# Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Authors: Ishfaq Ahmad Ganaie; Shelly Arora; V. K. Kukreja
Journal: International Journal of Engineering Mathematics
 (2014)

---

## Abstract

Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.


---

## Body

## 1. Introduction

Since time immemorial mathematicians are ardently pursuing the solution of linear or nonlinear two point boundary value problems (BVPs) of the type:(1)
d
2
y
d
x
2
+
α
1
(
x
)
d
y
d
x
+
α
2
(
x
)
y
=
f
(
x
)
,
x
∈
[
a
,
b
]
subject to Dirichlet, Neumann, and Robin’s boundary conditions.Such BVPs have wide application in astronomy, biology, boundary layer theory, deflection in cables, diffusion process, electromagnetism, heat transfer, and other topics. It is well known that closed form analytical solution of such problems cannot be obtained in many cases; therefore, numerical techniques such as collocation method [1, 2], B-spline interpolation [3], Hermite cubic collocation [4–6], finite difference method [7–9], nonlinear shooting method [10], geometric Hermite interpolation [11], quintic B-spline collocation method [12], polynomial and nonpolynomial spline approaches [13–15], quartic spline solution [16], cubic spline collocation method [17], and finite volume element method [18] are frequently used.In this study, cubic Hermite collocation method (CHCM) involves cubic Hermite basis function to reduce mathematical complexity. Different linear and nonlinear differential equations are solved subject to Dirichlet, Neumann, and Robin boundary conditions using the present method. Moreover, the decoupling technique [4] used to solve elliptic problems with Neumann and Dirichlet conditions is a particular case of present technique. In this paper, linear and nonlinear boundary value problems reported in recent papers [2, 3, 7, 9, 13, 14] are solved using CHCM. It is worth mentioning that CHCM is giving better results than finite difference method, finite element method, finite volume method, B-spline method, and polynomial and nonpolynomial spline approach with fourth order of convergence.The paper comprises five sections. Section1 deals with general introduction of the problem. Section 2 gives brief description of cubic Hermite collocation method. In Section 3, symbolic solution of (1) is presented. Seven numerical examples are discussed in Section 4 and finally overall conclusions are given in Section 5.


## 2. Proposed Technique

In the present method, the domain is divided into finite elements and then orthogonal collocation method with cubic Hermite as basis function is applied within each element. The cubic Hermite interpolant of the functionf relative to partition a
=
x
1
<
x
2
<
⋯
<
x
N
+
1
=
b is a function s that satisfies the following:(1)
on each subinterval[
x
k
,
x
k
+
1
],  s coincides with a cubic polynomial s
k
(
x
),
(2)
s interpolates f and f
(
1
) at x
1
,
x
2
,
…
,
x
N
+
1,
(3)
s and s
(
1
) are continuous on [
a
,
b
].The cubic Hermite interpolant off and its first derivative at x
=
x
k requires that
(2)
s
k
(
x
k
)
=
f
(
x
k
)
,
s
k
(
1
)
(
x
k
)
=
f
(
1
)
(
x
k
)
.
Combining the continuity of s and s
(
1
) at x
=
x
k
+
1 with interpolation of f and its first derivative at x
=
x
k
+
1, one gets
(3)
s
k
(
x
k
+
1
)
=
s
k
+
1
(
x
k
+
1
)
=
f
(
x
k
+
1
)
,
s
k
(
1
)
(
x
k
+
1
)
=
s
k
+
1
(
1
)
(
x
k
+
1
)
=
f
(
1
)
(
x
k
+
1
)
.
Hence, s
k
(
x
) is a third degree polynomial that interpolates both f and f
(
1
) at x
=
x
k and at x
=
x
k
+
1. Therefore, s
k
(
x
) can be written as
(4)
s
k
(
x
)
=
H
1
k
(
x
)
f
(
x
k
)
+
H
2
k
(
x
)
f
(
1
)
(
x
k
)
+
H
3
k
(
x
)
f
(
1
)
(
x
k
+
1
)
+
H
4
k
(
x
)
f
(
x
k
+
1
)
,
where
(5)
H
2
p
-
1
k
(
x
)
=
{
(
x
-
x
k
-
1
h
k
-
1
)
2
(
3
-
2
(
x
-
x
k
-
1
)
h
k
-
1
)
;
x
∈
[
x
k
-
1
,
x
k
]
(
1
-
x
-
x
k
h
k
)
2
(
1
-
2
(
x
-
x
k
)
h
k
)
;
x
∈
[
x
k
,
x
k
+
1
]
0
;
otherwise
,
H
2
p
k
(
x
)
=
{
-
h
k
-
1
(
x
-
x
k
-
1
h
k
-
1
)
2
(
1
-
x
-
x
k
-
1
h
k
-
1
)
;
x
∈
[
x
k
-
1
,
x
k
]
h
j
(
1
-
x
-
x
k
h
k
)
2
(
x
-
x
k
h
k
)
;
x
∈
[
x
k
,
x
k
+
1
]
0
;
otherwise
.
Here, p
=
1,2 and k
=
1,2
,
…
,
N.The grid points,x
k, are often called the “knots” of the piecewise polynomial since they are points where polynomials are “tied together.” The Hermite polynomials do not require the subsidiary condition to make first derivative continuous. This fact reduces the number of equations by (
N
-
1
), where N is the number of elements.The global variablex varies in the kth element, where k
=
1,2
,
…
,
N. A new variable u
=
(
x
-
x
k
)
/
h
k is introduced in kth element in such a way that as x varies from x
k to x
k
+
1, u varies from 0 to 1. Orthogonal collocation is applied on local variable u.Approximation of functiony
(
u
) in the kth element is given as [6]
(6)
y
-
(
u
)
=
∑
i
=
1
4
a
i
+
2
k
-
2
H
i
(
u
)
.To apply the collocation method, one must evaluate the trial function (6) and its derivatives at two internal collocation points u
=
u
j
(
j
=
1
,
2
). These are given by
(7)
y
-
(
u
j
)
=
∑
i
=
1
4
a
i
+
2
k
-
2
H
i
(
u
j
)
,
d
y
-
d
u
(
u
j
)
=
1
h
k
∑
i
=
1
4
a
i
+
2
k
-
2
A
j
i
,
d
2
y
-
d
u
2
(
u
j
)
=
1
h
k
2
∑
i
=
1
4
a
i
+
2
k
-
2
B
j
i
,
where the Hermite polynomials and their first and second derivatives are defined as
(8)
H
1
(
u
j
)

