Journal of Function Spaces
Volume 2015 (2015), Article ID 758410, 10 pages
http://dx.doi.org/10.1155/2015/758410

Research Article

An Approximation Method for Solving Volterra Integrodifferential Equations with a Weighted Integral Condition

A. Guezane-Lakoud, N. Bendjazia, and R. Khaldi

Laboratory of Advanced Materials, Department of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Received 19 February 2015; Revised 23 April 2015; Accepted 23 April 2015

Academic Editor: Richard I. Avery

Copyright © 2015 A. Guezane-Lakoud et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We apply the reproducing kernel Hilbert space (RKHS) method for getting analytical and approximate solutions for second-order hyperbolic integrodifferential equations with a weighted integral condition. The analytical solution is represented in the form of series; thus, the n-terms approximate solutions are obtained. The results of the numerical examples are compared with the exact solutions to illustrate the accuracy and the effectivity of this method.

1. Introduction

In functional analysis, a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. The subject was originally developed by Aronszajn in 1950 (see [1]).

Reproducing kernel theory has important applications in numerical analysis, differential equations, integral equations, probability, and statistics [2–4]. Recently, using the RKHS method, the authors in [5–21] have discussed singular linear two-point boundary value problems, singular nonlinear two-point periodic boundary value problems, nonlinear system of boundary value problems, initial value problems, singular integral equations, nonlinear partial differential equations, operator equations, and fourth-order integrodifferential equations (IDEs), differential equations with integral condition.

The purpose of this paper is to extend the application of the reproducing kernel Hilbert space method to solve a second-order boundary value problem of IDEs of the following Volterra type:

for all , subject to the initial conditions

and the weighted integral condition

where , , , , , and are given functions, such that

The operator is linear with respect to and continuous according to both variables and . The function is continuous and denote .

In order to put initial conditions and integral boundary conditions of (1) into the reproducing kernel space constructed in the following sections, we have to homogenize these conditions. For this, let ; then the problem can be converted into the following form:

for all , subject to the initial conditions

and the weighted integral condition

The theoretical aspects of (1) with conditions (2)–(4) have been studied by Galerkin method in [22]; however, no numerical method was presented. In fact, by this procedure, the authors have found several difficulties to find the discretization of this problem due to the presence of the integral condition. Therefore, to overcome these difficulties, in the present work, we propose the RKHS method which is a simple and effective method for obtaining the numerical solution that converges rapidly to the exact solution with a small error and with a good precision. Moreover, the partial derivatives of are also convergent to the partial of . So, from this work, we show the applicability of the RKHS method to solve this kind of problem without discretization such as Galerkin and Rothe’s method. Furthermore, the RKHS method has an advantage that it is the possibility to pick any point in the domain of integration as well as in the approximate solutions and all its partial derivatives up to order two that will be applicable.

In recent years, a growing interest has been devoted to the study of IDEs which are a combination of differential and Volterra-Fredholm integral equations. IDEs appear in various fields of science such as physics, biology, and engineering. We should mention also that integrodifferential equations are usually difficult to solve analytically.

Recently, various kinds of numerical methods have been used for efficient approximation solution. In [23], authors have presented a practical matrix method for solving nonlinear Volterra-Fredholm integrodifferential equations. In [24], authors have given a numerical solution of nonlinear Volterra-Fredholm integrodifferential equations using the spectral homotopy analysis method. In [25–28], authors have obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the reproducing kernel method.

To sum up, in this paper, we define several reproducing kernel spaces in Section 2, whereas, in Section 3, we introduce a linear operator, a complete normal orthogonal system, and some essential results. In Section 4, we provide the main results and the exact and approximate solution of (1)–(4) and we also develop an iterative method for this kind of problems. Finally, in Section 5, the results of some numerical examples are presented and are compared with the analytical solution to confirm the good accuracy of the presented method.

2. Reproducing Kernel Spaces

In this section, we define some useful reproducing kernel spaces.

Definition 1. Let

Define, respectively, the inner product and the norm in by

We have the following result.

Theorem 2. The space is a complete reproducing kernel space; that is, for each fixed , there exists simply and such that for any . The reproducing kernel is given by

where

Proof. Through several integrations by parts of (10), we obtain

Since , we have

Then

Note that the property of the reproducing kernel ; then is the solution of the following generalized differential equation:

with the boundary conditions

As , it follows that

Let us find the coefficients , , and .
Since , we have

Since , it follows that

Through (17)–(20), the unknown coefficients of (11) can be obtained.

Definition 3. Define the space

The inner product and the norm in are defined, respectively, by

The space is a complete reproducing kernel space, and its reproducing kernel is given by

Definition 4. Define

The inner product and the norm in are defined, respectively, by

is a reproducing kernel space and its reproducing kernel function is

such that, for any ,

Definition 5. Define

The inner product and the norm in are defined, respectively, by

Definition 6. Define

The inner product and the norm in are defined, respectively, by

is a complete reproducing kernel space; thus, its reproducing kernel function is (see [11])

where

3. A Bounded Linear Operator on

Define differential operator : by

therefore, (1) can be converted into an equivalent form as follows:

where ,  , and

It is easy to show that is a bounded linear operator from to . Now, we choose a countable subset in and we define

Lemma 7. Assume that is dense in ; then, is a complete system in , and

Applying Gram-Schmidt process, we obtain an orthogonal basis of , such that

where are orthogonal coefficients.

