# A Computational Study of the Boundary Value Methods and the Block Unification Methods fory

Authors: T. A. Biala
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

We derive a new class of linear multistep methods (LMMs) via the interpolation and collocation technique. We discuss the use of these methods as boundary value methods and block unification methods for the numerical approximation of the general second-order initial and boundary value problems. The convergence of these families of methods is also established. Several test problems are given to show a computational comparison of these methods in terms of accuracy and the computational efficiency.


---

## Body

## 1. Introduction

Linear multistep methods (LMMs) are widely used for the numerical integration of ordinary differential equations. They are a class ofk-step difference equations of the form(1)∑r=0kαryn+r=hμ∑r=0kβrfn+r,where α
r, β
r are coefficients to be uniquely determined, μ is the order of the differential equation whose solution is being sought, h is the constant stepsize, y
n
+
i
≡
y
(
x
n
+
i
), and f
n
+
i
≡
f
(
x
n
+
i
,
y
n
+
i
,
y
n
+
i
′
,
…
,
y
n
+
i
(
μ
-
1
)
).To be able to use (1), we need to impose k additional conditions. Initial value methods (IVMs) are methods whose additional conditions are specified as initial conditions so that they form discrete initial value problems. The IVMs are used for the numerical integration of initial value problems [1–4]. However, if these additional conditions are specified as initial and final conditions (or methods) so that they form a discrete analog of the continuous boundary value problems, we have the boundary value methods (BVMs). They are used for the approximation of both initial and boundary value problems [5–11]. The BVMs are a larger class of methods that contains the IVMs since the IVMs are BVMs with zero final conditions. Sometimes the additional conditions are given as a set of LMMs which together with the main method (1) forms the block methods. If the union of the methods in the block is obtained for n
=
0
(
k
)
(
N
-
k
), where N is the number of grid points, so that we have N difference equations in N unknowns (grid values) which can be easily solved, the resulting approach is termed the block unification methods (BUMs) [12]. The union of the methods in the block is taken to have a consistent equation.In what follows, we will consider the general second-order system of the form(2)y′′=fx,y,y′,x∈a,b,coupled with any of the initial or boundary conditions(3)ya=y0y′a=y0′,ya=y0yb=yN,y′a=y0′yb=yN,ya=y0y′b=yN′,y′a=y0′y′b=yN′,where f
:
R
m
→
R
m are continuous functions. The existence and uniqueness of the solutions of (2) subject to any of (3) have been given in Wend [13] or Ascher et al. [14].The BVMs have been used for the numerical integration of first-order initial and boundary value problems and their convergence and stability analysis have been fully discussed [5–10]. Aceto et al. [15] constructed P-stable LMMs which were used as BVMs for the special second-order problem y
′
′
=
f
(
x
,
y
). Recently, Biala and Jator [11] developed BVMs for the direct solution of the general second-order initial and boundary value problems arising from the semidiscretization of three-dimensional partial differential equations.The BUMs have also been successfully applied to solve initial and boundary value problems [12]. In this paper, we construct a new class of LMMs which we implement as boundary value methods and block unification methods. We compare the results of these two classes of methods in terms of accuracy and CPU time.The outline of the paper is as follows: In Section2, we derive a (
2
v
)-step continuous LMM (CLMM) via the interpolation and collocation technique [2–4, 11, 12]. In Section 3, we construct the BVMs by using the CLMM to derive a (
2
v
)-step discrete LMM which is to be used with some initial and final methods (also obtained from the CLMM). We also discuss the convergence and the use of the BVMs in this section. Section 4 details the BUMs. Their convergence analysis is carried out and an algorithm for their implementation is also discussed. In Section 5, we give several numerical test problems which were solved using both the BVMs and the BUMs. Their comparison in terms of accuracy and computational efficiency (CPU Time) was also shown. Finally, we give some concluding remarks on the methods in Section 6.


## 2. Derivation of the CLMM

In this section, we will construct a2
v-step CLMM using the interpolation and collocation technique. The CLMM will be used to generate the BVMs and the BUMs.We begin by constructing the CLMM of the form(4)Ux=αvxyn+v+αv-1xyn+v-1+α0xyn+h2∑r=02vβrxfn+r,where α
0
(
x
), α
v
-
1
(
x
), α
v
(
x
), and β
r
(
x
) are continuous coefficients. The next theorem discusses the construction of the CLMM.Theorem 1.
Let (4) satisfy the following equations:(5)Uxn+j=yn+jj=0,v-1,v,U′′xn+j=fn+jj=012v;then the continuous representation (4) is equivalent to(6)Ux=∑j=02v+3det⁡Wjdet⁡WPjx,where P
j
(
x
)
=
x
j; j
=
0
(
1
)
(
2
v
+
3
) are basis functions and the matrix W is defined as follows:(7)W=P0xnP1xn⋯P2v+3xnP0xn+v-1P1xn+v-1⋯P2v+3xn+v-1P0xn+vP1xn+v⋯P2v+3xn+vP0′′xnP1′′xn⋯P2v+3′′xnP0′′xn+1P1′′xn+1⋯P2v+3′′xn+1⋮⋮⋮⋮P0′′xn+2vP1′′xn+2v⋯P2v+3′′xn+2v;
W
j is obtained by replacing the jth column of W by V, where(8)V=yn,yn+v-1,yn+v,fn,fn+1,…,fn+2vT,where T denotes the transpose.Proof.
We begin the proof by assuming polynomial basis functions of the form(9)αjx=∑i=02v+3αi+1,jPix,j=0,v-1,v,h2βjx=∑i=02v+3h2βi+1,jPix,j=012v,where α
i
+
1
,
j, h
2
β
i
+
1
,
j are coefficients to be determined.
By substituting (9) into (4), we have(10)Ux=∑i=02v+3αi+1,0Pixyn+∑i=02v+3αi+1,v-1Pixyn+v-1+∑i=02v+3αi+1,vPixyn+v+∑j=02v∑i=02v+3h2βi+1,jPixfn+jwhich is simplified to(11)
U
x
=
∑
i
=
0
2
v
+
3
α
i
+
1,0
y
n
+
α
i
+
1
,
v
-
1
y
n
+
v
-
1
+
α
i
+
1
,
v
y
n
+
v
+
∑
j
=
0
2
v
h
2
β
i
+
1
,
j
f
n
+
j
P
i
xand expressed in the form(12)Ux=∑i=02v+3liPix,where(13)li=αi+1,0yn+αi+1,v-1yn+v-1+αi+1,vyn+v+∑j=02vh2βi+1,jfn+j.
Imposing conditions (5) on (12), we obtain a system of (
2
v
+
4
) equations which can be expressed as W
L
=
V, where L
=
(
l
0
,
l
1
,
…
,
l
2
v
+
3
)
T is a vector of (
2
v
+
4
) undetermined coefficients.
Using Cramer’s rule, the elements ofL are determined and given as(14)li=det⁡Wjdet⁡W,j=012v+3,where W
j is obtained by replacing the jth column of W by V. We rewrite (12) as (6) using the newly found elements of L.Remark 2.
It has been shown in [11] that symmetric schemes are the best candidates to be used as final methods. Thus, CLMM (4) is chosen to ensure that we have discrete symmteric schemes by evaluation at some points x
n
+
j.


## 3. The Boundary Value Methods

CLMM (4) is evaluated at x
n
+
j, j
=
1
(
1
)
(
2
v
), j
≠
v
-
1
,
v, to obtain the BVMs. The main method of the BVM, that is, U
(
x
n
+
2
v
), is of the form(15)yn+2v+αvyn+v+αv-1yn+v-1+α0yn=h2∑r=02vβrfn+rwhose derivative formula, obtained by evaluating U
′
x at x
n
+
2
v, is(16)hyn+2v′+αv′yn+v+αv-1′yn+v-1+α0′yn=h2∑r=02vβr′fn+r.


