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ISSN: 2229-8711
Global Journal of Technology and Optimization
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Solution of Voltra-Fredholm Integro-Differential Equations using Chebyshev Collocation Method

Deepmala1, Vishnu Narayan Mishra2*, HR Marasi3, H Shabanian4 and M Nosrati Sahlan4

1Mathematics Discipline, PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur-482005, India

2Applied Mathematics and Humanities Department, S.V. National Institute of Technology, Surat-395007, Gujarat, India

3Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

4Department of Mathematics, Faculty of Sciences, University of Bonab, Bonab, Iran

*Corresponding Author:
Vishnu Narayan Mishra
Applied Mathematics and Humanities Department
S.V. National Institute of Technology
Surat-395007, Gujarat, India
Tel: +91 99133 87604
E-mail: [email protected]

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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Abstract

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.

Keywords

Integro-differential equation; Chebyshev polynomial; Collocation method

Introduction

We consider the integro-differential equations of Fredholm, Volterra and Fredholm-Volterra types in the forms

image (1)

image (2)

and

image (3)

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 [1]. 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 [7], global spectral methods [8], methods for convolution equations [9], Newton-Gregory methods [10], Runge-Kutta methods [11,12], qualocation methods [13] and Galerkin methods [14] 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

image (4)

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

image (5)

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.

Methods and Approximations

Let D be a linear differential operator of order v with polynomial coefficients defined by

image (6)

We shall write for pr (x)

image (7)

Where αr is the degree of pr(x).

Let y(x) be the exact solution of the integro-differential equation

image (8)

with

image (9)

Where f(x) and k(x,t) are given continuous functions and λ, a, b, image and dj some given constants.

Matrix representation for Dy(x)

Let image be a polynomial basis given byimage whereimage and V is a non-singular lower triangular matrix with degree image According to [17] the effect of differentiation, shifting and integration on the coefficients vectorimage of a polynomialimage is the same as that of post-multiplication of image by the matrices Æ, rμ and l respectively,

image

Where

image

We recall now the following theorem given by Oritez and Samara [15].

Theorem 4.1.1 For any linear differential operator D defined by Eq. (6) and any series

image

we have

image

where

image

image

Function approximation

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)

where

image

Consequently, using Theorem 2.1 and substituting Eq. (10) in Eq. (1), we get

image (11)

Now using the chebyshev collocation points (5) in Eq. (11) we obtain the following new system of algebraic equations

image (12)

image

and so, unknown coefficients aj are found.

Definition 4.2.1 The polynomial image will be called an approximate solustion of Eqs. (8) and (9), if the vectorimage 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]

image

In this case we obtain the solution in terms of shifted Chebyshev polynomials T*j (x) in the form

image

where image Similar to the previous procedure and using the collocation points defined by

image (13)

one can get the following system of algebraic equations

image (14)

where image

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.

Results and Numerical Examples

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

image

y(0)=0,3mm

The exact solution is y(x) =cx. We assume the solution of y(x) as a truncated Chebyshev series

image (15)

Here, we have

image

image

The fundamental equation of the problem is defined by

image

where image

image

image

Therefore, using Theorem 4.1.1 we obtain

image

image

The system yields the solution

image

Substituting these values in (15), we get the exact solution of the problem

image

Example 5.2 We consider the following Fredholm-Volterra integrodifferential equation

image

y(0) = 0,3mm

The exact solution is y(x) = x2 . Let us suppose that y(x) is approximated by Chebyshev series

image

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 [16].

X The method discussed in [16] 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

image

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 [17]as shown in Table 2.

x Method in [13] presented method N=2 presented method N=4
0 0.0187621362 0.0036921151 0.002472602103
1/15 0.0200637354 0.0036087818 0.002472602236
2/15 0.0212780889 0.0035254484 0.002472602365
3/15 0.0215334191 0.0034421151 0.002472602492
4/15 0.0205212581 0.0033587818 0.002472602617
5/15 0.0192905931 0.0032754484 0.002472602740
6/15 0.0181294338 0.0031921151 0.002472602862
7/15 0.0170353464 0.0031087818 0.002472602984
8/15 0.0160143431 0.0030254484 0.002472603106
9/15 0.0150618428 0.0029421151 0.002472603229
10/15 0.0141627843 0.0028587818 0.002472603353
11/15 0.0133104506 0.0027754484 0.002472603479
12/15 0.0125010566 0.0026924451 0.002472603607
13/15 0.0117309887 0.0026087818 0.002472603788
14/15 0.0109968073 0.0025254484 0.002472603873
1.0 0.0102952225 0.0024421151 0.002472604012

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

image

The exact solution is y(x) = ex . See Table 3 for the numerical results.

X Exact Presented method N=11
-1 0.36787944 0.36787445212
-0.8 0.44932896 0.44932342567
-0.6 0.54881164 0.54881194564
-0.4 0.67032005 0.67032187409
-0.2 0.81873075 0.81897235137
0 1 0.99988319565
0.2 1.22140276 1.22140675944
0.4 1.49182470 1.49189543017
0.6 1.82211880 1.82211675430
0.8 2.22554093 2.22554075420
1.0 2.71828183 2.71828147693

Table 3: Comparison of numerical results for example (3.4).

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