(66)
∑
j
=
0
4
4
j
-
1
j
z
-
i
+
j
+
1
=
0
,
i
=
1,2
,
3,4
.Similarly, for the right end, we can define z
i
+
1
=
p
(
x
i
+
1
), i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4, and (67)
p
x
i
+
1
=
∑
j
=
1
4
4
j
-
1
j
+
1
z
i
-
j
+
1
.So we have the following boundary conditions at the right end:(68)
∑
j
=
0
4
4
j
-
1
j
z
i
-
j
+
1
=
0
,
i
=
N
+
1
,
N
+
2
,
N
+
3
,
N
+
4
.Finally, we get a following new system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}, in which N
+
1 equations are obtained from (41) and (47), four equations from boundary conditions (25), and eight from boundary conditions (66) and (68).


## 3.3. Stable Systems of Linear Equations

In this section, we present stable systems of linear equations for both interpolating and approximating collocation algorithms.


### 3.3.1. Stable System for Interpolating Collocation Algorithm

From (29) and (35), we get the following undetermined system of (
N
+
1
) equations with (
N
+
17
) unknowns {
z
i
}:(69)
A
1
Z
1
=
G
1
,where the matrices A
1, Z
1, and G
1 of orders (
N
+
1
)
×
(
N
+
17
), N
+
17, and N
+
1, respectively, are given by(70)
A
1
=
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
0
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
ϕ
8
i
v
⋯
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
1
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
⋯
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
2
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
⋯
0
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
3
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
7
′
′
′
ϕ
8
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
6
′
′
′
ϕ
7
′
′
′
ϕ
8
′
′
′
,
(71)
Z
1
=
z
-
8
,
z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
…
,
z
N
+
8
T
,
(72)
G
1
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
j
=
ϕ
0
i
v
-
a
j
h
4.For obtaining the unique solution of (69), we made some adjustment of boundary conditions in previous section which is defined in (53), (54), (60), and (62). By using this adjustment, we get the following system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, defined as(73)
D
1
Z
1
=
R
1
,where the coefficient matrix(74)
D
1
=
B
0
T
,
A
1
T
,
B
1
T
T
.
A
1 is defined by (70); B
0 and B
1 are defined as(75)
B
0
=
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
⋯
0
0
0
0
0
0
0
0
0
0
7381
N
2520
25200
N
2520
-
56700
N
2520
100800
N
2520
-
132300
N
2520
127008
N
2520
-
88200
N
2520
43200
N
2520
-
14175
N
2520
2800
N
2520
-
252
N
2520
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
.First six rows of B
0 are obtained from (60), second last row is obtained from (53), and last row is taken from given boundary conditions Z
1
(
0
) which is defined in (19) and(76)
B
1
=
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
N
10
-
10
N
9
45
N
8
-
120
N
7
35
N
-
252
N
5
105
N
2
-
40
N
45
N
2
-
10
N
7381
N
2520
0
0
0
0
0
0
0
0
0
0
⋯
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
0
⋯
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
.First row of B
1 is obtained from Z
1
(
N
) which is defined in (19), second row is obtained from (54), and the last six rows are obtained from (62) and Z
1 which is defined in (71) and R
1 is defined as (77)
R
1
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
1
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
1 is defined by (72).


### 3.3.2. Stable System for Approximating Collocation Algorithm

From (41) and (47), we get the following undetermined system of (
N
+
1
) equations with (
N
+
13
) unknowns {
z
i
}:(78)
A
2
Z
2
=
G
2
,where the matrices A
2, Z
2, and G
2 of orders (
N
+
1
)
×
(
N
+
13
), N
+
13, and N
+
1, respectively, are given by(79)
A
2
=
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
0
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
⋯
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
1
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
⋯
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
2
Φ
1
i
v
Φ
3
i
v
Φ
4
i
v
⋯
0
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
3
Φ
2
i
v
Φ
3
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
5
′
′
′
Φ
6
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
4
′
′
′
Φ
5
′
′
′
Φ
6
′
′
′
,
(80)
Z
2
=
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2
,
…
,
z
N
+
6
T
,
(81)
G
2
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
j
=
Φ
0
i
v
-
a
j
h
4.In order to get the unique solution of system (78), we have defined some extra conditions in (55), (56), (66), and (68). By using these extra conditions we get the following system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}:(82)
D
2
Z
2
=
R
2
,where the coefficient matrix(83)
D
2
=
B
0
T
,
A
2
T
,
B
1
T
T
.
A
2 is defined by (79); B
0 and B
1 are defined as(84)
B
0
=
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
4
1
0
0
0
0
⋯
0
0
0
0
0
0
0
0
-
251
N
12
48
N
12
-
36
N
12
16
N
12
-
3
N
12
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
⋯
0
0
.First four rows of B
0 are obtained from (66), second last row is obtained from (55), and the last row is taken from the given boundary conditions Z
2
(
0
) which is defined in (25) and(85)
B
1
=
0
0
0
0
⋯
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
3
N
12
-
16
N
12
36
N
12
-
48
N
12
25
N
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
.First row of B
1 is obtained from Z
2
(
N
) which is defined in (25), second row is obtained from (56), and the last four rows are obtained from (68); Z
2 is defined in (80) and R
2 is defined as (86)
R
2
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
2
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
2 is defined by (81). Hence to obtain the approximate solution of the fourth order boundary value problem (3) by interpolating and approximating collocation algorithms we need to solve systems (73) and (82), respectively.