### 3.1. Convergence of the BVMs

In this section, we will discuss the convergence of the BVMs. We emphasize that (4) is evaluated at x
n
+
j, j
=
1
(
1
)
(
2
v
), j
≠
v
-
1
,
v, to obtain(17)yn+1+αv-11yn+v-1+αv1yn+v+α01yn=h2∑i=02vβi1fn+iyn+2+αv-12yn+v-1+αv2yn+v+α02yn=h2∑i=02vβi2fn+i⋮yn+v-2+αv-1v-2yn+v-1+αvv-2yn+v+α0v-2yn=h2∑i=02vβiv-1fn+iyn+v+1+αv-1v+1yn+v-1+αvv+1yn+v+α0v+1yn=h2∑i=02vβiv+1fn+i⋮yn+2v+αv-12vyn+v-1+αv2vyn+v+α02vyn=h2∑i=02vβi2vfn+iand also, by evaluating U
′
(
x
) at x
n
+
i, i
=
0
(
1
)
(
2
v
), we obtain the derivative formulas(18)hyn′+αv-1′0yn+v-1+αv′0yn+v+α0′0yn=h2∑i=02vβi′0fn+ihyn+1′+αv-1′1yn+v-1+αv′1yn+v+α0′1yn=h2∑i=02vβi′1fn+i⋮hyn+2v′+αv-1′2vyn+v-1+αv′2vyn+v+α0′2vyn=h2∑i=02vβi′2vfn+i.We note that the formulas in (17) and (18) are of O
(
h
2
v
+
4
). We establish the convergence of the BVMs in the following theorem.Theorem 3.
LetY be an approximation of the solution vector Y
¯ for the system obtained on a partition π
N
≔
{
a
=
x
0
<
x
1
<
⋯
<
x
N
=
b
,
x
n
=
x
n
-
1
+
h
} from methods (17) and (18). If e
n
=
y
x
n
-
y
n, h
e
n
′
=
h
y
′
x
n
-
h
y
n
′, where the exact solution y
(
x
) is several times differentiable on [
a
,
b
], and if E
=
Y
-
Y
¯, then the BVM is convergent and of order 2
v
+
2, which implies that E
=
O
(
h
2
v
+
2
).Proof.
We compactly write (17) and (18) in matrix form by introducing the following matrix notations. Let A be a 2
N
×
2
N matrix defined by(19)A=A11A12A21A22,where A
i
j are N
×
N matrices given as(20)A11=αv-1′0αv′0α0′01αv-11αv1α01⋮⋮⋮⋮1αv-1v-2αvv-2α0v-21αv-1v+1αvv+1α0v+1⋮⋮⋮⋮1αv-12vαv2vα02v⋱⋱⋱⋱⋱⋱1αv-12vαv2v⋯α02v,A21=αv-1′1αv′1α0′1αv-1′2αv′2α0′2⋮⋮⋮αv-1′2vαv′2vα0′2vαv-1′2vαv′2v⋯α0′2v⋱⋱⋱αv-1′2vαv′2vα0′2v,
A
12 is an N
×
N null matrix, and A
22 is an N
×
N identity matrix.
Similarly, letB be a 2
N
×
2
N matrix defined by(21)B=B11B12B21B22,where B
i
j are N
×
N matrices given as(22)B11=h2β1′0β2′0⋯β2v′0β11β21⋯β2v1⋮⋮⋮⋮β1v-1β2v-1⋯β2vv-1β1v+1β2v+1⋯β2vv+1⋮⋮⋮⋮β12vβ22v⋯β2v2vβ0′0β1′0⋯β2v′0β01β1′1⋯β2v′1⋮⋮⋮⋮β0v-1β1v-1⋯β2vv-1β0v+1β1v+1⋯β2vv+1⋮⋮⋮⋮β02vβ12v⋯β2v2v⋱⋱⋱⋱⋱β02vβ12v⋯β2v2v,B21=h2β1′1β2′1⋯β2v′1⋮⋮⋮⋮β1′2vβ2′2v⋯β2v′2vβ0′1β1′1⋯β2v′1⋮⋮⋮⋮β0′1β1′1⋯β2v′1⋱⋱⋱β0′1β1′1⋯β2v′1and B
12, B
22 are N
×
N null matrices.
We also define the following vectors:(23)
Y
¯
=
y
1
,
…
,
y
N
,
h
y
1
′
,
…
,
h
y
N
′
T
,
Y
=
y
x
1
,
…
,
y
x
N
,
h
y
′
x
1
,
…
,
h
y
′
x
N
T
,
F
=
f
1
,
…
,
f
N
,
h
f
1
′
,
…
,
h
f
N
′
T
,
L
h
=
l
1
,
…
,
l
n
,
l
1
′
,
…
,
l
N
′
T
,
C
=
β
0
′
0
h
2
f
0
-
h
y
0
′
,
β
0
0
h
2
f
0
-
y
0
,
β
0
1
h
2
f
0
,
…
,
β
v
-
1
h
2
f
0
,
β
0
v
+
1
h
2
f
0
,
…
,
β
0
2
v
h
2
f
0
,
0
,
…
,
0
,
β
0
′
0
h
2
f
0
,
…
,
β
0
′
2
v
h
2
f
0
,
0
,
…
,
0
T
.The exact form of the system formed by (17) and (18) is given by(24)AY-BFY+C+Lh=0,where L
(
h
) is the truncation error vector of the formulas in (17) and (18). The approximate form of the system is given by(25)AY¯-BFY¯+C=0,where Y
¯ is the approximate solution of vector Y.
Subtracting (24) from (25) and letting E
=
Y
¯
-
Y
=
(
e
1
,
…
,
e
N
,
e
1
′
,
…
,
e
N
′
)
T and using the mean value theorem, we have the error system(26)A-BJE=Lh,where J is the Jacobian matrix and its entries J
11, J
12, J
21, and J
22 are defined as(27)J11=∂f1∂y1⋯∂f1∂yN⋮⋮⋮∂fN∂y1⋯∂fN∂yN,J12=∂f1∂y1′⋯∂f1∂yN′⋮⋮⋮∂fN∂y1′⋯∂fN∂yN′,J21=h∂f1′∂y1⋯∂f1′∂yN⋮⋮⋮∂fN′∂y1⋯∂fN′∂yN,J22=h∂f1′∂y1′⋯∂f1′∂yN′⋮⋮⋮∂fN′∂y1′⋯∂fN′∂yN′.Let N
=
-
B
J be a matrix of dimension 2
N so that (26) becomes(28)A+NE=Lh,and, for sufficiently small h, A
+
N is a monotone matrix and thus nonsingular (see [16]). Therefore(29)A+N-1=D=dij≥0,∑j=12Ndij=Oh-2,E=DLh,E=DLh=Oh-2Oh2v+4=Oh2v+2,which shows that the methods are convergent and the global error is of order O
(
h
2
v
+
2
).