=

(
1
+
2
u
j
)
(
1
-
u
j
)
2
,
H
2
(
u
j
)

=

u
j
(
1
-
u
j
)
2
h
k
,
H
3
(
u
j
)
=
u
j
2
(
3
-
2
u
j
)
,
H
4
(
u
j
)
=
u
j
2
(
u
j
-
1
)
h
k
,
A
j
1
(
u
j
)
=
6
u
j
2
-
6
u
j
,
A
j
2
(
u
j
)
=
(
1
-
4
u
j
+
3
u
j
2
)
h
k
,
A
j
3
(
u
j
)
=
6
u
j
-
6
u
j
2
,
A
j
4
(
u
j
)
=
(
3
u
j
2
-
2
u
j
)
h
k
,
B
j
1
(
u
j
)
=
12
u
j
-
6
,
B
j
2
(
u
j
)
=
(
6
u
j
-
4
)
h
k
,
B
j
3
(
u
j
)
=
6
-
12
u
j
,
B
j
4
(
u
j
)
=
(
6
u
j
-
2
)
h
k
,
where u
j’s are the zeros of shifted orthogonal Legendre polynomial P
2
(
0,0
)
(
u
) with u
1
=
0.2113248654 and u
2
=
0.7886751346, as shown in Figure 1.Figure 1
Subdivision of mesh points on the global domain. The four coefficients, in eachk element, are estimated by using four collocation points u
0, u
1, u
2, and u
3.