## 3.3.1. Stable System for Interpolating Collocation Algorithm

From (29) and (35), we get the following undetermined system of (
N
+
1
) equations with (
N
+
17
) unknowns {
z
i
}:(69)
A
1
Z
1
=
G
1
,where the matrices A
1, Z
1, and G
1 of orders (
N
+
1
)
×
(
N
+
17
), N
+
17, and N
+
1, respectively, are given by(70)
A
1
=
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
0
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
ϕ
8
i
v
⋯
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
1
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
⋯
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
2
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
⋯
0
0
0
0
0
0
ϕ
-
8
i
v
ϕ
-
7
i
v
ϕ
-
6
i
v
ϕ
-
5
i
v
ϕ
-
4
i
v
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
q
3
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
7
′
′
′
ϕ
8
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
⋯
⋯
ϕ
6
′
′
′
ϕ
7
′
′
′
ϕ
8
′
′
′
,
(71)
Z
1
=
z
-
8
,
z
-
7
,
z
-
6
,
z
-
5
,
z
-
4
,
…
,
z
N
+
8
T
,
(72)
G
1
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where ϕ
j
i
v
=
ϕ
i
v
(
j
) and q
j
=
ϕ
0
i
v
-
a
j
h
4.For obtaining the unique solution of (69), we made some adjustment of boundary conditions in previous section which is defined in (53), (54), (60), and (62). By using this adjustment, we get the following system of (
N
+
17
) linear equations with (
N
+
17
) unknowns {
z
i
}, defined as(73)
D
1
Z
1
=
R
1
,where the coefficient matrix(74)
D
1
=
B
0
T
,
A
1
T
,
B
1
T
T
.
A
1 is defined by (70); B
0 and B
1 are defined as(75)
B
0
=
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
⋯
0
0
0
0
0
0
0
0
0
0
7381
N
2520
25200
N
2520
-
56700
N
2520
100800
N
2520
-
132300
N
2520
127008
N
2520
-
88200
N
2520
43200
N
2520
-
14175
N
2520
2800
N
2520
-
252
N
2520
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
.First six rows of B
0 are obtained from (60), second last row is obtained from (53), and last row is taken from given boundary conditions Z
1
(
0
) which is defined in (19) and(76)
B
1
=
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
⋯
N
10
-
10
N
9
45
N
8
-
120
N
7
35
N
-
252
N
5
105
N
2
-
40
N
45
N
2
-
10
N
7381
N
2520
0
0
0
0
0
0
0
0
0
0
⋯
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
0
⋯
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
0
⋯
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
0
⋯
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
10
45
-
120
-
120
210
-
252
210
-
120
45
-
10
1
0
0
0
⋯
0
0
0
0
0
0
0
1
-
10
45
-
120
210
-
252
210
-
120
45
-
10
1
0
.First row of B
1 is obtained from Z
1
(
N
) which is defined in (19), second row is obtained from (54), and the last six rows are obtained from (62) and Z
1 which is defined in (71) and R
1 is defined as (77)
R
1
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
1
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
1 is defined by (72).


## 3.3.2. Stable System for Approximating Collocation Algorithm

From (41) and (47), we get the following undetermined system of (
N
+
1
) equations with (
N
+
13
) unknowns {
z
i
}:(78)
A
2
Z
2
=
G
2
,where the matrices A
2, Z
2, and G
2 of orders (
N
+
1
)
×
(
N
+
13
), N
+
13, and N
+
1, respectively, are given by(79)
A
2
=
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
0
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
⋯
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
1
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
⋯
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
2
Φ
1
i
v
Φ
3
i
v
Φ
4
i
v
⋯
0
0
0
0
0
0
0
Φ
-
6
i
v
Φ
-
5
i
v
Φ
-
4
i
v
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Υ
3
Φ
2
i
v
Φ
3
i
v
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
5
′
′
′
Φ
6
′
′
′
0
0
0
0
0
0
0
0
0
0
0
0
⋯
⋯
⋯
Φ
4
′
′
′
Φ
5
′
′
′
Φ
6
′
′
′
,
(80)
Z
2
=
z
-
6
,
z
-
5
,
z
-
4
,
z
-
3
,
z
-
2
,
…
,
z
N
+
6
T
,
(81)
G
2
=
b
0
h
4
,
b
1
h
4
,
b
2
h
4
,
b
4
h
3
,
…
,
b
N
h
4
T
,where Φ
j
i
v
=
Φ
i
v
(
j
) and Υ
j
=
Φ
0
i
v
-
a
j
h
4.In order to get the unique solution of system (78), we have defined some extra conditions in (55), (56), (66), and (68). By using these extra conditions we get the following system of (
N
+
13
) linear equations with (
N
+
13
) unknowns {
z
i
}:(82)
D
2
Z
2
=
R
2
,where the coefficient matrix(83)
D
2
=
B
0
T
,
A
2
T
,
B
1
T
T
.
A
2 is defined by (79); B
0 and B
1 are defined as(84)
B
0
=
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
4
1
0
0
0
0
⋯
0
0
0
0
0
0
0
0
-
251
N
12
48
N
12
-
36
N
12
16
N
12
-
3
N
12
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
⋯
0
0
.First four rows of B
0 are obtained from (66), second last row is obtained from (55), and the last row is taken from the given boundary conditions Z
2
(
0
) which is defined in (25) and(85)
B
1
=
0
0
0
0
⋯
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
3
N
12
-
16
N
12
36
N
12
-
48
N
12
25
N
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
0
0
0
0
0
⋯
0
0
0
0
0
0
0
0
0
0
1
-
4
6
-
4
1
0
0
0
.First row of B
1 is obtained from Z
2
(
N
) which is defined in (25), second row is obtained from (56), and the last four rows are obtained from (68); Z
2 is defined in (80) and R
2 is defined as (86)
R
2
=
0,0
,
0,0
,
0,0
,
β
,
α
,
G
2
T
,
γ
,
ω
,
0,0
,
0,0
,
0,0
T
,where G
2 is defined by (81). Hence to obtain the approximate solution of the fourth order boundary value problem (3) by interpolating and approximating collocation algorithms we need to solve systems (73) and (82), respectively.