### 3.2. Use of Methods

The BVMs can only be successfully implemented if used together with appropriate additional methods [5]. In this regard, we have proposed a main method and additional methods which are obtained from the same continuous scheme (the CLMM).To use LMM (15) as BVMs, we rewrite main method (15) as(30)yn+v-1+αvyn-1+αv-1yn-2+α0yn-v-1=h2∑i=-v-1v-1βi+v+1fn+i,n=v+1,…,N-v+1,with the derivative formula(31)hyn+v-1′+αv′yn-1+αv-1′yn-2+α0′yn-v-1=h2∑i=-v-1v-1βi+v+1′fn+i,n=v+1,…,N-v+1,which are to be used with some boundary conditions and U
′
(
x
r
), r
=
0
(
1
)
(
2
v
-
1
).The discrete solutions(32)y0,…,yv,yN-v+2,…,yNy0′,…,yv′,yN-v+2′,…,yN′are to be obtained for methods (30) and (31) to be useful. However, (3) provides two solution values so that we have to impose 4
v
-
2 additional conditions of which U
′
(
x
r
), r
=
0
(
1
)
(
2
v
-
1
), gives 2
v methods and the remaining 2
v
-
2 methods are given as a set of v
-
1 initial and v
-
1 final methods which is readily obtained from CLMM (4). Approximations (32) need be at least of order O
(
h
p
+
1
) accuracy if BVM (30) is of order p in order to have a solution accuracy of O
(
h
p
+
1
). Equations (30) and (31) with the 4
v
-
2 additional methods give a set of 2
N equations in 2
N unknowns which can be easily solved. We give below the BVMs of orders 6 and 8.BVM of Order 6
(
v
=
2
)
(33)
y
n
+
1
-
2
y
n
-
1
+
y
n
-
3
=
h
2
15
f
n
-
3
+
16
f
n
-
2
+
26
f
n
-
1
+
16
f
n
+
f
n
+
1
,
n
=
3
,
…
,
N
-
1
,with the derivative formulas(34)
h
y
n
+
1
′
=
-
107
42
y
n
-
1
+
128
21
y
n
-
2
-
149
42
y
n
-
3
+
h
2
1260
325
f
n
-
3
+
4048
f
n
-
2
+
1106
f
n
-
1
+
1744
f
n
+
397
f
n
+
1
,
h
y
0
′
=
-
107
42
y
2
+
128
21
y
1
-
149
42
y
0
+
h
2
1260
-
67
f
0
+
2256
f
1
+
434
f
2
-
48
f
3
+
5
f
4
,
h
y
1
′
=
41
21
y
2
-
61
21
y
1
+
20
21
y
0
+
h
2
10080
-
613
f
0
+
11464
f
1
-
2870
f
2
+
344
f
3
-
37
f
4
,
h
y
2
′
=
5
42
y
2
+
16
21
y
1
-
37
42
y
0
+
h
2
1260
73
f
0
+
1136
f
1
+
574
f
2
-
48
f
3
+
5
f
4
,
h
y
3
′
=
41
21
y
2
-
61
21
y
1
+
20
21
y
0
+
h
2
10080
-
725
f
0
-
7656
f
1
+
9898
f
2
+
410
f
3
-
149
f
4which are to be used with the initial method (35)y3-2y2+y1=h2240-f0+24f1+194f2+24f3-f4and the final method(36)
y
N
-
3
-
2
y
N
-
2
+
y
N
-
1
=
h
2
240
-
f
N
+
24
f
N
-
1
+
194
f
N
-
2
+
24
f
N
-
3
-
f
N
-
4
.BVM of Order 8
(
v
=
3
)
(37)
y
n
+
2
-
2
y
n
-
1
+
y
n
-
4
=
h
2
2240
141
f
n
-
4
+
2430
f
n
-
3
+
4131
f
n
-
2
+
6756
f
n
-
1
+
4131
f
n
+
2430
f
n
+
1
+
141
f
n
+
2
,
n
=
4
,
…
,
N
-
2
,with the derivative formulas(38)
h
y
n
+
2
′
=
-
233
30
y
n
-
1
+
243
20
y
n
-
2
-
263
60
y
n
-
3
+
h
2
44800
128089
f
n
-
4
+
216774
f
n
-
3
+
305847
f
n
-
2
+
122452
f
n
-
1
+
6327
f
n
+
69318
f
n
+
1
+
13113
f
n
+
2
,
h
y
0
′
=
-
233
30
y
3
+
243
20
y
2
-
263
60
y
0
+
h
2
44800
-
1031
f
0
+
147654
f
1
+
297207
f
2
+
35412
f
3
-
2313
f
4
+
198
f
5
-
7
f
6
,
h
y
1
′
=
5713
1920
y
3
-
5073
1280
y
2
+
3793
3840
y
0
+
h
2
77414400
-
3977083
f
0
-
113321
f
1
-
208639509
f
2
-
20080444
f
3
+
389931
f
4
+
130254
f
5
-
23611
f
6
,
h
y
2
′
=
7
30
y
3
+
3
20
y
2
-
23
60
y
0
+
h
2
1209600
26803
f
0
+
534018
f
1
+
323229
f
2
-
123236
f
3
+
31389
f
4
+
6654
f
5
+
691
f
6
,
h
y
3
′
=
3313
1920
y
3
-
2673
1280
y
2
+
1292
3840
y
0
+
h
2
2867200
-
60609
f
0
-
1189494
f
1
-
1182447
f
2
+
820748
f
3
-
90927
f
4
+
17802
f
5
-
1793
f
6
,
h
y
4
′
=
7
30
y
3
+
3
20
y
2
-
23
60
y
0
+
h
2
1209600
28403
f
0
+
510978
f
1
+
804189
f
2
+
1376924
f
3
+
512349
f
4
-
29694
f
5
+
2291
f
6
,
h
y
5
′
=
5713
19200
y
3
-
5073
1280
y
2
+
3793
3840
y
0
+
h
2
77414400
-
4632443
f
0
-
85305138
f
1
-
108369429
f
2
+
34314436
f
3
-
100660011
f
4
+
28146894
f
5
-
678971
f
6which are to be used with the initial methods(39)
y
1
+
95
64
y
3
-
349
128
y
2
+
31
128
y
0
=
h
2
77414400
91775
f
0
+
2787594
f
1
+
9305553
f
2
+
127768
f
3
+
109167
f
4
+
13578
f
5
-
961
f
6
,
y
4
-
2
y
3
+
y
2
=
h
2
60480
31
f
0
-
438
f
1
+
6513
f
2
+
48268
f
3
+
6513
f
4
-
438
f
5
+
31
f
6and the final methods(40)
y
N
-
4
-
2
y
N
-
3
+
y
N
-
2
=
h
2
60480
31
f
N
-
438
f
N
-
1
+
6513
f
N
-
2
+
48268
f
N
-
3
+
6513
f
N
-
4
-
438
f
N
-
5
+
31
f
N
-
6
,
y
N
-
5
-
223
64
y
N
-
3
+
349
128
y
N
-
2
-
31
128
y
N
-
2
=
h
2
25804800
-
f
N
-
724398
f
N
-
1
-
431259
f
N
-
2
-
4156156
f
N
-
3
+
2706981
f
N
-
4
+
200274
f
N
-
5
-
5141
f
N
-
6
.


