Journal of Applied Mathematics

Volume 2012 (2012), Article ID 803503, 11 pages

doi:10.1155/2012/803503

Research Article

Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

M. Tavassoli Kajani,¹ S. Vahdati,² Zulkifly Abbas,³ and Mohammad Maleki⁴

¹Department of Mathematics, Islamic Azad University, Khorasgan Branch, 81515-158 Isfahan, Iran

²Department of Mathematics, Khansar Faculty of Computer and Mathematics, University of Isfahan, Isfahan 81746-73441, Iran

³Department of Physics, Faculty of Science, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

⁴Department of Mathematics, Islamic Azad University, Mobarakeh Branch, 84819-14411 Isfahan, Iran

Received 22 September 2012; Accepted 8 December 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 M. Tavassoli Kajani 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.

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method.

1. Introduction

In recent years, there has been a growing interest in the system of integrodifferential equations (IDE), which arise frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, and so forth [1–8]. The systems of integrodifferential equations are generally difficult to solve analytically, thus finding efficient computational algorithms for obtaining numerical solution is required.

There are various techniques for solving systems of IDE, for example, operational Tau method [9, 10], Adomian decomposition method [11], Galerkin method [12], rationalized Haar functions method [13], He’s homotopy perturbation method (HPM) [14, 15], and Ghebyshev polynomial [16].

A number of problems arising in science and engineering are set in semi-infinite domains. One can apply different spectral methods that are used to solve problems in semi-infinite domains. The first approach is using Laguerre polynomials [17–20]. The second approach is replacing semi-infinite domain with interval by choosing , sufficiently large. This method is named domain truncation [21]. The third approach is reformulating original problem in semi-infinite domain to singular problem in bounded domain by variable transformation and then using the Jacobi polynomials to approximate the resulting singular problem [22]. The fourth approach of spectral method is based on rational orthogonal functions.

Boyd [23] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Guo et al. [24] introduced a new set of rational Legendre functions which are mutually orthogonal in . They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. [25] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre, and mapped Fourier sine.

The authors of [26–29] applied spectral method to solve nonlinear ordinary differential equations on semi-infinite intervals. Their approach was based on a rational Tau method. They obtained the operational matrices of derivative and product of rational Chebyshev and Legendre functions and then they applied these matrices together with the Tau method to reduce the solution of these problems to the solution of system of algebraic equations.

Zarebnia and Ali Abadi [30] used Sinc-Collocation method for solving system of nonlinear second-order integrodifferential equations of the Fredholm type. Rational second (third) kind Chebyshev (RSC) functions, for the first time, were proposed by Tavassoli Kajani and Ghasemi Tabatabaei [31] to find the numerical solution of Lane-Emden equation.

This paper outlines the application of rational second kind Chebyshev functions and Galerkin method to the following system of linear high-order integrodifferential equations on the interval . Two problems of such equations are solved to make clear the application of the proposed method. One has

where .

2. Properties of RSC Functions

In this section, we present some properties of RSC functions.

2.1. RSC Functions

The second kind Chebyshev polynomials , are orthogonal in the interval with respect to the weight function and we find that satisfies the recurrence relation [32]

The RSC functions are defined by [31, 33]

thus RSC functions satisfy

2.2. Function Approximation

Let denotes a nonnegative, integrable, real-valued function over the interval . We define

where

is the norm induced by the scalar product

Thus denote a system which is mutually orthogonal under (2.6), that is,

where is the Kronecker delta function. This system is complete in ; as a result, any function can be expanded as follows:

with

The ’s are the expansion coefficients associated with the family . If the infinite series in (2.8) is truncated, then it can be written as

where and .

We can also approximate the function in as follows:

where is an matrix that

Moreover, from recurrence relation in (2.3) we have

2.3. Product Integration of the RSC Functions

We also use the matrix as follows:

To illustrate the calculation we choose and , then we obtain

2.4. Operational Matrix of Derivative

The derivative of the vector defined in (2.10) can be approximated by

where is named the operational matrix of derivative. Differentiating (2.3) we get

By using (2.17) the matrix can be calculated. The matrix is a lower Hessenberg matrix and can be expressed as , where is a tridiagonal matrix which is obtained from

and the elements of matrix are obtained from

2.5. The Product Operational Matrix

The following property of the product of two rational Chebyshev vectors will also be used:

where is called product operational matrix for the vector . Using (2.20) and the orthogonal property, the elements of the matrix can be calculated from

where is given by

3. Solving System of Integrodifferential Equations over Semi-Infinite Interval

Consider the following system of integrodifferential equations:

Using (2.10) and (2.11) to approximate , , , and when and , we have

According to the operational matrix of derivative we can approximate as

With substituting these approximations in (3.1) we have

Then using (2.20) we obtain

which can be simplified using (2.14)

By solving this linear system of algebraic equations we can find vectors , , and then approximate the solutions as

4. Numerical Examples

Example 4.1. Consider the following system of linear integrodifferential equations:

The exact solution of this example is and .

We solved Example 4.1 using the present method with , and we obtained and , which imply

that are the exact solutions.

Example 4.2. Next, consider the following system of integrodifferential equations with the exact solution and :

We solved this example by using the method described in Section 3 for and . Results are shown in Table 1 and Figures 1 and 2. The errors for large values of are shown in Table 2. It is seen that the proposed method provides accurate results even for large values of .

Table 1: Numerical results of Example 4.2.

Exact

Exact

0.0
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000

0.2
0.82435
0.81643
0.81873
0.67203
0.66964
0.67032

0.4
0.67955
0.66654
0.67032
0.45203
0.44825
0.44933

0.6
0.56024
0.54411
0.54881
0.30436
0.29994
0.30119

0.8
0.46199
0.44410
0.44933
0.20516
0.20062
0.20190

1.0
0.38119
0.36238
0.36788
0.13844
0.13413
0.13534

1.2
0.31473
0.29559
0.30119
0.09351
0.08965
0.09072

1.4
0.26007
0.24100
0.24660
0.06321
0.05991
0.06081

1.6
0.21512
0.19637
0.20190
0.04274
0.04006
0.04076

1.8
0.17819
0.15987
0.16530
0.02886
0.02681
0.02732

2.0
0.14787
0.13003
0.13534
0.01942
0.01798
0.01832

2.2
0.12300
0.10562
0.11080
0.01297
0.01211
0.01228

2.4
0.10261
0.08565
0.09072
0.00855
0.00822
0.00823

2.6
0.08589
0.06932
0.07427
0.00550
0.00565
0.00552

2.8
0.07218
0.05596
0.06081
0.00339
0.00395
0.00370

3.0
0.06094
0.04503
0.04970
0.00193
0.00285
0.00248

Table 2: Absolute error for large values of .

Figure 1: Graph of the exact and numerical solutions of for ; symbols correspond to the numerical solution.

Figure 2: Graph of the exact and numerical solutions of for ; symbols correspond to the numerical solution.

5. Conclusion

The fundamental goal of this paper has been to construct an approximation to the solution of the integrodifferential equations system in a semi-infinite interval. In the above discussion, the Galerkin method with RSC functions, which have the property of orthogonality, is employed to achieve this goal. Advantages of this method is that we do not reform the problem to a finite domain, and with a small value of accurate results are obtained. There is a good agreement between obtained results and exact values that demonstrates the validity of the present method for this type of problems and gives the method a wider applicability.

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