## 3. Symbolic Solution

Equation (1) can be discretized using (7) as follows:
(9)
1
h
k
2
∑
i
=
1
4
a
i
+
2
k
-
2
B
j
i
+
β
1
(
u
)
1
h
k
∑
i
=
1
4
a
i
+
2
k
-
2
A
j
i
+
β
2
(
u
)
×
∑
i
=
1
4
a
i
+
2
k
-
2
H
j
i
=
f
(
u
)
,
where, in the 1st element, a
i’s vary from 1 to 4, 2nd element varies from 3 to 6, and so on. Symbolic solution is given subject to Neumann boundary conditions, for a
=
0 and b
=
1:
(10)
y
(
1
)
(
0
)
=
c
0
,
(11)
y
(
1
)
(
1
)
=
c
L
,
where c
0 and c
L are finite real constants.In the first element,u
j
=
0
⇒
A
j
1
=
A
j
3
=
A
j
4
=
0; therefore, (10) becomes
(12)
1
h
k
∑
i
=
1
4
a
i
A
j
i
(
u
j
)
=
c
0
⟹
a
2
=
c
0
.
In the last element, u
j
=
1
⇒
A
j
1
=
A
j
2
=
A
j
3
=
0; therefore, (11) becomes
(13)
1
h
k
∑
i
=
1
4
a
i
+
2
N
-
2
A
j
i
(
u
j
)
=
c
L
⟹
a
2
N
+
1
=
c
L
.
A system of (
2
N
+
2
) equations is obtained from (9) to (13). It includes all parameters of the system and the dependent variables at the boundaries. The support of each Hermite cubic basis function spans at most two subintervals; therefore, a band matrix is obtained with bandwidth two (Figure 2). Of these, two unknowns are found using boundary conditions and rest 2
N are from discretized system of (9), using Mathematica. After substituting the appropriate values of a’s in (7), the result can be obtained for any element.Figure 2
Pattern of nonzero elements of banded matrix arising from CHCM forN
=
6, where ×’s represent nonzero elements and 0’s are represented by dots.


### 3.1. Error Analysis

Suppose the functionf
(
x
) and its four derivatives are continuous on [
a
,
b
] and there is a positive constant which satisfies |
f
4
(
x
)
|
≤
M for all x
∈
[
a
,
b
]. If H
(
x
) is the cubic Hermite interpolant of f at a and b, then according to de Boor [19], |
f
(
x
)
-
H
(
x
)
|
≤
ε
h
4, where h
=
b
-
a, ε
=
M
/
384.Also the placement of the collocation points plays a critical role in obtaining theO
(
h
4
) estimate [20, 21]. For Gauss Legendre roots, as collocation points, an error estimate of O
(
h
4
) is obtained whereas for other choices of the collocation points, only second-order accuracy is obtained.


### 3.2. Algorithm of the Method

Step 1.
Divide the domain0
≤
x
≤
1 into a mesh 0
=
x
1
<
x
2
<
⋯
<
x
N
+
1
=
1.Step 2.
Transform the global variablex into local variable u
=
(
x
-
x
k
)
/
h
k.Step 3.
Approximate the solution atu
=
u
j.Step 4.
Obtain the trial functiony
-
(
u
j
) and the derivatives (
d
y
-
/
d
u
)
(
u
j
), (
d
2
y
-
/
d
u
2
)
(
u
j
).Step 5.
Carry out discretization of the model using Step4.Step 6.
Evaluate Step5 at collocation points u
1
=
0.2113248654 and u
2
=
0.7886751346.Step 7.
The obtained system in Step6 is solved using any software.


## 3.1. Error Analysis

Suppose the functionf
(
x
) and its four derivatives are continuous on [
a
,
b
] and there is a positive constant which satisfies |
f
4
(
x
)
|
≤
M for all x
∈
[
a
,
b
]. If H
(
x
) is the cubic Hermite interpolant of f at a and b, then according to de Boor [19], |
f
(
x
)
-
H
(
x
)
|
≤
ε
h
4, where h
=
b
-
a, ε
=
M
/
384.Also the placement of the collocation points plays a critical role in obtaining theO
(
h
4
) estimate [20, 21]. For Gauss Legendre roots, as collocation points, an error estimate of O
(
h
4
) is obtained whereas for other choices of the collocation points, only second-order accuracy is obtained.


## 3.2. Algorithm of the Method

Step 1.
Divide the domain0
≤
x
≤
1 into a mesh 0
=
x
1
<
x
2
<
⋯
<
x
N
+
1
=
1.Step 2.
Transform the global variablex into local variable u
=
(
x
-
x
k
)
/
h
k.Step 3.
Approximate the solution atu
=
u
j.Step 4.
Obtain the trial functiony
-
(
u
j
) and the derivatives (
d
y
-
/
d
u
)
(
u
j
), (
d
2
y
-
/
d
u
2
)
(
u
j
).Step 5.
Carry out discretization of the model using Step4.Step 6.
Evaluate Step5 at collocation points u
1
=
0.2113248654 and u
2
=
0.7886751346.Step 7.
The obtained system in Step6 is solved using any software.


