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The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.

In this paper we solve the Fredholm Linear Integro-Differential Equations as

where

Systems of integro-differential equations have a major role in the fields of science, physical phenomena, and engineering, such as nano-hydrodynamics, glass-forming process, dropwise condensation, wind ripple in the de- sert, and modeling the competition between tumor cells and the immune system. The concept of a system of integro-differential equations has motivated a huge amount of research work in recent years. Alot of attention has been devoted to the study of differential-difference equations, e.g. equations containing shifts of the un- known function and its derivates, and also integro-differential-difference equations. For instance, see [

Our approach consists of reducing the problem to a set of linear equations by trigonometric scaling functions which is constructed for Hermite interpolation. A difficulty of using wavelet for the representation of integral operators is that quadrature leads to potentially high cost with sparse matrix. This fact particularly encourages us in efforts to devote to some appropriate wavelet bases to simplify the computation expense of the reoresentation matrix, which is importent to improve the wavelet method. Recently, the trigonometric interpolant wavelet has arisen in the approximation of operators [

The organization of the rest of this paper is as follows: Section (0) describe the trigonometric scaling function on

In this section, we will give a brief introduction of Quak’s work on the construction of Hermite interpolatory trigonometric wavelets and their basic properties. More details can be found in (see [

For all

where the Dirichlet kernel

Obviously,

exceeding l. The equally spaced nodes on the interval

any

Lemma 1 (See [

and their derivations are given by

Theorem 1 (Interpolatory properties of the scaling functions). (See [

From above we can take wavelet functions

Definition 2 (Scaling functions space). For all

As a first step of studying the spaces

Now a Hermite-type project operator can be introduced by means of the scaling functions. For all

where

Theorem 3 Let

where C is a positive constant value.

Proof. See [

Lemma 2 (The operational matrix of scaling function derivative). (See [

where

where

where

where

Using 7 and

and

for

In this section, we first give the computational schemes for Equation (1) with the Newton-Cotes formulas. For either one of these rules, we can make a more accurate approximation by breaking up the interval

where the subintervals have the form

basis

where C is

Thus we have

By using Lemma 2 and after summarizing Equation (13) can be rewritten as the matrix form

where the matrix

To support our theoretical discussion, we applied the method presented in this paper to several examples. The main objective here is to solve these two examples using the trigonometric scaling function and compare our results with exact solution.

Example 4 Consider the second-order the Fredholm Linear Integro-Differential Equation

with the mixed conditions

apply the suggested method with

In

Example 5 Consider the following second-order the Fredholm Linear Integro-Differential Equation

with the initial conditions

J | |||||
---|---|---|---|---|---|

Exact solution | ||||
---|---|---|---|---|

0 | ||||

0.125 | ||||

0.2500 | ||||

0.3750 | ||||

0.5000 | ||||

0.6250 | ||||

0.7500 | ||||

0.8750 |

J | |||||
---|---|---|---|---|---|

Exact solution | ||||
---|---|---|---|---|

0 | 1.00000000 | 1.00104921 | 1.00002430 | 1.00000013 |

0.125 | 1.13310019 | 1.13048310 | 1.13307891 | 1.13310007 |

0.2500 | 1.28400186 | 1.28178029 | 1.28314720 | 1.28400134 |

0.3750 | 1.45500374 | 1.45304829 | 1.45456781 | 1.45500352 |

0.5000 | 1.64870391 | 1.64910273 | 1.64820756 | 1.64870372 |

0.6250 | 1.86820582 | 1.86705912 | 1.86835405 | 1.86820565 |

0.7500 | 2.11704076 | 2.11078316 | 2.11527804 | 2.11704052 |

0.8750 | 2.39893108 | 2.39719802 | 2.39884240 | 2.39893214 |

Our results indicate that the method with the trigonometric scaling bases can be regarded as a structurally simple algorithm that is conventionally applicable to the numerical solution of IDEs. In addition, although we have re- stricted our attention to linear Fredholm IDEs, we expect the method to be easily extended to more general IDEs. the presented method which is based on the trigonometric scaling function is proposed to find the approximate solution. A comparison of the exact solution reveals that the presented method is very effective and convenient. Nevertheless, as

The authors are very grateful to the editor for carefully reading the paper and for their comments and sugges- tions which have improved the paper.