## 3.4.  Existence of the Solution

In this section, we discuss the nonsingularity of the coefficient matricesD
1 and D
2 defined in (74) and (83), respectively. We observe that the coefficient matrices D
1 and D
2 are neither symmetric nor diagonally dominant. However it can be shown that D
1 and D
2 are nonsingular. Since D
1 and D
2 are band matrices with half bandwidth 9 and 7, numerical complexities for solving the linear systems using Gaussian elimination are about 81
(
N
+
17
) and 49
(
N
+
13
) multiplications, respectively. For large N, the matrices are almost symmetric except the first and last eight rows and columns of D
1 and first and last four rows and columns of D
2 due to the boundary conditions. Therefore we first consider their symmetric part, that is, square symmetric matrices E
1 and E
2 of orders N
+
1 defined as(87)
E
1
=
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
ϕ
8
i
v
⋯
0
0
0
0
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
ϕ
7
i
v
⋯
0
0
0
0
ϕ
-
2
i
v
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
ϕ
6
i
v
⋯
0
0
0
0
ϕ
-
3
i
v
ϕ
-
2
i
v
ϕ
-
1
i
v
ϕ
0
i
v
ϕ
1
i
v
ϕ
2
i
v
ϕ
3
i
v
ϕ
4
i
v
ϕ
5
i
v
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
ϕ
4
i
v
ϕ
3
i
v
ϕ
2
i
v
ϕ
1
i
v
0
0
0
0
0
0
0
0
0
⋯
ϕ
3
i
v
ϕ
2
i
v
ϕ
1
i
v
ϕ
0
i
v
,
E
2
=
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
0
0
⋯
0
0
0
0
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
0
⋯
0
0
0
0
Φ
-
2
i
v
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
Φ
6
i
v
⋯
0
0
0
0
Φ
-
3
i
v
Φ
-
2
i
v
Φ
-
1
i
v
Φ
0
i
v
Φ
1
i
v
Φ
2
i
v
Φ
3
i
v
Φ
4
i
v
Φ
5
i
v
⋯
0
0
0
0
0
0
0
0
0
0
0
0
0
⋯
Φ
4
i
v
Φ
3
i
v
Φ
2
i
v
Φ
1
i
v
0
0
0
0
0
0
0
0
0
⋯
Φ
3
i
v
Φ
2
i
v
Φ
1
i
v
Φ
0
i
v
.So E
1 and E
2 are symmetric matrices obtained from D
1 and D
2, respectively. It can be shown that E
1 and E
2 are nonsingular when N increase. The nonsingularity of the matrices E
1 and E
2 is shown in Table 5 by finding their determinants. The matrices E
1 and E
2 remain nonsingular for N
⩽
500 and N
⩽
100, respectively. For large N the determinants of the matrices may or may not be equal to zero. The nonsingularity of the matrices D
1 and D
2 has been checked by finding their eigenvalues for N
⩽
500 and for N
⩽
100, respectively. Since the eigenvalues for both the matrices are nonzero, then by [17] matrices D
1 and D
2 are nonsingular. However the matrices D
1 for N
⩾
500 and D
2 for N
⩾
100 may or may not be nonsingular. Therefore we claim that systems of (73) and (82) are stable.Table 5
Determinants of the matrices.
N
E
1
E
2
10
-
8667
/
56
1.11401292
×
10
3
50
-
177183
3.2517495
×
10
2
100
-
552709050
0.753776508473953
200
-
5.033491472
×
1
0
36
⋯
300
-
4.477989536
×
1
0
71
⋯
400
3987757210720454
⋯
500
3987757210720454
⋯