## 3.1. Convergence of the BVMs

In this section, we will discuss the convergence of the BVMs. We emphasize that (4) is evaluated at x
n
+
j, j
=
1
(
1
)
(
2
v
), j
≠
v
-
1
,
v, to obtain(17)yn+1+αv-11yn+v-1+αv1yn+v+α01yn=h2∑i=02vβi1fn+iyn+2+αv-12yn+v-1+αv2yn+v+α02yn=h2∑i=02vβi2fn+i⋮yn+v-2+αv-1v-2yn+v-1+αvv-2yn+v+α0v-2yn=h2∑i=02vβiv-1fn+iyn+v+1+αv-1v+1yn+v-1+αvv+1yn+v+α0v+1yn=h2∑i=02vβiv+1fn+i⋮yn+2v+αv-12vyn+v-1+αv2vyn+v+α02vyn=h2∑i=02vβi2vfn+iand also, by evaluating U
′
(
x
) at x
n
+
i, i
=
0
(
1
)
(
2
v
), we obtain the derivative formulas(18)hyn′+αv-1′0yn+v-1+αv′0yn+v+α0′0yn=h2∑i=02vβi′0fn+ihyn+1′+αv-1′1yn+v-1+αv′1yn+v+α0′1yn=h2∑i=02vβi′1fn+i⋮hyn+2v′+αv-1′2vyn+v-1+αv′2vyn+v+α0′2vyn=h2∑i=02vβi′2vfn+i.We note that the formulas in (17) and (18) are of O
(
h
2
v
+
4
). We establish the convergence of the BVMs in the following theorem.Theorem 3.
LetY be an approximation of the solution vector Y
¯ for the system obtained on a partition π
N
≔
{
a
=
x
0
<
x
1
<
⋯
<
x
N
=
b
,
x
n
=
x
n
-
1
+
h
} from methods (17) and (18). If e
n
=
y
x
n
-
y
n, h
e
n
′
=
h
y
′
x
n
-
h
y
n
′, where the exact solution y
(
x
) is several times differentiable on [
a
,
b
], and if E
=
Y
-
Y
¯, then the BVM is convergent and of order 2
v
+
2, which implies that E
=
O
(
h
2
v
+
2
).Proof.
We compactly write (17) and (18) in matrix form by introducing the following matrix notations. Let A be a 2
N
×
2
N matrix defined by(19)A=A11A12A21A22,where A
i
j are N
×
N matrices given as(20)A11=αv-1′0αv′0α0′01αv-11αv1α01⋮⋮⋮⋮1αv-1v-2αvv-2α0v-21αv-1v+1αvv+1α0v+1⋮⋮⋮⋮1αv-12vαv2vα02v⋱⋱⋱⋱⋱⋱1αv-12vαv2v⋯α02v,A21=αv-1′1αv′1α0′1αv-1′2αv′2α0′2⋮⋮⋮αv-1′2vαv′2vα0′2vαv-1′2vαv′2v⋯α0′2v⋱⋱⋱αv-1′2vαv′2vα0′2v,
A
12 is an N
×
N null matrix, and A
22 is an N
×
N identity matrix.
Similarly, letB be a 2
N
×
2
N matrix defined by(21)B=B11B12B21B22,where B
i
j are N
×
N matrices given as(22)B11=h2β1′0β2′0⋯β2v′0β11β21⋯β2v1⋮⋮⋮⋮β1v-1β2v-1⋯β2vv-1β1v+1β2v+1⋯β2vv+1⋮⋮⋮⋮β12vβ22v⋯β2v2vβ0′0β1′0⋯β2v′0β01β1′1⋯β2v′1⋮⋮⋮⋮β0v-1β1v-1⋯β2vv-1β0v+1β1v+1⋯β2vv+1⋮⋮⋮⋮β02vβ12v⋯β2v2v⋱⋱⋱⋱⋱β02vβ12v⋯β2v2v,B21=h2β1′1β2′1⋯β2v′1⋮⋮⋮⋮β1′2vβ2′2v⋯β2v′2vβ0′1β1′1⋯β2v′1⋮⋮⋮⋮β0′1β1′1⋯β2v′1⋱⋱⋱β0′1β1′1⋯β2v′1and B
12, B
22 are N
×
N null matrices.
We also define the following vectors:(23)
Y
¯
=
y
1
,
…
,
y
N
,
h
y
1
′
,
…
,
h
y
N
′
T
,
Y
=
y
x
1
,
…
,
y
x
N
,
h
y
′
x
1
,
…
,
h
y
′
x
N
T
,
F
=
f
1
,
…
,
f
N
,
h
f
1
′
,
…
,
h
f
N
′
T
,
L
h
=
l
1
,
…
,
l
n
,
l
1
′
,
…
,
l
N
′
T
,
C
=
β
0
′
0
h
2
f
0
-
h
y
0
′
,
β
0
0
h
2
f
0
-
y
0
,
β
0
1
h
2
f
0
,
…
,
β
v
-
1
h
2
f
0
,
β
0
v
+
1
h
2
f
0
,
…
,
β
0
2
v
h
2
f
0
,
0
,
…
,
0
,
β
0
′
0
h
2
f
0
,
…
,
β
0
′
2
v
h
2
f
0
,
0
,
…
,
0
T
.The exact form of the system formed by (17) and (18) is given by(24)AY-BFY+C+Lh=0,where L
(
h
) is the truncation error vector of the formulas in (17) and (18). The approximate form of the system is given by(25)AY¯-BFY¯+C=0,where Y
¯ is the approximate solution of vector Y.
Subtracting (24) from (25) and letting E
=
Y
¯
-
Y
=
(
e
1
,
…
,
e
N
,
e
1
′
,
…
,
e
N
′
)
T and using the mean value theorem, we have the error system(26)A-BJE=Lh,where J is the Jacobian matrix and its entries J
11, J
12, J
21, and J
22 are defined as(27)J11=∂f1∂y1⋯∂f1∂yN⋮⋮⋮∂fN∂y1⋯∂fN∂yN,J12=∂f1∂y1′⋯∂f1∂yN′⋮⋮⋮∂fN∂y1′⋯∂fN∂yN′,J21=h∂f1′∂y1⋯∂f1′∂yN⋮⋮⋮∂fN′∂y1⋯∂fN′∂yN,J22=h∂f1′∂y1′⋯∂f1′∂yN′⋮⋮⋮∂fN′∂y1′⋯∂fN′∂yN′.Let N
=
-
B
J be a matrix of dimension 2
N so that (26) becomes(28)A+NE=Lh,and, for sufficiently small h, A
+
N is a monotone matrix and thus nonsingular (see [16]). Therefore(29)A+N-1=D=dij≥0,∑j=12Ndij=Oh-2,E=DLh,E=DLh=Oh-2Oh2v+4=Oh2v+2,which shows that the methods are convergent and the global error is of order O
(
h
2
v
+
2
).


## 3.2. Use of Methods

The BVMs can only be successfully implemented if used together with appropriate additional methods [5]. In this regard, we have proposed a main method and additional methods which are obtained from the same continuous scheme (the CLMM).To use LMM (15) as BVMs, we rewrite main method (15) as(30)yn+v-1+αvyn-1+αv-1yn-2+α0yn-v-1=h2∑i=-v-1v-1βi+v+1fn+i,n=v+1,…,N-v+1,with the derivative formula(31)hyn+v-1′+αv′yn-1+αv-1′yn-2+α0′yn-v-1=h2∑i=-v-1v-1βi+v+1′fn+i,n=v+1,…,N-v+1,which are to be used with some boundary conditions and U
′
(
x
r
), r
=
0
(
1
)
(
2
v
-
1
).The discrete solutions(32)y0,…,yv,yN-v+2,…,yNy0′,…,yv′,yN-v+2′,…,yN′are to be obtained for methods (30) and (31) to be useful. However, (3) provides two solution values so that we have to impose 4
v
-
2 additional conditions of which U
′
(
x
r
), r
=
0
(
1
)
(
2
v
-
1
), gives 2
v methods and the remaining 2
v
-
2 methods are given as a set of v
-
1 initial and v
-
1 final methods which is readily obtained from CLMM (4). Approximations (32) need be at least of order O
(
h
p
+
1
) accuracy if BVM (30) is of order p in order to have a solution accuracy of O
(
h
p
+
1
). Equations (30) and (31) with the 4
v
-
2 additional methods give a set of 2
N equations in 2
N unknowns which can be easily solved. We give below the BVMs of orders 6 and 8.BVM of Order 6
(
v
=
2
)
(33)
y
n
+
1
-
2
y
n
-
1
+
y
n
-
3
=
h
2
15
f
n
-
3
+
16
f
n
-
2
+
26
f
n
-
1
+
16
f
n
+
f
n
+
1
,
n
=
3
,
…
,
N
-
1
,with the derivative formulas(34)
h
y
n
+
1
′
=
-
107
42
y
n
-
1
+
128
21
y
n
-
2
-
149
42
y
n
-
3
+
h
2
1260
325
f
n
-
3
+
4048
f
n
-
2
+
1106
f
n
-
1
+
1744
f
n
+
397
f
n
+
1
,
h
y
0
′
=
-
107
42
y
2
+
128
21
y
1
-
149
42
y
0
+
h
2
1260
-
67
f
0
+
2256
f
1
+
434
f
2
-
48
f
3
+
5
f
4
,
h
y
1
′
=
41
21
y
2
-
61
21
y
1
+
20
21
y
0
+
h
2
10080
-
613
f
0
+
11464
f
1
-
2870
f
2
+
344
f
3
-
37
f
4
,
h
y
2
′
=
5
42
y
2
+
16
21
y
1
-
37
42
y
0
+
h
2
1260
73
f
0
+
1136
f
1
+
574
f
2
-
48
f
3
+
5
f
4
,
h
y
3
′
=
41
21
y
2
-
61
21
y
1
+
20
21
y
0
+
h
2
10080
-
725
f
0
-
7656
f
1
+
9898
f
2
+
410
f
3
-
149
f
4which are to be used with the initial method (35)y3-2y2+y1=h2240-f0+24f1+194f2+24f3-f4and the final method(36)
y
N
-
3
-
2
y
N
-
2
+
y
N
-
1
=
h
2
240
-
f
N
+
24
f
N
-
1
+
194
f
N
-
2
+
24
f
N
-
3
-
f
N
-
4
.BVM of Order 8
(
v
=
3
)
(37)
y
n
+
2
-
2
y
n
-
1
+
y
n
-
4
=
h
2
2240
141
f
n
-
4
+
2430
f
n
-
3
+
4131
f
n
-
2
+
6756
f
n
-
1
+
4131
f
n
+
2430
f
n
+
1
+
141
f
n
+
2
,
n
=
4
,
…
,
N
-
2
,with the derivative formulas(38)
h
y
n
+
2
′
=
-
233
30
y
n
-
1
+
243
20
y
n
-
2
-
263
60
y
n
-
3
+
h
2
44800
128089
f
n
-
4
+
216774
f
n
-
3
+
305847
f
n
-
2
+
122452
f
n
-
1
+
6327
f
n
+
69318
f
n
+
1
+
13113
f
n
+
2
,
h
y
0
′
=
-
233
30
y
3
+
243
20
y
2
-
263
60
y
0
+
h
2
44800
-
1031
f
0
+
147654
f
1
+
297207
f
2
+
35412
f
3
-
2313
f
4
+
198
f
5
-
7
f
6
,
h
y
1
′
=
5713
1920
y
3
-
5073
1280
y
2
+
3793
3840
y
0
+
h
2
77414400
-
3977083
f
0
-
113321
f
1
-
208639509
f
2
-
20080444
f
3
+
389931
f
4
+
130254
f
5
-
23611
f
6
,
h
y
2
′
=
7
30
y
3
+
3
20
y
2
-
23
60
y
0
+
h
2
1209600
26803
f
0
+
534018
f
1
+
323229
f
2
-
123236
f
3
+
31389
f
4
+
6654
f
5
+
691
f
6
,
h
y
3
′
=
3313
1920
y
3
-
2673
1280
y
2
+
1292
3840
y
0
+
h
2
2867200
-
60609
f
0
-
1189494
f
1
-
1182447
f
2
+
820748
f
3
-
90927
f
4
+
17802
f
5
-
1793
f
6
,
h
y
4
′
=
7
30
y
3
+
3
20
y
2
-
23
60
y
0
+
h
2
1209600
28403
f
0
+
510978
f
1
+
804189
f
2
+
1376924
f
3
+
512349
f
4
-
29694
f
5
+
2291
f
6
,
h
y
5
′
=
5713
19200
y
3
-
5073
1280
y
2
+
3793
3840
y
0
+
h
2
77414400
-
4632443
f
0
-
85305138
f
1
-
108369429
f
2
+
34314436
f
3
-
100660011
f
4
+
28146894
f
5
-
678971
f
6which are to be used with the initial methods(39)
y
1
+
95
64
y
3
-
349
128
y
2
+
31
128
y
0
=
h
2
77414400
91775
f
0
+
2787594
f
1
+
9305553
f
2
+
127768
f
3
+
109167
f
4
+
13578
f
5
-
961
f
6
,
y
4
-
2
y
3
+
y
2
=
h
2
60480
31
f
0
-
438
f
1
+
6513
f
2
+
48268
f
3
+
6513
f
4
-
438
f
5
+
31
f
6and the final methods(40)
y
N
-
4
-
2
y
N
-
3
+
y
N
-
2
=
h
2
60480
31
f
N
-
438
f
N
-
1
+
6513
f
N
-
2
+
48268
f
N
-
3
+
6513
f
N
-
4
-
438
f
N
-
5
+
31
f
N
-
6
,
y
N
-
5
-
223
64
y
N
-
3
+
349
128
y
N
-
2
-
31
128
y
N
-
2
=
h
2
25804800
-
f
N
-
724398
f
N
-
1
-
431259
f
N
-
2
-
4156156
f
N
-
3
+
2706981
f
N
-
4
+
200274
f
N
-
5
-
5141
f
N
-
6
.