4. The Exact and Approximate Solution

Lemma 8. If is dense in , then the exact solution of (35) is

Proof. The exact solution can be expanded in the Fourier series in terms of normal orthogonal basis in :

4.1. Iterative Method

We construct an iterative method to obtain the approximate solutions of (1) in the reproducing kernel space . Let

Then

Define the initial function and the -term approximation to by

where the coefficients are given as

Theorem 9. If is dense in , then

where is given by (44).

Theorem 10. Suppose that and ; then

Proof. From the preceding theorem, we have

so

From and (49), we get

Proposition 11. Suppose that as ; then

Now, the approximate solution can be obtained by taking finitely many terms in the series representation of and

5. Numerical Examples

In order to calculate the approximate solution , we take

Example 1. Consider the following hyperbolic integrodifferential equation:

for all , subject to the initial conditions

and the weighted integral condition

where

The exact solution is given by . In order to homogenize the initial conditions, we put ; then the problem can be converted into the following form:

where

The numerical results are presented in Tables 1, 2, 3, and 4 and Figures 1 and 2.

Table 1: Comparison of results for (, ).

)

(0.00, 0.01)
−0.0000333333
−0.0000333637

−0.0000333333

(0.10, 0.05)
−0.000820406
−0.00082375

−0.000820406

(0.15, 0.10)
−0.00321499
−0.00323799
0.000022998
−0.00321499

(0.35, 0.20)
−0.0105706
−0.0106773
0.000106649
−0.0105706
0.0000198708

(0.49, 0.31)
−0.0183567
−0.0185277
0.000170909
−0.0183885
0.0000317253

(0.57, 0.45)
−0.0273509
−0.0275692
0.000218296
−0.0273509
0.0000385202

(0.69, 0.52)
−000800865
−0.00804727
0.0000386212
−0.00800874

(0.73, 0.71)
0.0059218
0.00598197
0.0000601728
0.00594941
0.0000276115

(0.81, 0.73)
0.0557278
0.0560604
0.000332616
0.0558007
0.0000728933

(0.99, 0.99)
0.36058
0.362213
0.00163333
0.360756
0.000176542

Table 2: Comparison of results for (, ).

(0.00, 0.01)
0.00000
0.0000
0.00000
−0.00666667
−0.00667043

(0.10, 0.05)
0.000262927
0.000263362

−0.0328162
−0.0328162
0.000065

(0.15, 0.10)
0.00161834
0.0016219

−0.0642998
−0.0644248
0.000124

(0.35, 0.20)
0.0167627
0.0167986
0.0000359083
−0.105706
−0.105901
0.000194

(0.49, 0.31)
0.0607656
0.0608914
0.000125813
−0.118431
−0.118615
0.00018

(0.57, 0.45)
0.155574
0.155866
0.000291519
−0.12156
−0.121689
0.00012

(0.69, 0.52)
0.268701
0.269116
0.000415711
−0.0308025
−0.0307829
0.000019

(0.73, 0.71)
0.541948
0.54252
0.00057215
0.0166811
0.0167237
0.000042

(0.81, 0.73)
0.66501
0.665538
0.000527674
0.152679
0.152725
0.000045

(0.99, 0.99)
1.65758
1.659
0.00142569
0.728444
0.727913
0.00053

Table 3: Comparison of results for (, ).

(0.00, 0.01)
0.0001000
0.000100171

−0.666667
−0.667391
0.000724

(0.10, 0.05)
0.00276293
0.00277046

−0.656325
−0.658253
0.001928

(0.15, 0.10)
0.0116183
0.0116374
0.0000190
−0.642998
−0.64296
0.0000379

(0.35, 0.20)
0.0567627
0.0568428
0.0000801
−0.528532
−0.528362
0.000169

(0.49, 0.31)
0.156866
0.157201
0.0003349
−0.382034
−0.382073
0.000038

(0.57, 0.45)
0.358074
0.35823
0.0001562
−0.270133
−0.270318
0.0001851

(0.69, 0.52)
0.539101
0.539354
0.0002531
−0.0592356
−0.0059133
0.0001026

(0.73, 0.71)
1.04605
1.04597
0.0000733
0.0234945
0.0232241
0.00027045

(0.81, 0.73)
1.19791016
1.19798
0.00006967
0.209149
0.208289
0.00085993

(0.99, 0.99)
2.63768
2.64076
0.003076
0.735802
0.735802
0.00426859

Table 4: Comparison of results for (, ).