## 3.5. Error Estimation

In this section, we discuss the approximation properties of the interpolating and approximating collocation algorithms. Since schemes (1) and (2) reproduce polynomial curves of degree nine and three, so by [10] schemes have approximation orders ten and four, respectively. Here we present our main results for error estimation. Proof of these results is similar to the proof of Proposition [12, 14].Proposition 7.
Suppose the exact solutiony
(
x
)
∈
C
4
[
0,1
] and {
z
i
} are obtained by (73); then absolute error by interpolating collocation algorithm is (88)
e
r
r
1
x
∞
=
Z
1
l
x
-
y
l
x
∞
=
O
h
3
-
l
,
l
=
0,1
,
2,3
,where l denotes the order of derivative.Proposition 8.
Suppose the exact solutiony
(
x
)
∈
C
4
[
0,1
] and {
z
i
} are obtained by (82); then absolute error by approximating collocation algorithm is (89)
e
r
r
2
x
∞
=
Z
2
l
x
-
y
l
x
∞
=
O
h
3
-
l
,
l
=
0,1
,
2,3
.


## 4. Numerical Examples and Comparison

In this section, the interpolating and approximating collocation algorithms described in Section3 are tested on the problems given below. Absolute errors between exact and approximate solutions are also calculated. For the sake of comparisons, we also tabulated the results in this section. Graphical illustrations of solutions are presented.


### 4.1. Numerical Examples

Here we find the numerical solutions of some of the boundary value problems arising in the mathematical modeling of viscoelastic and inelastic flows and so forth.Example 1.
Consider the fourth order linear boundary value problem(90)
y
i
v
x
+
x
y
=
-
8
+
7
x
+
x
3
e
x
,
0
<
x
<
1
,subject to the boundary conditions(91)
y
0
=
y
1
=
0
,
y
′
0
=
1
,
y
′
1
=
-
e
.By comparing the above problem with (3), we have a
(
x
)
=
-
x and b
(
x
)
=
-
(
8
+
7
x
+
x
3
)
e
x. The exact solution for the above problem is y
=
x
(
1
-
x
)
e
x
.
Here we present the numerical solution of the above problem by interpolating and approximating collocation algorithms.
Solution by Interpolating Collocation Algorithm. In this method, by solving the system of linear equations (73) at N
=
10, we obtain the approximate solution (17) of (90) where {
z
i
}, -
8
≤
i
≤
18, are(92)
-
0.736798818
,
-
0.643536669
,
-
0.554623724
,
-
0.467529093
,
-
0.379799398
,
-
0.289680649
,
-
0.196215973
,
-
0.099344276
,
0.0000
,
0.0997855
,
0.196780435
,
0.286445566
,
0.362823930
,
0.4184258296
,
0.444109480
,
0.428957387
,
0.360149377
,
0.222833881
,
0.000
,
-
0.327646133
,
-
0.781795936
,
-
1.386637856
,
-
2.168934169
,
-
3.158065904
,
-
4.386034900
,
-
5.887410047
,
-
7.698656102
.Solution by Approximating Collocation Algorithm. In this method, we solve the system of linear equations (82) at N
=
10 and get solution (23) of (90) where z
i, -
6
≤
i
≤
16, are(93)
240576495.97007
,
346838.29496
,
518.9347067
,
-
0.27530396
,
-
0.192116155
,
-
0.098802082
,
0.0000
,
0.099651832
,
0.195515154
,
0.282951708
,
0.357323235
,
0.412290577
,
0.439106327
,
0.426338357