## 4. The Block Unification Methods

The BUMs are also a class of methods for the numerical integration of both initial and boundary value problems. CLMM (4) is a 2
v-step continuous scheme and as such the BUM requires a set of 4
v methods so that, on the partition π
N
,
h
>
o, x
n
=
x
0
+
n
h, n
=
0
(
1
)
N, the solution of the 2
v step [
x
n
,
y
n
,
y
n
′
]
↦
[
x
n
+
2
v
,
y
n
+
2
v
,
y
n
+
2
v
′
] is obtained. CLMM (4) is used to generate (
4
v
-
1
) methods by evaluating (4) at x
=
{
x
n
+
1
,
…
,
x
n
+
v
-
2
,
x
n
+
v
+
1
,
…
,
x
n
+
2
v
} and also evaluating U
′
(
x
) at x
=
{
x
n
,
x
n
+
1
,
…
,
x
n
+
2
v
}. CLMM (4) can only be used to construct 4
v
-
1 methods and as such we give the last method as (since it is also of O
(
h
2
v
+
4
))(41)yn+2v+αvyn+v+αv-1yn+v-1=h2∑i=12vβifn+i.The BUM (4
v methods) is of the form(42)yn+1+αvyn+v+αv-1yn+v-1+α0yn=h2∑i=02vβifn+iyn+2+αvyn+v+αv-1yn+v-1+α0yn=h2∑i=02vβifn+i⋮yn+v-2+αvyn+v+αv-1yn+v-1+α0yn=h2∑i=02vβifn+iyn+v+1+αvyn+v+αv-1yn+v-1+α0yn=h2∑i=02vβifn+i⋮yn+2v+αvyn+v+αv-1yn+v-1+α0yn=h2∑i=02vβifn+iyn+2v+αvyn+v+αv-1yn+v-1=h2∑i=02vβifn+ihyn′+αv′yn+v+αv-1′yn+v-1+α0′yn=h2∑i=02vβi′fn+ihyn+1′+αv′yn+v+αv-1′yn+v-1+α0′yn=h2∑i=02vβi′fn+i⋮hyn+2v′+αv′yn+v+αv-1′yn+v-1+α0′yn=h2∑i=02vβi′fn+i,n=02vN-2v.Formulas (42) that form the BUM are all weighted the same unlike the BVMs that have main methods (15) and (16). We give below the BUMs of orders 6 and 8.BUM of Order 6
(43)
y
n
+
3
-
2
y
n
+
2
+
y
n
+
1
=
h
2
240
-
f
n
+
24
f
n
+
1
+
194
f
n
+
2
+
24
f
n
+
3
-
f
n
+
4
,
y
n
+
4
-
2
y
n
+
2
+
y
n
=
h
2
15
f
n
+
16
f
n
+
1
+
26
f
n
+
2
+
16
f
n
+
3
+
f
n
+
4
,
y
n
+
4
-
3
y
n
+
2
+
2
y
n
+
1
=
h
2
240
-
3
f
n
+
52
f
n
+
1
+
402
f
n
+
2
+
252
f
n
+
3
+
17
f
n
+
4
,
h
y
n
′
=
-
107
42
y
n
+
2
+
128
21
y
n
+
1
-
149
42
y
n
+
h
2
1260
-
67
f
n
+
2256
f
n
+
1
+
434
f
n
+
2
-
48
f
n
+
3
+
5
f
n
+
4
,
h
y
n
+
1
′
=
41
21
y
n
+
2
-
61
21
y
n
+
1
+
20
21
y
n
+
h
2
10080
-
613
f
n
+
11464
f
n
+
1
-
2870
f
n
+
2
+
344
f
n
+
3
-
37
f
n
+
4
,
h
y
n
+
2
′
=
5
42
y
n
+
2
+
16
21
y
n
+
1
-
37
42
y
n
+
h
2
1260
73
f
n
+
1136
f
n
+
1
+
574
f
n
+
2
-
48
f
n
+
3
+
5
f
n
+
4
,
h
y
n
+
3
′
=
41
21
y
n
+
2
-
61
21
y
n
+
1
+
20
21
y
n
+
h
2
10080
-
725
f
n
-
7656
f
n
+
1
+
9898
f
n
+
2
+
410
f
n
+
3
-
149
f
n
+
4
,
h
y
n
+
4
′
=
-
107
42
y
n
+
2
+
128
21
y
n
+
1
-
149
42
y
n
+
h
2
1260
325
f
n
+
4048
f
n
+
1
+
1106
f
n
+
2
+
1744
f
n
+
3
+
397
f
n
+
4
,
n
=
0
4
N
-
4
.BUM of Order 8
(44)
y
n
+
1
+
95
64
y
n
+
3
-
349
128
y
n
+
2
+
31
128
y
n
=
h
2
77414400
91775
f
n
+
2787594
f
n
+
1
+
9305553
f
n
+
2
+
127768
f
n
+
3
+
109167
f
n
+
4
+
13578
f
n
+
5
-
961
f
n
+
6
,
y
n
+
4
-
2
y
n
+
3
+
y
n
+
2
=
h
2
60480
31
f
n
-
438
f
n
+
1
+
6513
f
n
+
2
+
48268
f
n
+
3
+
6513
f
n
+
4
-
438
f
n
+
5
+
31
f
n
+
6
,
y
n
+
5
-
223
64
y
n
+
3
+
349
128
y
n
+
2
-
31
128
y
n
=
h
2
25804800
-
f
n
-
724398
f
n
+
1
-
431259
f
n
+
2
-
4156156
f
n
+
3
+
2706981
f
n
+
4
+
200274
f
n
+
5
-
5141
f
n
+
6
,
y
n
+
6
-
2
y
n
+
3
+
y
n
=
h
2
2240
141
f
n
+
2430
f
n
+
1
+
4131
f
n
+
2
+
6756
f
n
+
3
+
4131
f
n
+
4
+
2430
f
n
+
5
+
141
f
n
+
6
,
y
n
+
6
-
4
y
n
+
3
+
3
y
n
+
2
=
h
2
10080
-
11
f
n
-
18
f
n
+
1
+
2523
f
n
+
2
+
27268
f
n
+
3
+
19323
f
n
+
4
+
10734
f
n
+
5
+
661
f
n
+
6
,
h
y
n
′
=
-
233
30
y
n
+
3
+
243
20
y
n
+
2
-
263
60
y
n
+
h
2
44800
-
1031
f
n
+
147654
f
n
+
1
+
297207
f
n
+
2
+
35412
f
n
+
3
-
2313
f
n
+
4
+
198
f
n
+
5
-
7
f
n
+
6
,
h
y
n
+
1
′
=
5713
1920
y
n
+
3
-
5073
1280
y
n
+
2
+
3793
3840
y
n
+
h
2
77414400
-
3977083
f
n
-
113321
f
n
+
1
-
208639509
f
n
+
2
-
20080444
f
n
+
3
+
389931
f
n
+
4
+
130254
f
n
+
5
-
23611
f
n
+
6
,
h
y
n
+
2
′
=
7
30
y
n
+
3
+
3
20
y
n
+
2
-
23
60
y
n
+
h
2
1209600
26803
f
n
+
534018
f
n
+
1
+
323229
f
n
+
2
-
123236
f
n
+
3
+
31389
f
n
+
4
+
6654
f
n
+
5
+
691
f
n
+
6
,
h
y
n
+
3
′
=
3313
1920
y
n
+
3
-
2673
1280
y
n
+
2
+
1292
3840
y
n
+
h
2
2867200
-
60609
f
n
-
1189494
f
n
+
1
-
1182447
f
n
+
2
+
820748
f
n
+
3
-
90927
f
n
+
4
+
17802
f
n
+
5
-
1793
f
n
+
6
,
h
y
n
+
4
′
=
7
30
y
n
+
3
+
3
20
y
n
+
2
-
23
60
y
n
+
h
2
1209600
28403
f
n
+
510978
f
n
+
1
+
804189
f
n
+
2
+
1376924
f
n
+
3
+
512349
f
n
+
4
-
29694
f
n
+
5
+
2291
f
n
+
6
,
h
y
n
+
5
′
=
5713
19200
y
n
+
3
-
5073
1280
y
n
+
2
+
3793
3840
y
n
+
h
2
77414400
-
4632443
f
n
-
85305138
f
n
+
1
-
108369429
f
n
+
2
+
34314436
f
n
+
3
-
100660011
f
n
+
4
+
28146894
f
n
+
5
-
678971
f
n
+
6
,
h
y
n
+
6
′
=
-
233
30
y
n
+
3
+
243
20
y
n
+
2
-
263
60
y
n
+
h
2
44800
128089
f
n
+
216774
f
n
+
1
+
305847
f
n
+
2
+
122452
f
n
+
3
+
6327
f
n
+
4
+
69318
f
n
+
5
+
13113
f
n
+
6
,
n
=
0
6
N
-
6
.