(0.00, 0.01)
0.00000
0.0000
0.00000

(0.10, 0.05)
0.0105171
0.0105402
0.0000231397

(0.15, 0.10)
0.0323668
0.0324379
0.0000710837

(0.35, 0.20)
0.167627
0.167991
0.000364067

(0.49, 0.31)
0.392036
0.392836
0.000799951

(0.57, 0.45)
0.69144
0.692595
0.00115446

(0.69, 0.52)
1.03346
1.03456
0.00109998

(0.73, 0.71)
1.52661
1.52694
0.000326263

(0.81, 0.73)
1.82195
1.82199
0.000041932

(0.99, 0.99)
3.34864
3.34924
0.000598899

Figure 1: Consider .

Figure 2: Consider .

Example 2. Consider the following hyperbolic integrodifferential equation:

where

The exact solution is given by .

The numerical results are presented in Tables 5, 6, 7, and 8 and Figures 3 and 4.

Table 5: Comparison of results for (, ).

(0.00, 0.01)
−0.000050000
−0.0000500038

0.0000762474

(0.10, 0.05)
−0.001225000
−0.00122559

0.000482156

(0.15, 0.10)
−0.004775
−0.0047785

0.000732779

(0.35, 0.20)
−0.01510000
−0.0151099

0.000656861

(0.49, 0.31)
−0.0249764
−0.0249918
0.0000153856
0.000616005

(0.57, 0.45)
−0.0354578
−0.0354835
0.0000257761
0.000726954

(0.69, 0.52)
−0.00646256
−0.00645257

0.00154632

(0.73, 0.71)
0.0165849
0.0166669
0.0000819828
0.00494322

(0.81, 0.73)
0.0831857
0.083387
0.000201347
0.00242045

(0.99, 0.99)
0.470546
0.471707
0.00116079
0.00246689

Table 6: Comparison of results for (, ).

(0.00, 0.01)
0.0000
0.0000
0.000
−0.010
−0.0100011

(0.10, 0.05)
0.0005
0.000499228

−0.049
−0.049031
0.00003

(0.15, 0.10)
0.003
0.00300832

−0.0955
−0.0955913
0.00009

(0.35, 0.20)
0.028
0.0280282
0.000028
−0.151
−0.151122
0.00012

(0.49, 0.31)
0.094178
0.0942695
0.00009
−0.161138
−0.161259
0.00012

(0.57, 0.45)
0.23085
0.231107
0.00025
−0.15759
−0.157779
0.00018

(0.69, 0.52)
0.373152
0.37359
0.00043
−0.024856
−0.0247779
0.00007

(0.73, 0.71)
0.735986
0.737218
0.0012
0.046718
0.0471139
0.00039

(0.81, 0.73)
0.863298
0.864699
0.0014
0.227906
0.228805
0.00089

(0.99, 0.99)
1.9406
1.94427
0.0036
0.950598
0.954343
0.0037

Table 7: Comparison of results for (, ).

(0.00, 0.01)
0.0002
0.000196032

−1.0000
−1.00022
0.0002

(0.10, 0.05)
0.005
0.00504296
0.00004
−0.98
−0.981028
0.00102

(0.15, 0.10)
0.02
0.0201771
0.0001
−0.955
−0.956444
0.0014

(0.35, 0.20)
0.08
0.0799329
0.00006
−0.755
0.0007
0.0001

(0.49, 0.31)
0.1922
0.191932
0.00026
−0.5198
−0.520859
0.0010

(0.57, 0.45)
0.405
0.404939
0.00006
−0.3502
−0.351163
0.0009

(0.69, 0.52)
0.5408
0.541206
0.0004
−0.0478
−0.0473431
0.0004

(0.73, 0.71)
1.0082
1.00936
0.0011
0.0658
0.0670897
0.0012

(0.81, 0.73)
1.0658
1.06519
0.0006
0.3122
0.314928
0.0027

(0.99, 0.99)
1.9406
1.94427
0.0036
0.950598
0.954343
0.0037

Table 8: Comparison of results for (, ).

(0.00, 0.01)
0.000
0.000
0.000

(0.10, 0.05)
0.02
0.0199786
0.00002

(0.15, 0.10)
0.06
0.0602084
0.0002

(0.35, 0.20)
0.28
0.280396
0.00039

(0.49, 0.31)
0.6076
0.608402
0.0008

(0.57, 0.45)
1.026
1.02799
0.0019

(0.69, 0.52)
1.4352
1.43788
0.002

(0.73, 0.71)
2.0732
2.07878
0.005

(0.81, 0.73)
2.3652
2.37112
0.005

(0.99, 0.99)
3.9204
3.93184
0.011

Figure 3: Consider .

Figure 4: Consider .

6. Conclusion

In this paper, we have applied RKHS method to solve a hyperbolic integrodifferential equation with a weighted integral condition. The numeric results have demonstrated that only a small number of iterations can be used to obtain numeric results with a good precision. From this work, we see that the absolute errors are monotonically decreasing if increase. The numerical examples show that the approximate solution and its partial derivatives of a superior order or equal to two converge to the exact solution and its partial derivatives, respectively. Consequently, we confirm that this method is effective for the integrodifferential equation with weighted integral conditions. All the computations are performed using Mathematica 9.0.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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