,
0.359800241
,
0.223138586
,
0
,
-
0.325968911
,
-
0.771121538
,
-
1.351811276
,
759.3687002
,
508693.855464
,
352843031.1188856
.Example 2.
Consider the following fourth order linear boundary value problem,(94)
y
i
v
x
=
x
4
+
14
x
3
+
49
x
2
+
32
x
-
12
e
x
0
≤
x
≤
1with(95)
y
0
=
y
1
=
y
′
0
=
y
′
1
,corresponds to the bending of a thin beam clamped at both ends. The unique solution of (94) is (96)
y
x
=
x
2
1
-
x
2
e
x
.Solution by Interpolating Collocation Algorithm. By using this method, we solve system (73) at N
=
10 and get solution (17) of (94) where {
z
i
}, -
8
≤
i
≤
18, are (97)
-
1.292188418
,
0.538942241
,
0.376822099
,
0.254979738
,
0.157378182
,
0.084436111
,
0.035355124
,
0.008209932
,
0
,
0.006713642
,
0.023450361
,
0.044645810
,
0.064445941
,
0.077282873
,
0.078714604
,
0.066604028
,
0.042729735
,
0.014941769
,
0
,
0.027260994
,
0.143411251
,
0.418480429
,
0.953407669
,
1.889477424
,
3.419988190
,
5.826254985
,
6.993262174
.Solution by Approximating Collocation Algorithm. In this method, by solving system of linear equations (82) at N
=
10, we obtain the approximate solution (23) of (94) where {
z
i
}, -
6
≤
i
≤
16, are(98)
1.711850528
×
1
0
11
,
-
2.46996092
×
1
0
8
,
3.5607957013
×
1
0
5
,
-
5.335463543
×
1
0
5
,
0.06034270857
,
0.013560666
,
0
,
0.01051064293055
7
,
0.035942526106682
,
0.067145581105052
,
0.094969739502347
,
0.110264932875244
,
0.107216514892050
,
0.086571522615334
,
0.056424246024606
,
0.024868975122817
,
0
,
0.01008838960215368
,
0.002698096580563341
,
-
3.898592698
×
1
0
3
,
2.597091086
×
1
0
6
,
-
1.801597056
×
1
0
9
,
1.2486272867
×
1
0
12
.Example 3.
Consider the boundary value problem(99)
y
i
v
-
y
=
-
4
2
x
cos
⁡
x
+
3
sin
⁡
xwith boundary conditions(100)
y
0
=
0
,
y
1
=
0
,
y
′
0
=
-
1
,
y
′
1
=
2
sin
⁡
1
.The exact solution of this problem is y
=
(
x
2
-
1
)
s
i
n
(
x
).
Solution by Interpolating Collocation Algorithm. Here, we solve system (73) at N
=
10 and get solution (17) of (99) where {
z
i
}, -
8
≤
i
≤
18, are(101)
0.0600152940
,
0.18160335240
,
0.25833493990
,
0.29216208360
,
0.2870204837
,
0.1824477485
,
0.09700293
,
0.000000000
,
-
0.1001836945
,
-
0.1951709144
,
-
0.2768975252
,
-
0.3379193763
,
-
0.3717065517
,
-
0.372918743
,
-
0.3376558661
,
-
0.26367852
,
-
0.1505934691
,
0.000000000
,
0.1844062122
,
0.3967742453
,
0.6290856272
,
0.87118909250
,
1.11088175850
,
1.33403484280
,
1.52476071733
,
1.6661398769
.Solution by Approximating Collocation Algorithm. By solving system of linear equations (82) at N
=
10, we obtain the approximate solution (23) of (99) where {
z
i
}, -
6
≤
i
≤
16, are(102)
115585876195.8271
,
166975210.2723412
,
240725.36249556
,
360.797052040
,
0.1912504195
,
0.0989171622
,
0.000000
,
-
0.0988737231
,
-
0.1910766633
,
-
0.2699814765
,
-
0.328960819
,
-
0.361387346
,
-
0.3620271786
,
-
0.3276693903
,
-
0.2568529513
,
-
0.1481168313
,
0.00000
,
0.1889585727
,
0.4202199172
,
648.2401207
,
432617.012
,
300071307.86
,
207719711144.92
.