### 4.1. Convergence and Use of the BUMs

BUMs (42) are weighted the same as each formula in (42) is used the same number of times as others. Their convergence was established in a similar way to Theorem 3 with some changes in the coefficients of the matrices and the global error is also of O
(
h
2
v
+
2
).The BUM is implemented efficiently by using the following algorithm.Step 1.
Use the block unification of (42) for n
=
0 to obtain Y
1 in the interval [
y
n
,
y
n
+
2
v
]; for n
=
1, Y
2 is obtained in the interval [
y
n
+
2
v
,
y
n
+
4
v
]; and in the intervals y
n
+
4
v
,
y
n
+
6
v
,
y
n
+
6
v
,
y
n
+
8
v
,
…
,
[
y
N
-
2
v
,
y
N
] for n
=
2,3
,
…
,
(
Γ
-
1
), we obtain Y
3
,
…
,
Y
Γ where N
=
2
v
×
Γ.Step 2.
The unified block given by the systemY
1
∪
Y
2
∪
⋯
∪
Y
Γ
-
1
∪
Y
Γ obtained in Step 1 results in a system of 2
N equations in 2
N unknowns which can be easily solved.Step 3.
The values of the solution and the first derivatives of (2) are generated by the sequence of {
y
n
}, {
y
n
′
}, n
=
0
,
…
,
N, obtained as the solution in Step 2.


## 4.1. Convergence and Use of the BUMs

BUMs (42) are weighted the same as each formula in (42) is used the same number of times as others. Their convergence was established in a similar way to Theorem 3 with some changes in the coefficients of the matrices and the global error is also of O
(
h
2
v
+
2
).The BUM is implemented efficiently by using the following algorithm.Step 1.
Use the block unification of (42) for n
=
0 to obtain Y
1 in the interval [
y
n
,
y
n
+
2
v
]; for n
=
1, Y
2 is obtained in the interval [
y
n
+
2
v
,
y
n
+
4
v
]; and in the intervals y
n
+
4
v
,
y
n
+
6
v
,
y
n
+
6
v
,
y
n
+
8
v
,
…
,
[
y
N
-
2
v
,
y
N
] for n
=
2,3
,
…
,
(
Γ
-
1
), we obtain Y
3
,
…
,
Y
Γ where N
=
2
v
×
Γ.Step 2.
The unified block given by the systemY
1
∪
Y
2
∪
⋯
∪
Y
Γ
-
1
∪
Y
Γ obtained in Step 1 results in a system of 2
N equations in 2
N unknowns which can be easily solved.Step 3.
The values of the solution and the first derivatives of (2) are generated by the sequence of {
y
n
}, {
y
n
′
}, n
=
0
,
…
,
N, obtained as the solution in Step 2.


## 5. Test Problems

We consider five numerical examples. The examples were solved using the BVMs and the BUMs of different order derived in this paper. Comparisons are made between the BVMs and BUMs by obtaining the maximum errors in the interval of integration. We note that the number of function evaluations (NFEs) involved in implementing the two methods isN
×
2
v in the entire range of integration. In order to show the competitiveness of the derived methods with some existing methods in the literature, we compared our methods with the Extended Trapezoidal Rules (ETRs), Extended Trapezoidal Rules of the second kind (ETR2s), and the Top Order Methods (TOMs) of orders 6, 8, and 10, respectively, given in [6]. For linear problems, we solve the resulting system of equations using Gaussian elimination with partial pivoting and, for nonlinear problems, we use a modified Newton-Raphson method.Example 1.
We consider the boundary value problem given in [6]: (45)x3u′′′′=1,1<x<2,u1=u′′1=u2=u′′2=0,Exact:ux=1410log⁡2-31-x+12x-1+3+xlog⁡x-x.Example 2.
We consider the nonlinear Fehlberg problem given in [12]:(46)y1′′=-4x2y1-2y12+y22y2,π2<x<10,y2′′=-4x2y2-2y12+y22y1,y1x0=0,y1′x0=-2π,y2x0=1,y2′x0=0,x0=π2,Exact:y1x=cos⁡x2,y2x=sin⁡x2.Example 3.
We consider the nonlinear BVP with mixed boundary conditions given in [17]:(47)y′′=y′2+y22ex,0<x<1,y0-y′0=0,y1+y′1=2e,Exact:yx=ex.Example 4.
We consider the nonlinear BVP given in [18]:(48)d2y1dx2+20y1′+4cos⁡xy1+sin⁡y1y2=f1x,0<x<1,d2y2dx2+5exy2′+6sinh⁡xy2+cos⁡y2=f2x,y10=1,y20=0,y11=e,y21=sinh⁡1,where(49)f1x=21ex+4excos⁡x+sin⁡exsinh⁡x,f2x=xcos⁡sinhx+5excosh⁡x+sinhx+6sinh2⁡x,Exact:y1x=ex,y2x=sinh⁡x.Example 5.
Lastly, we consider the following BVP forx
,
y
∈
[
-
1,1
] given in [19]:(50)uxx+uyy=-32π2sin⁡4πx,u±1,y=ux,±1=0,Exact:sin⁡4πxsin⁡4πy.


