Journal of Applied Mathematics, Islamic Azaduniversity of Lahijan, Vol.8, No.1(28), Spring

Journal of Applied Mathematics, Islamic AzadUniversity of Lahijan, Vol.8, No.1(28), Spring 2011, pp 55-60

Discrete Collocation Method for Solving Fredholm–Volterra Integro–Differential Equations

Mohsen Rabbani1*, S.H.Kiasoltani2

1 Department of Mathematics, Sari Branch, Islamic AzadUniversity, Sari, Iran

2 Department of Mathematics, Qaemshahr Branch, Islamic AzadUniversity, Qaemshahr, Iran

Received:29 November 2010

Accepted:31 January 2011

Abstract

In this article we use discrete collocation method for solving Fredholm–Volterra integro– differential equations, because these kinds of integral equations are used in applied sciences and engineering such as models of epidemic diffusion, population dynamics, reaction–diffusion in small cells. Also the above integral equations with convolution kernel will be solved by discrete collocation method. In this method we approximate solution of problem by no smooth piecewise polynomial. Numerical results show a high accuracy and validity discrete collocation method.

Keywords:Discrete CollocationMethod, Fredholm–Volterra Integro, Differential Equations.

1 Introduction

For solving Valterra integral equations see. Also for chemical absorption kinetics was used from Volterra integral equations in. Several numerical methods for solving Fredholm integral equations are in. Fredholm integro–differential equation by using Petrove-Galerkin is solved in also by spline wavelet and spline is solved respectively. Fredholm-Volterra integro–differential equation particular with convolution kernel is important and we consider this equation:

whereand for are known functions also can be convolution kernel, and is unknown function.

We decide to approximate by Lagrange polynomial interpolation such that it obtain by non- smooth piecewise polynomial space. So we use uniform mesh on

Also,

where is an element of the non-smooth piecewise polynomial with degree less than , in other words:

For finding in , we use Lagrange interpolation with points for and where is approximation and are -points Gauss as collocation parameters that they can be found by roots of Legendre polynomial on , for example in the case of they are obtained the following form:

also we choose as end-point of
By considering -points Gauss, we define mesh points as follows:

According to condition of interpolation we can write

and,

by choosing then , in this way is restricted to subintervals such as .

For solving it can be written the following form:

Now, we introduce discrete form Eq.,

For numerical solution of Eqs.we use as follows:

also,

whereis introduced by . At first are computed by , then they are used in system for obtainings.

In for we have :

From and we can write

By substitutingand ins are obtained by the following system :

where so and

By solving algebraic system we find S, then by, are found and alsoas a approximation of is introduced.

3 Application

In this section, we solve two examples of Fredholm–Volterra integro–differential equations by discrete collocation method.

Example1

Consider Fredholm-Volterra integro–differential equations with convolution kernel :

with exact solution

In the case of we use discrete collocation method and (12-14)then interpolation is obtained the following form and numerical results are shown in table1,

Table1. Numerical results for example 1.

According numerical results, it is obvious that the proposed method has a high accuracy.

Example2

In this example we solve Fredholm-Volterra integro-differential equations as follows:

with exact solution

By choosing in the discrete collocation method and use interpolation is introduced by,

and absolute errors in some points are given in table 2

Table2. Numerical results for example 2

By considering numerical results are shown in the table 2, we conclude the discrete collocation method has a high accuracy and it can be used for other problems.

4Conclusion

In this work, we discrete integral equations to use collocation method, thus we could obtain the numerical results with high accuracy. Also these results are shown ability and validity the proposed method.

References

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