### 4.2. Comparison

The numerical results of Examples1, 2, and 3 by interpolating and approximating collocation algorithms are presented in Tables 6, 7, and 8, respectively. The maximum absolute errors in the solution of Examples 1, 2, and 3 obtained by interpolating and approximating collocation algorithms are given in Table 9. Graphical representation of these results is shown in Figures 1, 2, and 3. In these figures solid curve represents the exact solutions, dashed lines represent approximate solutions obtained by (73), and dotted lines represent the approximate solutions obtained by (82). Following is the comparison of the numerical solutions obtained by proposed algorithms and other approaches of this type of boundary value problems:(i)
From the above results we see that approximating schemes based collocation algorithms give better results than interpolating schemes based collocation algorithms.
(ii)
Example1 is also solved by [15]. He solved this problem by second order finite difference method and obtained the maximum absolute errors at different step sizes h. We observe that the maximum absolute error at the step size h
=
1
/
10 by the proposed approximating collocation algorithm is better than the maximum absolute error obtained by [15] at step size h
=
1
/
16. The comparison of proposed methods with second order finite difference method of [15] at difference step sizes is shown in Table 10.
(iii)
Example2 is also solved by [18] by quintic spline based collocation methods. We observe that order of error approximation obtained by [18] and proposed approximating collocation algorithm is the same.Table 6
Numerical results of Example1.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating    collocation algorithm
Approximate solutionZ
2 by approximating    collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
0.09946538
0.0997855152
0.0996518317
0.00032013
0.00018645
0.2
0.19542444
0.1967804015
0.1955151540
0.00135596
0.00009071
0.3
0.28347035
0.2864455108
0.2829517080
0.00297516
0.00051864
0.4
0.35803793
0.3628238578
0.3573232345
0.00478593
0.00071469
0.5
0.41218032
0.4184257512
0.4122905765
0.00624543
0.00011025
0.6
0.43730851
0.4441094075
0.4391063274
0.00680090
0.00179781
0.7
0.42288807
0.4289573314
0.4263383569
0.00606926
0.00345029
0.8
0.35608655
0.3601493430
0.3598002408
0.00406279
0.00371369
0.9
0.22136428
0.2228338692
0.2231385861
0.00146959
0.00177431
1.0
0
0
0
0
0Table 7
Numerical results of Example2.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating  collocation algorithm
Approximate solutionZ
2 by approximating  collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
0.008951884
0.006713642
0.010510643
0.002238243
0.001558758
0.2
0.0312679106
0.023450361
0.035942526
0.007817550
0.004674615
0.3
0.059528773
0.044645810
0.067145581
0.0148829636
0.007616808
0.4
0.085929102
0.064445941
0.094969740
0.021483162
0.009040637
0.5
0.103045079
0.077282873
0.110264933
0.025762206
0.007219853
0.6
0.104954043
0.078714604
0.107216515
0.026239439
0.002262472
0.7
0.088806494
0.066604028
0.086571523
0.022202467
0.002234972
0.8
0.056973848
0.042729735
0.056424256
0.014244112
0.000549602
0.9
0.019922785
0.014941769
0.024868975
0.004981016
0.0049461810
1.0
0
0
0
0
0Table 8
Numerical results of Example3.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating  collocation algorithm
Approximate solutionZ
2 by approximating  collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
-
0.09883508
-
0.10018369
-
0.098873723
0.0013486
0.000038641
0.2
-
0.19072256
-
0.19517091
-
0.1910766633
0.0044484
0.00035411
0.3
-
0.26892339
-
0.27689752
-
0.2699814765
0.0079741
0.001058
0.4
-
0.32711141
-
0.33791938
-
0.328960819
0.010808
0.0018494
0.5
-
0.35956915
-
0.37170655
-
0.361387346
0.012137
0.0018182
0.6
-
0.36137118
-
0.37291874
-
0.3620271786
0.011548
0.00065600
0.7
-
0.32855102
-
0.33765587
-
0.3276693903
0.0091048
0.00088163
0.8
-
0.25824819
-
0.26367852
-
0.2568529513
0.0054303
0.0013952
0.9
-
0.14883211
-
0.15059347
-
0.1481168313
0.0017614
0.00071528
1.0
0
0
0
0
0Table 9
Maximum absolute errors of Examples1, 2, and 3.
Example
Max. absolute errors by interpolating  collocation algorithm
Max. absolute errors by approximating collocation algorithm
1
6.8010
×
1
0
-
3
3.7137
×
1
0
-
3
2
2.6239
×
1
0
-
2
0.9041
×
1
0
-
2
3
1.2137
×
1
0
-
2
0.18494
×
1
0
-
2Table 10
Comparison of Example1 with different methods.
h
Max. absolute errors by interpolating  collocation algorithm
Max. absolute errors by approximating collocation algorithm
Second order finite difference method [15]
1
4
⋯
⋯
8.50
×
1
0
-
2
1
8
⋯
1.4098
×
1
0
-
2
2.09
×
1
0
-
2
1
10
6.8010
×
1
0
-
3
3.7137
×
1
0
-
3
⋯
1
16
6.7469
×
1
0
3
4.4572
×
1
0
-
3
5.27
×
1
0
-
3Figure 1
Comparison between exact and approximate solutions of Example1 obtained by interpolating and approximating collocation algorithms.Figure 2
Comparison between exact and approximate solutions of Example2 obtained by interpolating and approximating collocation algorithms.Figure 3
Comparison between exact and approximate solutions of Example3 obtained by interpolating and approximating collocation algorithms.


