^{1}

^{*}

^{1}

^{*}

A numerical technique is presented for solving integration operator of Green’s function. The approach is based on Hermite trigonometric scaling function on [0,2 π], which is constructed for Hermite interpolation. The operational matrices of derivative for trigonometric scaling function are presented and utilized to reduce the solution of the problem. One test problem is presented and errors plots show the efficiency of the proposed technique for the studied problem.

In mathematics, a Green’s function is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions. Via the superposition principle, the con- volution of a Green’s function with an arbitrary function

For concreteness, we assume that all functions are defined on the interval

where functions

where

In most situations, it is difficult to obtain exact solution of the above integration. Hence, various approxi- mation methods have been proposed and studied. The purpose of the present paper is to develop a trigonometric Hermite wavelet approximation for the computing the Green’s function of the problem 2.

Recently, the arisen wavelet Galerkin method has demonstrated its advantages for the treatment of integral operators [

The outline of this paper is as follows. In Section 2, we 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 (see [

Obviously,

The equally spaced nodes on the interval

Definition 1 (Scaling functions). (See [

Lemma 1 (See [

and their derivations are given by

Theorem 2 (Interpolatory properties of the scaling functions). (See [

From above we can take wavelet functions

Definition 3 (Scaling functions space). For all

As a first step of studying the spaces

Theorem 4 For any

consequently

Definition 5 For any

where

The following properties of the operators

Theorem 6 Let

where C is a positive constant value.

Proof. See [

The differentiation of vector

where

where

where

where

Using Equation (7) and

and

for

In this section, we give the concrete computational schemes for this integral Equation (2) with the Green’s function kernel. The discretization form of (2) is given in the following subsection.

By introducing a basis

where

We tries to solve the above function by picking approximate values for

The abscissas

Applying Equation (10) in Equation (12) we have

By substituting

Thus, we have system of linear equation

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

So, the unknown function

To support our theoretical discussion, we applied the method presented in this paper to several examples. All the generalized Green’s function kernels in this numerical example are solved by trigonometric wavelet. Our me- thod compared with exact solution.

Example. Consider the inhomogeneous differential equation with the following coditions:

The exact solution is

descent method and results are shown in

The above example states

1. Our numerical method is also efficient when the wave number J is very large, that is to say, the wave number J can hardly affect the convergence rate,

2. Our numerical method is very fast, for example, the run time is only 2.000 s as J = 8, for which the corresponding matrix

The trigonometric scaling function is used to solve the Green’s function of an inhomogeneous differential equation. Some properties of trigonometric scaling function are presented and the operational matrices of derivative for trigonometric scaling function are utilized to reduce the solution of Green’s function to the solution of linear

Lightaqua | Exact solution | Absolute error | |
---|---|---|---|

0 | |||

Lightaqua | Exact solution | Absolute error | |
---|---|---|---|

0 | |||

system of equations with sparse matrix of coefficients. Applications of the wavelets allow the creation of more effective and faster algorithms than the ordinary ones. Illustrative examples are included to demonstrate the vali- dity and applicability of the technique. The main advantage of this method is its simplicity and small com- putation costs.

The authors are very grateful to both refrees for carefully reading the paper and for comments and suggestions which have improved the paper.