### 5.1. Numerical Results and Discussion

Example1 is a variable coefficient fourth-order BVP. The fourth-order BVP is transformed to a system of second-order BVP. We solved the system using the BVM and BUM of orders 6 and 8. The problem is also solved using the ETRs and ETR2s of orders 6 and 8, respectively. Tables 1 and 2 show the computational results for this example. While the BUM produces solutions of approximate accuracy with the BVM, it uses shorter CPU Time. Example 2 is the well-known nonlinear Fehlberg problem. It was solved for v
=
3,4 and the maximum of the Euclidean norm of the errors in y
1 and y
2 was obtained in the interval of integration. Example 2 was also solved using the ETR2s and the TOMs of orders 8 and 10, respectively. Tables 3 and 4 show that both methods produce solutions of approximate accuracy with the BUM using shorter CPU Time. Example 3 was chosen to demonstrate the use of the methods on a nonlinear BVP with mixed boundary conditions. The computational results were given in Tables 5 and 6. Example 4 was chosen to show the performance of the schemes on systems of nonlinear BVPs. The maximum of the Euclidean norm of the errors in y
1 and y
2 is given in Tables 7 and 8. Lastly, we show the performance of the methods on a Poisson equation with boundary conditions. The partial differential equation is transformed into a system of second-order ordinary differential equations with boundary conditions using the method of lines. Table 9 shows the computational result for this example. Also, Figures 1–4 show the efficiency curves of these methods for the different examples where we have denoted BVM and BUM with v
=
2 as BVM2 and BUM2, respectively.Table 1
Computational results forv
=
2 for Example 1.
N
BVM
BUM
ETRs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
2.184
e
-
02
0.281
6.182
e
-
05
0.297
3.641
e
-
05
0.296
8
3.357
e
-
06
0.267
1.225
e
-
05
0.328
2.193
e
-
06
0.312
16
1.867
e
-
08
0.329
2.656
e
-
07
0.328
8.754
e
-
08
0.365
32
6.634
e
-
10
0.389
4.351
e
-
09
0.359
2.249
e
-
09
0.389
64
2.884
e
-
12
0.422
6.837
e
-
11
0.391
4.564
e
-
11
0.441
128
6.456
e
-
14
0.626
1.070
e
-
12
0.546
8.156
e
-
13
0.625Table 2
Computational results forv
=
3 for Example 1.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
6
9.982
e
-
01
0.281
3.228
e
-
06
0.234
1.837
e
-
06
0.328
12
4.437
e
-
08
0.312
3.948
e
-
07
0.313
2.245
e
-
08
0.342
24
1.913
e
-
10
0.358
2.627
e
-
09
0.344
1.353
e
-
10
0.359
48
3.812
e
-
13
0.436
1.142
e
-
11
0.390
4.333
e
-
13
0.391
96
4.442
e
-
16
0.577
4.573
e
-
14
0.499
1.030
e
-
15
0.514
192
5.660
e
-
16
0.843
6.145
e
-
16
0.689
6.161
e
-
15
0.811Table 3
Computational results forv
=
3 for Example 2.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
150
1.326
e
-
02
1.266
6.027
e
-
02
1.078
1.631
e
-
02
1.178
300
4.268
e
-
05
2.391
1.133
e
-
03
2.000
6.889
e
-
05
2.196
600
1.511
e
-
07
4.470
4.756
e
-
05
3.749
2.714
e
-
07
4.461
1200
5.520
e
-
10
9.171
1.968
e
-
08
7.343
1.067
e
-
09
8.786
2400
3.132
e
-
12
19.125
7.834
e
-
11
15.578
4.204
e
-
12
17.468
4800
3.677
e
-
12
42.936
2.549
e
-
12
35.236
8.294
e
-
14
40.312Table 4
Computational results forv
=
4 for Example 2.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
150
5.048
e
-
04
2.142
2.261
e
-
03
1.671
2.993
e
-
03
1.985
300
2.979
e
-
07
3.952
1.343
e
-
05
3.156
4.373
e
-
06
3.766
600
2.560
e
-
10
7.704
1.447
e
-
08
5.890
4.752
e
-
09
3.735
1200
2.201
e
-
13
16.577
1.499
e
-
11
11.860
4.791
e
-
12
15.017
2400
5.283
e
-
13
33.172
6.877
e
-
13
25.750
4.292
e
-
14
28.250
4800
1.500
e
-
12
77.532
1.505
e
-
12
59.281
4.447
e
-
15
76.687Table 5
Computational results forv
=
2 for Example 3.
N
BVM
BUM
ETRs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
4.495
e
-
01
0.312
2.536
e
-
06
0.266
5.631
e
-
06
0.297
8
4.144
e
-
09
0.297
3.737
e
-
08
0.328
6.782
e
-
08
0.343
16
4.680
e
-
11
0.313
5.753
e
-
10
0.328
9.270
e
-
10
0.344
32
5.347
e
-
13
0.375
8.955
e
-
12
0.328
1.518
e
-
11
0.366
64
7.105
e
-
15
0.734
1.399
e
-
13
0.500
2.430
e
-
13
0.546
128
4.441
e
-
16
2.031
2.442
e
-
15
1.188
3.553
e
-
15
1.156Table 6
Computational results forv
=
4 for Example 3.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
8
2.399
e
-
00
0.297
4.823
e
-
12
0.234
1.031
e
-
11
0.273
16
2.220
e
-
14
0.297
5.773
e
-
15
0.276
7.994
e
-
15
0.297
32
5.940
e
-
14
0.313
1.332
e
-
14
0.281
6.661
e
-
16
0.343
64
3.055
e
-
13
0.453
2.398
e
-
14
0.374
1.332
e
-
15
0.374
128
9.859
e
-
14
0.532
5.063
e
-
14
0.423
1.110
e
-
15
0.453
256
4.596
e
-
13
0.797
6.672
e
-
14
0.625
8.882
e
-
15
0.671Table 7
Computational results forv
=
2 for Example 4.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
6.230
e
-
02
0.282
6.379
e
-
07
0.273
5.431
e
-
06
0.297
8
2.632
e
-
09
0.359
1.303
e
-
08
0.297
8.024
e
-
08
0.327
16
6.021
e
-
11
0.390
1.865
e
-
10
0.405
1.307
e
-
09
0.468
32
5.291
e
-
12
0.703
2.668
e
-
12
0.563
2.078
e
-
11
0.828
64
5.991
e
-
13
1.782
4.157
e
-
14
1.281
3.269
e
-
13
2.624
128
5.361
e
-
12
9.156
1.633
e
-
15
5.280
5.336
e
-
15
10.532Table 8
Computational results forv
=
3 for Example 4.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
6
2.582
e
-
00
0.343
1.044
e
-
09
0.298
3.834
e
-
10
0.375
12
8.390
e
-
13
0.375
3.563
e
-
12
0.267
1.449
e
-
12
0.453
24
1.242
e
-
13
0.375
1.460
e
-
14
0.328
4.158
e
-
15
0.656
48
3.999
e
-
11
0.455
1.629
e
-
14
0.421
4.578
e
-
16
0.861
96
1.035
e
-
11
0.703
2.517
e
-
14
0.593
4.578
e
-
16
1.040
192
1.775
e
-
13
1.077
6.846
e
-
14
0.842
4.965
e
-
16
1.235Table 9
Computational results forv
=
2 for Example 5.
N
BVM
BUM
l
∞ error
CPU Time
l
∞ error
CPU Time
16
9.662
e
-
00
0.483
1.251
e
-
01
0.531
32
2.582
e
-
02
1.235
2.578
e
-
02
1.031
64
6.433
e
-
03
5.358
6.459
e
-
03
5.516
128
1.607
e
-
03
43.641
1.607
e
-
03
46.923
256
2.00
e
-
00
512.843
4
.
016
e
-
04
532.657Figure 1
Efficiency curves for Example1.Figure 2
Efficiency curves for Example2.Figure 3
Efficiency curves for Example3.Figure 4
Efficiency curves for Example4.From the foregoing, it can be concluded that the BUM and the BVM produce solutions of approximate accuracy with the BUM using shorter CPU Time. However, a2
v-step BVM performs poorly when the number of steps, N, is 2
v. This is because the main method together with the initial and final methods does not form a good discrete analog or approximation of the continuous boundary value problem. Also, the BUM has the drawback that it is only implemented for any N which is a multiple of 2
v.