## 4.1. Numerical Examples

Here we find the numerical solutions of some of the boundary value problems arising in the mathematical modeling of viscoelastic and inelastic flows and so forth.Example 1.
Consider the fourth order linear boundary value problem(90)
y
i
v
x
+
x
y
=
-
8
+
7
x
+
x
3
e
x
,
0
<
x
<
1
,subject to the boundary conditions(91)
y
0
=
y
1
=
0
,
y
′
0
=
1
,
y
′
1
=
-
e
.By comparing the above problem with (3), we have a
(
x
)
=
-
x and b
(
x
)
=
-
(
8
+
7
x
+
x
3
)
e
x. The exact solution for the above problem is y
=
x
(
1
-
x
)
e
x
.
Here we present the numerical solution of the above problem by interpolating and approximating collocation algorithms.
Solution by Interpolating Collocation Algorithm. In this method, by solving the system of linear equations (73) at N
=
10, we obtain the approximate solution (17) of (90) where {
z
i
}, -
8
≤
i
≤
18, are(92)
-
0.736798818
,
-
0.643536669
,
-
0.554623724
,
-
0.467529093
,
-
0.379799398
,
-
0.289680649
,
-
0.196215973
,
-
0.099344276
,
0.0000
,
0.0997855
,
0.196780435
,
0.286445566
,
0.362823930
,
0.4184258296
,
0.444109480
,
0.428957387
,
0.360149377
,
0.222833881
,
0.000
,
-
0.327646133
,
-
0.781795936
,
-
1.386637856
,
-
2.168934169
,
-
3.158065904
,
-
4.386034900
,
-
5.887410047
,
-
7.698656102
.Solution by Approximating Collocation Algorithm. In this method, we solve the system of linear equations (82) at N
=
10 and get solution (23) of (90) where z
i, -
6
≤
i
≤
16, are(93)
240576495.97007
,
346838.29496
,
518.9347067
,
-
0.27530396
,
-
0.192116155
,
-
0.098802082
,
0.0000
,
0.099651832
,
0.195515154
,
0.282951708
,
0.357323235
,
0.412290577
,
0.439106327
,
0.426338357
,
0.359800241
,
0.223138586
,
0
,
-
0.325968911
,
-
0.771121538
,
-
1.351811276
,
759.3687002
,
508693.855464
,
352843031.1188856
.Example 2.
Consider the following fourth order linear boundary value problem,(94)
y
i
v
x
=
x
4
+
14
x
3
+
49
x
2
+
32
x
-
12
e
x
0
≤
x
≤
1with(95)
y
0
=
y
1
=
y
′
0
=
y
′
1
,corresponds to the bending of a thin beam clamped at both ends. The unique solution of (94) is (96)
y
x
=
x
2
1
-
x
2
e
x
.Solution by Interpolating Collocation Algorithm. By using this method, we solve system (73) at N
=
10 and get solution (17) of (94) where {
z
i
}, -
8
≤
i
≤
18, are (97)
-
1.292188418
,
0.538942241
,
0.376822099
,
0.254979738
,
0.157378182
,
0.084436111
,
0.035355124
,
0.008209932
,
0
,
0.006713642
,
0.023450361
,
0.044645810
,
0.064445941
,
0.077282873
,
0.078714604
,
0.066604028
,
0.042729735
,
0.014941769
,
0
,
0.027260994
,
0.143411251
,
0.418480429
,
0.953407669
,
1.889477424
,
3.419988190
,
5.826254985
,
6.993262174
.Solution by Approximating Collocation Algorithm. In this method, by solving system of linear equations (82) at N
=
10, we obtain the approximate solution (23) of (94) where {
z
i
}, -
6
≤
i
≤
16, are(98)
1.711850528
×
1
0
11
,
-
2.46996092
×
1
0
8
,
3.5607957013
×
1
0
5
,
-
5.335463543
×
1
0
5
,
0.06034270857
,
0.013560666
,
0
,
0.01051064293055
7
,
0.035942526106682
,
0.067145581105052
,
0.094969739502347
,
0.110264932875244
,
0.107216514892050
,
0.086571522615334
,
0.056424246024606
,
0.024868975122817
,
0
,
0.01008838960215368
,
0.002698096580563341
,
-
3.898592698
×
1
0
3
,
2.597091086
×
1
0
6
,
-
1.801597056
×
1
0
9
,
1.2486272867
×
1
0
12
.Example 3.
Consider the boundary value problem(99)
y
i
v
-
y
=
-
4
2
x
cos
⁡
x
+
3
sin
⁡
xwith boundary conditions(100)
y
0
=
0
,
y
1
=
0
,
y
′
0
=
-
1
,
y
′
1
=
2
sin
⁡
1
.The exact solution of this problem is y
=
(
x
2
-
1
)
s
i
n
(
x
).
Solution by Interpolating Collocation Algorithm. Here, we solve system (73) at N
=
10 and get solution (17) of (99) where {
z
i
}, -
8
≤
i
≤
18, are(101)
0.0600152940
,
0.18160335240
,
0.25833493990
,
0.29216208360
,
0.2870204837
,
0.1824477485
,
0.09700293
,
0.000000000
,
-
0.1001836945
,
-
0.1951709144
,
-
0.2768975252
,
-
0.3379193763
,
-
0.3717065517
,
-
0.372918743
,
-
0.3376558661
,
-
0.26367852
,
-
0.1505934691
,
0.000000000
,
0.1844062122
,
0.3967742453
,
0.6290856272
,
0.87118909250
,
1.11088175850
,
1.33403484280
,
1.52476071733
,
1.6661398769
.Solution by Approximating Collocation Algorithm. By solving system of linear equations (82) at N
=
10, we obtain the approximate solution (23) of (99) where {
z
i
}, -
6
≤
i
≤
16, are(102)
115585876195.8271
,
166975210.2723412
,
240725.36249556
,
360.797052040
,
0.1912504195
,
0.0989171622
,
0.000000
,
-
0.0988737231
,
-
0.1910766633
,
-
0.2699814765
,
-
0.328960819
,
-
0.361387346
,
-
0.3620271786
,
-
0.3276693903
,
-
0.2568529513
,
-
0.1481168313
,
0.00000
,
0.1889585727
,
0.4202199172
,
648.2401207
,
432617.012
,
300071307.86
,
207719711144.92
.


