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In this article we have considered Fredholm integro-differential equation type second-order boundary value problems and proposed a rational difference method for numerical solution of the problems. The composite trapezoidal quadrature and non-standard difference method are used to convert Fredholm integro-differential equation into a system of equations. The numerical results in experiment on some model problems show the simplicity and efficiency of the method. Numerical results showed that the proposed method is convergent and at least second-order of accurate.

The occurrences of differential equations and integral equations are common in many areas of studies in particular sciences and engineering. However, there are many mathematical formulations in science where both differential and integral operators appear together in the same equation. These equations were termed as integro-dif- ferential equations. The integro-differential equations have gained importance in the literature for the variety of their applications and in general it is impossible to obtain solutions of these problems using analytical methods. So it is required to obtain an efficient approximate solution. There are different methods and approaches for approximate numerical solution such as difference and compact finite difference method [

In this article we consider a method for the numerical solution of the following linear Fredholm integro-dif- ferrential equations of the form

subject to the boundary conditions

where

The emphasis in this article will be on the development of an efficient numerical method to deal with approximate numerical solution of the integro-differential equation and then to prove theoretical concepts of convergence and existence. The theorems of uniqueness, existence and convergence are important and can be found in the literature [

Last few decades have seen substantial progress in the development of approximate solution by non-conven- tional methods. One such method, a non-standard finite difference method has increasingly been recognized as a efficient method for the numerical solution of initial value problems in ordinary differential equation [

We have presented our work in this article as follows. In the next section we derived a non-standard finite difference method. In Section 3, we have discussed local truncation error in propose method and convergence under appropriate condition in Section 4. The applications of the proposed method to the model problems and illustrative numerical results have been produced to show the efficiency in Section 5. Discussion and conclusion on the performance of the new method are presented in Section 6.

Let us assume that

We define N finite nodal points of the domain [a, b], in which the solution of the problem (1) is desired, as

where

We approximate the integral that appeared in Equation (2) by the repeated/composite trapezoidal quadrature method [

where

and

Thus with the application of (4), the considered problem (1) at node

subject to the given boundary conditions.

Let us assume a local assumption as in [

and following the ideas [

where

For the computational purpose in Section 4, we have used the following finite difference approximation in place of

Thus from (7) we can write (6) as,

which is a nonlinear system of equations. We have to solve a nonlinear system with a large number of equations. So there is some complexity in the system and computation is difficult. However we have applied an iterative method to solve above system of nonlinear Equation (8).

The local truncation error at the node

At the nodal point

writing the Taylor series expansion for

Thus we obtain a truncation error at each node of

Consider the difference Method (6),

Let us ignore the third and other terms on right side of the above expression. After replacing

Thus

Let us define

So we can write (10) as,

Let us define

Let us define column matrix

The difference Method (11) represents a system of nonlinear equations in unknown

where

is tridiagonal matrix. Let Y be the exact solution of (11), so it will satisfy matrix equation

where Y is column matrix of order

Let us define

After linearization of

where

Similarly, we can linearize

By Taylor series expansion of

where

Let us assume that the solution of difference Equation (11) has no round off error. So from (12), (13) and (17) we have

Let us define

Then

We further define

Let there exist some positive constant

Let

Neglecting the higher order terms i.e.

where

It is easy to prove after neglecting higher order terms i.e.

Then

Thus from (18) and (19), we have

It follows from (9) and (20) that

To illustrate our method and demonstrate its computational efficiency, we have considered four model problems. In each model problem, we took uniform step size h. In Tables 1-4, we have shown MAY the maximum absolute error in the solution y of the problems (1) for different values of N. We have used the following formula in computation of MAY,

The order of convergence

where m can be estimated by considering the ratio of different values of N.

We use Newton-Raphson iteration method to solve the system of nonlinear equations arising from Equation (9). All computations are performed on a Windows 2007 Ultimate operating system in the GNU FORTRAN environment version 99 compiler (2.95 of gcc) on Intel Core i3-2330M, 2.20 Ghz PC. The solutions are computed on N nodes and iteration is continued until either the maximum difference between two successive iterates is less than 10^{−10} or the number of iterations reaches 10^{3}.

Problem 1. The model linear problem given by

Maximum absolute error | ||||
---|---|---|---|---|

N = 8 | N = 16 | N = 32 | N = 64 | |

MAY | 0.19123437 (−3) | 0.47540005 (−4) | 0.95266687 (−5) | 0.79055745 (−7) |

Iter. | 43 | 113 | 166 | 6 |

Maximum absolute error | ||||
---|---|---|---|---|

N = 8 | N = 16 | N = 32 | N = 64 | |

MAY | 0.35524368 (−4) | 0.76293945 (−5) | 0.19073486 (−5) | 0.47683716 (−6) |

Iter. | 36 | 80 | 64 | 3 |

Maximum absolute error | ||||
---|---|---|---|---|

N = 8 | N = 16 | N = 32 | N = 64 | |

MAY | 0.13130903 (−3) | 0.32335520 (−4) | 0.57816505 (−5) | 0.59064645 (−7) |

Iter. | 45 | 116 | 158 | 3 |

Maximum absolute error | ||||
---|---|---|---|---|

N = 8 | N = 16 | N = 32 | N = 64 | |

MAY | 0.14352799 (−4) | 0.58977230 (−6) | 0.41181391 (−7) | 0.38045517 (−7) |

Iter. | 24 | 10 | 2 | 2 |

subject to boundary conditions

where

Problem 2. The model linear problem given by

subject to boundary conditions

The analytical solution is

Problem 3. The model nonlinear problem [

subject to boundary conditions

The analytical solution is

Problem 4. The model nonlinear problem given by

subject to boundary conditions

where

We have described a numerical method for numerical solution of Fredholm integro-differential type boundary value problem and four model problems considered to illustrate the preciseness and effectiveness of the proposed method. Numerical results for example 1 which is presented in

A non-standard difference method to find the numerical solution of Fredholm integro-differential equation type boundary value problems has been developed. This method has been used for transforming Fredholm integro- differential equation into system of algebraic equations i.e. each nodal point

Pramod Kumar Pandey, (2015) Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems. Open Access Library Journal,02,1-10. doi: 10.4236/oalib.1101465