## 4. Numerical Examples and Discussion

In this section, seven examples demonstrate the efficiency and accuracy of the method. Following formulae are used for estimation of error in this study.Rate of convergence of CHCM is calculated usingρ
=
(
ln
⁡
(
L
∞
(
N
2
)
/
L
∞
(
N
1
)
)
)
/
(
ln
⁡
(
N
2
/
N
1
)
).Relative error is obtained by(
y
ex
-
y
nm
)
/
y
ex.Max norm is found byL
∞
=
max
⁡
i
=
1
n
|
y
(
u
i
)
-
y
-
(
u
i
)
|.Example 1.
Solve (1) for
(14)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
-
H
2
,
f
(
x
)
=
0
,
subject to Robin’s boundary conditions
(15)
y
(
0
)
=
1
,
y
(
1
)
(
1
)
=
0
,
which has exact solution [9], for planer geometry
(16)
y
=
cosh
⁡
H
(
1
-
x
)
cosh
⁡
H
.
The problem is solved for different values of dimensionless parameterH by taking 10 to 40 elements. The exact and numeric results are plotted in Figure 3. The results reported by [9] for 10 elements using finite difference method are matching with exact solution up to 3 decimal places, whereas using CHCM the results are matching up to 9 decimal places. This shows the superiority of cubic Hermite collocation method over the finite difference method. Relative error between CHCM and exact values is presented in Table 1.Table 1
Relative error for different values of parameterH for Example 1.
X
H
=
0
H
=
0.5
H
=
1
H
=
2
H
=
3
H
=
5
H
=
10
N
=
10
N
=
10
N
=
10
N
=
20
N
=
20
N
=
30
N
=
40
0.0
0
0
0
0
0
0
0
0.1
1.0000
E
-
10
5.963
E
-
11
2.708
E
-
9
5.1850
E
-
9
3.6328
E
-
8
8.9622
E
-
8
9.0871
E
-
7
0.2
1.0000
E
-
10
1.145
E
-
10
5.337
E
-
9
1.0504
E
-
8
7.3297
E
-
8
1.7941
E
-
7
1.8174
E
-
6
0.3
1.0000
E
-
10
1.640
E
-
10
7.842
E
-
9
1.5952
E
-
8
1.1120
E
-
7
2.6957
E
-
7
2.7261
E
-
6
0.4
1.0000
E
-
10
2.079
E
-
10
1.017
E
-
8
2.1492
E
-
8
1.5038
E
-
7
3.6059
E
-
7
3.6349
E
-
6
0.5
1.0000
E
-
10
2.457
E
-
10
1.229
E
-
8
2.7024
E
-
8
1.9106
E
-
7
4.5343
E
-
7
4.5440
E
-
6
0.6
1.0000
E
-
10
2.773
E
-
10
1.413
E
-
8
3.2361
E
-
8
2.3301
E
-
7
5.4982
E
-
7
5.4547
E
-
6
0.7
1.0000
E
-
10
3.020
E
-
10
1.561
E
-
8
3.7203
E
-
8
2.7477
E
-
7
6.5192
E
-
7
6.3745
E
-
6
0.8
1.0000
E
-
10
3.199
E
-
10
1.673
E
-
8
4.1147
E
-
8
3.1262
E
-
7
7.5863
E
-
7
7.3351
E
-
6
0.9
1.0000
E
-
10
3.307
E
-
10
1.741
E
-
8
4.3753
E
-
8
3.4019
E
-
7
8.5360
E
-
7
8.3951
E
-
6
1.0
1.0000
E
-
10
3.344
E
-
10
1.766
E
-
8
4.4668
E
-
8
3.5045
E
-
7
4.2434
E
-
7
9.0872
E
-
6Figure 3
Temperature profiles in a rectangular fin for dimensionless heat transfer coefficientH.Example 2.
Solve (1) for
(17)
α
1
(
x
)
=
-
1
,
α
2
(
x
)
=
0
,
f
(
x
)
=
-
e
x
-
1
-
1
,
subject to Dirichlet boundary conditions
(18)
y
(
0
)
=
y
(
1
)
=
0
,
0
<
x
<
1
,
which has exact solution [3, 7]
(19)
y
(
x
)
=
x
(
1
-
e
x
-
1
)
.
The exact and numeric results are plotted in Figure4. The CHCM results are matching up to 11 decimal places with the exact ones in Table 2. On comparing present results with the results of [3, 7], shown in Table 3, a big difference of errors between CHCM with finite difference method, finite element method, finite volume method, and B-spline method is observed. This indicates the supremacy of the present method.Table 2
Relative error between CHCM and exact values for Example2.
