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This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.

Mathematical models describe a variety of physical and engineering problems and processes which can be represented by integral equations (IEs). The homogeneous Fredholm IE is written as follows:

There exist two different methods to solve IEs numerically. The first one is to expand the equation by the appropriate set of basis functions, such as the classical orthogonal polynomials [

These methods have their own advantages and disadvantages. The main advantage of our approach over the existing wavelet methods is that the wavelet expansion coefficients can be computed analytically. In addition, it will be shown that the computational cost of our approach is low and the accuracy is high. It is worth to be mentioned that the application of wavelets takes a special place in the modern computational methods thanks to quick convergence of a series of wavelets and the possibility to find the solution with a low approximation error.

The pioneering contribution into the wavelet approach for solution of IEs belongs to Beylkin et al. [

The most part of the existing research programs is devoted to solution of the Fredholm and Volterra-type IEs. The Galerkin and collocation methods are mainly used in such papers [

In our opinion, the attention to the PHW and its application for solution of IEs have not been sufficiently paid, although there were attempts to use this basis for solution of partial differential equations (e.g., [

It is known that the wavelet approach offers an alternative route for a signal and function decomposition in the time-frequency domain. Recent applications of the wavelet transform to engineering and applied problems can be found in several studies [

In addition, for the convenience of the further analysis it would be better to operate with such functions, whose Fourier transform was compact and which could, if possible, be constructed from simple functions. The wavelets considered in our paper are called PHW and they possess all mentioned properties and constitute a specific but a representative example of wavelets in general.

The main purpose of the present work is to propose for numerical solution of IEs a simple approach based on periodized harmonic wavelets. This technique is also applicable with minor changes to the Fredholm, Volterra, and integro-differential equations. In Section

It is known [

Real (solid line) and imaginary (dashed line) parts of the periodic harmonic wavelets

According to [

Thus, we have a set of functions, which form basis for the

Let us consider the homogeneous Fredholm integral equation

We obtained the projection of the solution of the unknown eigenfunctions

Let us now consider the approximation error for the periodic wavelets. Let

The approximation error (

Using the wavelet periodic expansion (

Define

Let us compare the approximation error of wavelets with the error of the Fourier approximation for

A smooth function is a function that has continuous derivatives up to some desired order

According to [

In this work we have proposed PHWs as basis functions for solution of IEs. The approach was verified by solving a test problem and its approximation error was analytically estimated for periodized wavelets. The assumption of

There are several important facts to remember about the wavelet approximation.

The goal of the wavelet expansion of a function or signal is to obtain the coefficients of the expansion

The second goal is to have the most zero coefficients or very small. This is called a sparse representation and it is very important in applications for statistical estimation and detection, data compression, noise reduction, and fast algorithms.

The fact that the error is restricted to a small neighborhood of the discontinuity is the result of the “locality” of wavelets. The behavior of

Most of the linear part of

We can infer from the example that the present approach is applicable to a large class of problems, where the expected solution is a periodic function. It should be also mentioned that any differential equation can be transformed into an integral equation. It means that it might be solved a large class of eigenvalue equations derived by differential equations.

The work of A. Kudreyko is supported by the Istituto Nazionale di Alta Matematica

Francesco Severi (Rome-IT) under scholarship U 2008/000564, 21/11/2008.