Abstract
Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.
Highlights
The study of finite-dimensional linear systems is well developed
Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods
We prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations
Summary
The study of finite-dimensional linear systems is well developed. As an infinite-dimensional counter part of finite-dimensional linear systems, one can view integral equations as extensions of linear systems of algebraic equations. An integral equation maybe interpreted as an analogue of a matrix equation which is easier to solve. There are many different ways to transform integral equations to linear systems. Many different methods have been used for solving Volterra integral equations and Freholm-Velterra integral equations numerically. We first recall the method of scaling function interpolation. We solve linear Volterra integral equation of the form: f. We will use scaling function interpolation method to solve integral equations. Convergence properties and how this method would compare with other methods. We will prove two convergence theorems and present several examples
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