x
Exact solution
CHCM (100 elements)
Relative error
0.0
0
0
0
0.1
0.059343034025940
0.059343034024546
1.3944
E
-
12
0.2
0.110134207176555
0.110134207173893
2.6620
E
-
12
0.3
0.151024408862577
0.151024408858821
3.7560
E
-
12
0.4
0.182725813258852
0.182725813254164
4.6880
E
-
12
0.5
0.197560538965947
0.197560538960736
5.2110
E
-
12
0.6
0.196995306556119
0.196995306550776
5.3430
E
-
12
0.7
0.178732867019218
0.178732867014236
4.9820
E
-
12
0.8
0.145015397537614
0.145015397533471
4.1430
E
-
12
0.9
0.085646323767636
0.085646323765122
2.5130
E
-
12
1.0
0
0
0Table 3
Max norm of errors for five methods with respect to exact solution.
Methods
h
Max norm/h
2
FDM
0.1
8.24
E
-
3
0.01
8.31
E
-
3
FEM
0.1
6.35
E
-
3
0.01
6.36
E
-
3
FVM
0.1
3.18
E
-
3
0.01
3.18
E
-
3
B-spline
0.1
2.9
E
-
4
0.01
2.89
E
-
6
CHCM
0.1
7.240
E
-
4
0.01
5.352
E
-
8Figure 4
Comparison of CHCM with exact result for Example2.Example 3.
Solve (1) for
(20)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
-
ϕ
2
,
f
(
x
)
=
0
,
subject to Dirichlet boundary conditions
(21)
y
(
0
)
=
0.1
,
y
(
1
)
=
1
,
which has exact solution [2], for ϕ
2
=
10 and
(22)
y
=
g
sinh
⁡
ϕ
x
+
f
sinh
⁡
ϕ
(
1
-
x
)
sinh
⁡
ϕ
.
The exact and numeric results are plotted in Figure5. The results are matching up to 9 decimal places as shown in Table 4. From Table 5, order of convergence is found to be 4.Table 4
Relative error between CHCM and exact values for Example3.
x
Exact solution
CHCM (60 elements)
Relative error
0.1
0.100042548871756
0.100042549016137
1.4400
E
-
10
0.2
0.110172999814179
0.110173000081150
2.6700
E
-
10
0.3
0.131412868154409
0.131412868530861
3.7600
E
-
10
0.4
0.165903899721862
0.165903900198008
4.7600
E
-
10
0.5
0.217124036179286
0.217124036742860
5.6400
E
-
10
0.6
0.290238117151979
0.290238117780922
6.2900
E
-
10
0.7
0.392618682595762
0.392618683248041
6.5200
E
-
10
0.8
0.534589391091750
0.534589391690484
5.9900
E
-
10
0.9
0.730466017481367
0.730466017892698
4.1100
E
-
10
1.0
1
1
0Table 5
Rate of convergence of CHCM for Example3.
N
k
E
(
N
k
)
ρ
0.0666667
1.6720
E
-
07
—
0.0333333
1.0447
E
-
08
4.000345
0.0166667
6.8968
E
-
10
3.921095Figure 5
Comparison of CHCM with exact result for Example3.Example 4.
Solve (1) for
(23)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
1
,
f
(
x
)
=
-
1
,
subject to Neumann boundary conditions
(24)
y
(
1
)
(
0
)
=
1
-
cos
⁡
(
1
)
sin
(
1
)
,
y
(
1
)
(
1
)
=
-
1
-
cos
⁡
(
1
)
sin
(
1
)
,
which has exact solution [13, 14] as
(25)
y
(
x
)
=
cos
⁡
x
+
1
-
cos
⁡
(
1
)
sin
(
1
)
sin
x
-
1
.Example 5.
Solve (1) for
(26)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
x
,
f
(
x
)
=
(
3
-
x
-
x
2
+
x
3
)
sin
x
+
4
cos
⁡
x
,
subject to Neumann boundary conditions
(27)
y
(
1
)
(
0
)
=
-
1
,
y
(
1
)
(
1
)
=
2
sin
(
1
)
,
which has exact solution [13, 14] as
(28)
y
(
x
)
=
(
x
2
-
1
)
sin
x
.
For different number of elementsN
=
8, 16, 32, 64, and 128, Examples 4 and 5 are solved using the present method. Maximum absolute errors of the numerical solutions are calculated and compared with those reported by [13, 14], for different methods, in Tables 6 and 7. The present approach is giving much more accurate results than the others. From Table 8, the order of convergence is found to be 4.Table 6
