Solution of Voltra-Fredholm Integro-Differential Equations using Chebyshev Collocation MethodDeepmala1, Vishnu Narayan Mishra2*, HR Marasi3, H Shabanian4 and M Nosrati Sahlan4
Received date: February 23, 2017; Accepted date: April 20, 2017; Published date: April 26, 2017
Citation: Deepmala, Mishra VN, Marasi H, Shabanian H, Nosrati Sahlan M (2017) Solution of Voltra-Fredholm Integro-Differential Equations using Chebyshev Collocation Method. Global J Technol Optim 8:210. doi: 10.4172/2229-8711.1000210
Copyright: © 2017 Deepmala, et al. This is an open-access article distributedunder the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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In this paper, we use chebyshev polynomial basis functions to solve the Fredholm and Volterra integro-differential equations. We directly calculate integrals and other terms are calculated by approximating the functions with the Chebyshev polynomials. This method affects the computational aspect of the numerical calculations. Application of the method on some examples show its accuracy and efficiency.
Integro-differential equation; Chebyshev polynomial; Collocation method
We consider the integro-differential equations of Fredholm, Volterra and Fredholm-Volterra types in the forms
where λ, λ1, and λ2 are real parameters. The functions f(x), k(x,t), k1(x,t) and k2(x,t) are known, y(x) is the unknown function to be determined and D is a linear differential operator. We suppose, without loss of generality, that the interval of integration is [-1,1]. Many problems in engineering and mechanics can be transformed into integral equations. For example it is usually required to solve Fredholm integral equations(FIE) in the calculations of plasma physics . The numerical solution of these equations is a well-studied problem and a large variety of numerical methods have been developed to rapidly and accurately obtain approximations to y(x). Overviews and references to the literature for many existing methods are available in [2,3]. Collocation methods [2-6], Sinc methods , global spectral methods , methods for convolution equations , Newton-Gregory methods , Runge-Kutta methods [11,12], qualocation methods  and Galerkin methods  are several of the many approaches that have previously been considered. In this paper the aim is to obtain the solution as a truncated Chebyshev series defined by
where Tj(x) denotes the Chebyshev polynomials of the frist kind, aj are unknown Chebyshev coefficients and N is any chosen positive integer. The Chebyshev collocation points are defined by
The paper is organized as follows: In Section Approximations we describe numerical approximations for differential operator and functions of integro-differential equation. The numerical results are presented in Section Numerical examples.
Let D be a linear differential operator of order v with polynomial coefficients defined by
We shall write for pr (x)
Where αr is the degree of pr(x).
Let y(x) be the exact solution of the integro-differential equation
Where f(x) and k(x,t) are given continuous functions and λ, a, b, and dj some given constants.
Matrix representation for Dy(x)
Let be a polynomial basis given by where and V is a non-singular lower triangular matrix with degree According to  the effect of differentiation, shifting and integration on the coefficients vector of a polynomial is the same as that of post-multiplication of by the matrices Æž, rμ and l respectively,
We recall now the following theorem given by Oritez and Samara .
Theorem 4.1.1 For any linear differential operator D defined by Eq. (6) and any series
The solution of Eqs. (1), (2) and (3) can be expressed as a truncated Chebyshev series. Therefore, the approximate solution (4) can be written in the matrix from
y (x) = TT (x)A, (10)
Consequently, using Theorem 2.1 and substituting Eq. (10) in Eq. (1), we get
Now using the chebyshev collocation points (5) in Eq. (11) we obtain the following new system of algebraic equations
and so, unknown coefficients aj are found.
Definition 4.2.1 The polynomial will be called an approximate solustion of Eqs. (8) and (9), if the vector is the solution of the system of liner algebraic equations (12).
Similarly we can develop the method for the problem defined in the domain [0, 1]
In this case we obtain the solution in terms of shifted Chebyshev polynomials T*j (x) in the form
where Similar to the previous procedure and using the collocation points defined by
one can get the following system of algebraic equations
Solving the nonlinear system, unknown coefficients aj are found. Similarly, we obtain the fundamental equation for Volterra and Fredholm-Volterra integral equation. In this study, instead of approximating integral terms, we directly calculate integrals. Examples show that this method affects the computational aspect of the numerical calculations.
The results obtained in previous sections are used to introduce a direct efficient and simple method to solve integro-differential equations of Volterra and Fredholm type.
Example 5.1 We consider the following Fredholm integrodifferential equation of the second kind
The exact solution is y(x) =cx. We assume the solution of y(x) as a truncated Chebyshev series
Here, we have
The fundamental equation of the problem is defined by
Therefore, using Theorem 4.1.1 we obtain
The system yields the solution
Substituting these values in (15), we get the exact solution of the problem
Example 5.2 We consider the following Fredholm-Volterra integrodifferential equation
y(0) = 0,3mm
The exact solution is y(x) = x2 . Let us suppose that y(x) is approximated by Chebyshev series
Using the procedure in section Approximations, we obtain the approximate solution of the problem.
In Table 1, we compare the numerical results of the problem by the proposed method of N=3 with the method discussed in an earlier study .
|X||The method discussed in ||Presented method N=3|
|0||4.930 × 10-4||3.3590 × 10-15|
|0.1||2.240 × 10-3||3.7764 × 10-15|
|0.2||1.571 × 10-3||4.2889 × 10-15|
|0.3||1.514 × 10-3||4.8711 × 10-15|
|0.4||7.015 × 10-3||5.5788 × 10-15|
|0.5||1.6336 × 10-2||6.2727 × 10-15|
|0.6||1.1862 × 10-2||7.0499 × 10-15|
|0.7||4.971 × 10-3||7.8825 × 10-15|
|0.8||4.338 × 10-3||8.9928 × 10-15|
|0.9||1.6068 × 10-2||9.8809 × 10-15|
|1.0||----||1.0880 × 10-15|
Table 1: Comparison of the absolute errors of example (3.2).
Example 5.3 We consider the following Fredholm integro-differential equation of the second kind
The exact solution is y(x) = x2 + 2x + 2 . Talking N = 2,4, the approximate solutions are obtained by this method. Results are compared with those of the methods in literature as shown in Table 2.
|x||Method in ||presented method N=2||presented method N=4|
Table 2: Comparison of the absolute errors of example (3.3).
Example 5.4 We consider the following Volterra integro-differential equation of the second kind
The exact solution is y(x) = ex . See Table 3 for the numerical results.
|X||Exact||Presented method N=11|
Table 3: Comparison of numerical results for example (3.4).