## 4.2. Comparison

The numerical results of Examples1, 2, and 3 by interpolating and approximating collocation algorithms are presented in Tables 6, 7, and 8, respectively. The maximum absolute errors in the solution of Examples 1, 2, and 3 obtained by interpolating and approximating collocation algorithms are given in Table 9. Graphical representation of these results is shown in Figures 1, 2, and 3. In these figures solid curve represents the exact solutions, dashed lines represent approximate solutions obtained by (73), and dotted lines represent the approximate solutions obtained by (82). Following is the comparison of the numerical solutions obtained by proposed algorithms and other approaches of this type of boundary value problems:(i)
From the above results we see that approximating schemes based collocation algorithms give better results than interpolating schemes based collocation algorithms.
(ii)
Example1 is also solved by [15]. He solved this problem by second order finite difference method and obtained the maximum absolute errors at different step sizes h. We observe that the maximum absolute error at the step size h
=
1
/
10 by the proposed approximating collocation algorithm is better than the maximum absolute error obtained by [15] at step size h
=
1
/
16. The comparison of proposed methods with second order finite difference method of [15] at difference step sizes is shown in Table 10.
(iii)
Example2 is also solved by [18] by quintic spline based collocation methods. We observe that order of error approximation obtained by [18] and proposed approximating collocation algorithm is the same.Table 6
Numerical results of Example1.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating    collocation algorithm
Approximate solutionZ
2 by approximating    collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
0.09946538
0.0997855152
0.0996518317
0.00032013
0.00018645
0.2
0.19542444
0.1967804015
0.1955151540
0.00135596
0.00009071
0.3
0.28347035
0.2864455108
0.2829517080
0.00297516
0.00051864
0.4
0.35803793
0.3628238578
0.3573232345
0.00478593
0.00071469
0.5
0.41218032
0.4184257512
0.4122905765
0.00624543
0.00011025
0.6
0.43730851
0.4441094075
0.4391063274
0.00680090
0.00179781
0.7
0.42288807
0.4289573314
0.4263383569
0.00606926
0.00345029
0.8
0.35608655
0.3601493430
0.3598002408
0.00406279
0.00371369
0.9
0.22136428
0.2228338692
0.2231385861
0.00146959
0.00177431
1.0
0
0
0
0
0Table 7
Numerical results of Example2.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating  collocation algorithm
Approximate solutionZ
2 by approximating  collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
0.008951884
0.006713642
0.010510643
0.002238243
0.001558758
0.2
0.0312679106
0.023450361
0.035942526
0.007817550
0.004674615
0.3
0.059528773
0.044645810
0.067145581
0.0148829636
0.007616808
0.4
0.085929102
0.064445941
0.094969740
0.021483162
0.009040637
0.5
0.103045079
0.077282873
0.110264933
0.025762206
0.007219853
0.6
0.104954043
0.078714604
0.107216515
0.026239439
0.002262472
0.7
0.088806494
0.066604028
0.086571523
0.022202467
0.002234972
0.8
0.056973848
0.042729735
0.056424256
0.014244112
0.000549602
0.9
0.019922785
0.014941769
0.024868975
0.004981016
0.0049461810
1.0
0
0
0
0
0Table 8
Numerical results of Example3.
x
i
Analytic solutionY
Approximate solutionZ
1 by interpolating  collocation algorithm
Approximate solutionZ
2 by approximating  collocation algorithm
er
r
1
x
i
∞
er
r
2
x
i
∞
0.0
0
0
0
0
0
0.1
-
0.09883508
-
0.10018369
-
0.098873723
0.0013486
0.000038641
0.2
-
0.19072256
-
0.19517091
-
0.1910766633
0.0044484
0.00035411
0.3
-
0.26892339
-
0.27689752
-
0.2699814765
0.0079741
0.001058
0.4
-
0.32711141
-
0.33791938
-
0.328960819
0.010808
0.0018494
0.5
-
0.35956915
-
0.37170655
-
0.361387346
0.012137
0.0018182
0.6
-
0.36137118
-
0.37291874
-
0.3620271786
0.011548
0.00065600
0.7
-
0.32855102
-
0.33765587
-
0.3276693903
0.0091048
0.00088163
0.8
-
0.25824819
-
0.26367852
-
0.2568529513
0.0054303
0.0013952
0.9
-
0.14883211
-
0.15059347
-
0.1481168313
0.0017614
0.00071528
1.0
0
0
0
0
0Table 9
Maximum absolute errors of Examples1, 2, and 3.
Example
Max. absolute errors by interpolating  collocation algorithm
Max. absolute errors by approximating collocation algorithm
1
6.8010
×
1
0
-
3
3.7137
×
1
0
-
3
2
2.6239
×
1
0
-
2
0.9041
×
1
0
-
2
3
1.2137
×
1
0
-
2
0.18494
×
1
0
-
2Table 10
Comparison of Example1 with different methods.
h
Max. absolute errors by interpolating  collocation algorithm
Max. absolute errors by approximating collocation algorithm
Second order finite difference method [15]
1
4
⋯
⋯
8.50
×
1
0
-
2
1
8
⋯
1.4098
×
1
0
-
2
2.09
×
1
0
-
2
1
10
6.8010
×
1
0
-
3
3.7137
×
1
0
-
3
⋯
1
16
6.7469
×
1
0
3
4.4572
×
1
0
-
3
5.27
×
1
0
-
3Figure 1
Comparison between exact and approximate solutions of Example1 obtained by interpolating and approximating collocation algorithms.Figure 2
Comparison between exact and approximate solutions of Example2 obtained by interpolating and approximating collocation algorithms.Figure 3
Comparison between exact and approximate solutions of Example3 obtained by interpolating and approximating collocation algorithms.


## 5. Conclusion

In this paper, we have presented interpolating and approximating collocation algorithms based on interpolating and approximating subdivision schemes for the solution of linear fourth order boundary value problems. The proposed algorithms have been applied on different linear fourth order boundary value problems. Results show that the approximating collocation algorithm gives better results comparative to interpolating collocation algorithm. We have also observed that the accuracy of the solution can be improved by choosing different subdivision schemes with the proper adjustment of boundary conditions.Approximating subdivision scheme based collocation algorithm gives better results comparative to second order finite difference method. However, approximating subdivision scheme based collocation algorithm and quintic spline based collocation algorithm have the same order of approximation.



---

*Source: 240138-2015-07-14.xml*