Maximum absolute errors obtained by different methods in Example4.
Methods
N
=
8
N
=
16
N
=
32
N
=
64
N
=
128
Quadratic spline method [14]
7.70
E
-
04
1.93
E
-
04
4.83
E
-
05
1.21
E
-
05
3.02
E
-
06
Cubic spline method [14]
7.13
E
-
04
1.78
E
-
04
4.45
E
-
05
1.11
E
-
05
2.78
E
-
06
Nonpolynomial spline method [14]
1.75
E
-
04
2.16
E
-
05
2.68
E
-
06
3.33
E
-
07
4.15
E
-
08
Polynomial spline approach [13]
2.68
E
-
05
4.53
E
-
07
8.42
E
-
09
2.21
E
-
10
6.41
E
-
12
Present approach
1.53
E
-
07
9.35
E
-
09
6.11
E
-
10
3.00
E
-
11
9.99
E
-
13Table 7
Maximum absolute errors obtained by different methods in Example5.
Methods
N
=
8
N
=
16
N
=
32
N
=
64
N
=
128
Quadratic spline method [14]
4.50
E
-
02
3.08
E
-
03
7.70
E
-
04
1.93
E
-
04
4.80
E
-
05
Cubic spline method [14]
1.15
E
-
02
2.88
E
-
03
7.21
E
-
04
1.80
E
-
04
4.50
E
-
05
Nonpolynomial spline method [14]
2.67
E
-
03
3.24
E
-
04
3.99
E
-
05
4.94
E
-
06
6.16
E
-
07
Polynomial spline approach [13]
2.22
E
-
04
5.05
E
-
06
1.63
E
-
07
5.58
E
-
09
1.89
E
-
10
Present approach
2.34
E
-
06
1.43
E
-
07
9.99
E
-
09
8.00
E
-
10
6.00
E
-
11Table 8
Rate of convergence of CHCM for Examples4 and 5.
N
k
Example4
Example5
E
(
N
k
)
ρ
E
(
N
k
)
ρ
1/8
1.53
E
-
07
—
2.34
E
-
06
—
1/16
9.35
E
-
09
4.03
1.43
E
-
07
4.04
1/32
6.11
E
-
10
3.93
9.99
E
-
09
3.84
1/64
3.00
E
-
11
4.35
8.00
E
-
10
3.64Example 6.
Consider a nonlinear problem, that is, solving (1) for
(29)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
-
y
,
f
(
x
)
=
2
π
2
cos
⁡
(
2
π
x
)
-
si
n
4
(
π
x
)
,
subject to Neumann boundary conditions
(30)
y
(
1
)
(
0
)
=
0
,
y
(
1
)
(
1
)
=
0
,
which has exact solution [13]
(31)
y
(
x
)
=
si
n
2
(
π
x
)
.
The exact and CHCM results are showing good agreement in Figure6 for N
=
50.Figure 6
Comparison of CHCM with exact result for Example6.Example 7.
Consider a nonlinear problem, that is, solving (1) for
(32)
α
1
(
x
)
=
0
,
α
2
(
x
)
=
e
-
2
y
y
,
f
(
x
)
=
0
,
subject to Neumann boundary conditions
(33)
y
(
1
)
(
0
)
=
1
,
y
(
1
)
(
1
)
=
1
2
,
which has exact solution [13]
(34)
y
(
x
)
=
ln
⁡
(
1
+
x
)
.
An excellent matching is found between the exact and CHCM results in Figure7 for N
=
50. In Table 9, the order of convergence is again found to be 4 for Examples 6 and 7.Table 9
Rate of convergence of CHCM for Examples6 and 7.
N
k
Example6
Example7
E
(
N
k
)
ρ
E
(
N
k
)
ρ
1/25
2.4810
E
-
03
—
3.3211
E
-
03
—
1/50
1.8600
E
-
04
3.74
1.9900
E
-
04
4.07
1/100
1.1000
E
-
05
4.08
1.2000
E
-
05
4.06Figure 7
Comparison of CHCM with exact result for Example7.


## 5. Conclusion

In this paper, cubic Hermite collocation method is tested for seven problems. The numerical results obtained are quite satisfactory and comparable with the existing solution available in the literature. The superiority over the finite difference method, finite element method, finite volume method, B-spline method, and polynomial and nonpolynomial spline approach shows the strength of this method. The convergence of the CHCM technique is of order 4.



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