## 5.1. Numerical Results and Discussion

Example1 is a variable coefficient fourth-order BVP. The fourth-order BVP is transformed to a system of second-order BVP. We solved the system using the BVM and BUM of orders 6 and 8. The problem is also solved using the ETRs and ETR2s of orders 6 and 8, respectively. Tables 1 and 2 show the computational results for this example. While the BUM produces solutions of approximate accuracy with the BVM, it uses shorter CPU Time. Example 2 is the well-known nonlinear Fehlberg problem. It was solved for v
=
3,4 and the maximum of the Euclidean norm of the errors in y
1 and y
2 was obtained in the interval of integration. Example 2 was also solved using the ETR2s and the TOMs of orders 8 and 10, respectively. Tables 3 and 4 show that both methods produce solutions of approximate accuracy with the BUM using shorter CPU Time. Example 3 was chosen to demonstrate the use of the methods on a nonlinear BVP with mixed boundary conditions. The computational results were given in Tables 5 and 6. Example 4 was chosen to show the performance of the schemes on systems of nonlinear BVPs. The maximum of the Euclidean norm of the errors in y
1 and y
2 is given in Tables 7 and 8. Lastly, we show the performance of the methods on a Poisson equation with boundary conditions. The partial differential equation is transformed into a system of second-order ordinary differential equations with boundary conditions using the method of lines. Table 9 shows the computational result for this example. Also, Figures 1–4 show the efficiency curves of these methods for the different examples where we have denoted BVM and BUM with v
=
2 as BVM2 and BUM2, respectively.Table 1
Computational results forv
=
2 for Example 1.
N
BVM
BUM
ETRs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
2.184
e
-
02
0.281
6.182
e
-
05
0.297
3.641
e
-
05
0.296
8
3.357
e
-
06
0.267
1.225
e
-
05
0.328
2.193
e
-
06
0.312
16
1.867
e
-
08
0.329
2.656
e
-
07
0.328
8.754
e
-
08
0.365
32
6.634
e
-
10
0.389
4.351
e
-
09
0.359
2.249
e
-
09
0.389
64
2.884
e
-
12
0.422
6.837
e
-
11
0.391
4.564
e
-
11
0.441
128
6.456
e
-
14
0.626
1.070
e
-
12
0.546
8.156
e
-
13
0.625Table 2
Computational results forv
=
3 for Example 1.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
6
9.982
e
-
01
0.281
3.228
e
-
06
0.234
1.837
e
-
06
0.328
12
4.437
e
-
08
0.312
3.948
e
-
07
0.313
2.245
e
-
08
0.342
24
1.913
e
-
10
0.358
2.627
e
-
09
0.344
1.353
e
-
10
0.359
48
3.812
e
-
13
0.436
1.142
e
-
11
0.390
4.333
e
-
13
0.391
96
4.442
e
-
16
0.577
4.573
e
-
14
0.499
1.030
e
-
15
0.514
192
5.660
e
-
16
0.843
6.145
e
-
16
0.689
6.161
e
-
15
0.811Table 3
Computational results forv
=
3 for Example 2.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
150
1.326
e
-
02
1.266
6.027
e
-
02
1.078
1.631
e
-
02
1.178
300
4.268
e
-
05
2.391
1.133
e
-
03
2.000
6.889
e
-
05
2.196
600
1.511
e
-
07
4.470
4.756
e
-
05
3.749
2.714
e
-
07
4.461
1200
5.520
e
-
10
9.171
1.968
e
-
08
7.343
1.067
e
-
09
8.786
2400
3.132
e
-
12
19.125
7.834
e
-
11
15.578
4.204
e
-
12
17.468
4800
3.677
e
-
12
42.936
2.549
e
-
12
35.236
8.294
e
-
14
40.312Table 4
Computational results forv
=
4 for Example 2.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
150
5.048
e
-
04
2.142
2.261
e
-
03
1.671
2.993
e
-
03
1.985
300
2.979
e
-
07
3.952
1.343
e
-
05
3.156
4.373
e
-
06
3.766
600
2.560
e
-
10
7.704
1.447
e
-
08
5.890
4.752
e
-
09
3.735
1200
2.201
e
-
13
16.577
1.499
e
-
11
11.860
4.791
e
-
12
15.017
2400
5.283
e
-
13
33.172
6.877
e
-
13
25.750
4.292
e
-
14
28.250
4800
1.500
e
-
12
77.532
1.505
e
-
12
59.281
4.447
e
-
15
76.687Table 5
Computational results forv
=
2 for Example 3.
N
BVM
BUM
ETRs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
4.495
e
-
01
0.312
2.536
e
-
06
0.266
5.631
e
-
06
0.297
8
4.144
e
-
09
0.297
3.737
e
-
08
0.328
6.782
e
-
08
0.343
16
4.680
e
-
11
0.313
5.753
e
-
10
0.328
9.270
e
-
10
0.344
32
5.347
e
-
13
0.375
8.955
e
-
12
0.328
1.518
e
-
11
0.366
64
7.105
e
-
15
0.734
1.399
e
-
13
0.500
2.430
e
-
13
0.546
128
4.441
e
-
16
2.031
2.442
e
-
15
1.188
3.553
e
-
15
1.156Table 6
Computational results forv
=
4 for Example 3.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
8
2.399
e
-
00
0.297
4.823
e
-
12
0.234
1.031
e
-
11
0.273
16
2.220
e
-
14
0.297
5.773
e
-
15
0.276
7.994
e
-
15
0.297
32
5.940
e
-
14
0.313
1.332
e
-
14
0.281
6.661
e
-
16
0.343
64
3.055
e
-
13
0.453
2.398
e
-
14
0.374
1.332
e
-
15
0.374
128
9.859
e
-
14
0.532
5.063
e
-
14
0.423
1.110
e
-
15
0.453
256
4.596
e
-
13
0.797
6.672
e
-
14
0.625
8.882
e
-
15
0.671Table 7
Computational results forv
=
2 for Example 4.
N
BVM
BUM
TOMs
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
4
6.230
e
-
02
0.282
6.379
e
-
07
0.273
5.431
e
-
06
0.297
8
2.632
e
-
09
0.359
1.303
e
-
08
0.297
8.024
e
-
08
0.327
16
6.021
e
-
11
0.390
1.865
e
-
10
0.405
1.307
e
-
09
0.468
32
5.291
e
-
12
0.703
2.668
e
-
12
0.563
2.078
e
-
11
0.828
64
5.991
e
-
13
1.782
4.157
e
-
14
1.281
3.269
e
-
13
2.624
128
5.361
e
-
12
9.156
1.633
e
-
15
5.280
5.336
e
-
15
10.532Table 8
Computational results forv
=
3 for Example 4.
N
BVM
BUM
ETR2s
l
∞ error
CPU Time
l
∞ error
CPU Time
l
∞ error
CPU Time
6
2.582
e
-
00
0.343
1.044
e
-
09
0.298
3.834
e
-
10
0.375
12
8.390
e
-
13
0.375
3.563
e
-
12
0.267
1.449
e
-
12
0.453
24
1.242
e
-
13
0.375
1.460
e
-
14
0.328
4.158
e
-
15
0.656
48
3.999
e
-
11
0.455
1.629
e
-
14
0.421
4.578
e
-
16
0.861
96
1.035
e
-
11
0.703
2.517
e
-
14
0.593
4.578
e
-
16
1.040
192
1.775
e
-
13
1.077
6.846
e
-
14
0.842
4.965
e
-
16
1.235Table 9
Computational results forv
=
2 for Example 5.
N
BVM
BUM
l
∞ error
CPU Time
l
∞ error
CPU Time
16
9.662
e
-
00
0.483
1.251
e
-
01
0.531
32
2.582
e
-
02
1.235
2.578
e
-
02
1.031
64
6.433
e
-
03
5.358
6.459
e
-
03
5.516
128
1.607
e
-
03
43.641
1.607
e
-
03
46.923
256
2.00
e
-
00
512.843
4
.
016
e
-
04
532.657Figure 1
Efficiency curves for Example1.Figure 2
Efficiency curves for Example2.Figure 3
Efficiency curves for Example3.Figure 4
Efficiency curves for Example4.From the foregoing, it can be concluded that the BUM and the BVM produce solutions of approximate accuracy with the BUM using shorter CPU Time. However, a2
v-step BVM performs poorly when the number of steps, N, is 2
v. This is because the main method together with the initial and final methods does not form a good discrete analog or approximation of the continuous boundary value problem. Also, the BUM has the drawback that it is only implemented for any N which is a multiple of 2
v.


## 6. Conclusion

In this paper, we have developed a new class of LMMs and implemented the LMMs via two approaches, the boundary value approach and the block unification strategy, which are used to solve initial and boundary value problems. The comparison of the two approaches was carried out in terms of accuracy and computational efficiency. The results given in Section5 show that both approaches perform very well with the BUM using shorter CPU Time. Our future research will be to develop a variable stepsize version of the BVM and the BUM and a study of the conditioning of the matrices arising from the discretization of the continuous second-order problems.



---

*Source: 8465103-2016-04-11.xml*