Matrix method with Legendre wavelets for solving some nonlinear integral equations of the first kind with error analysis and existence of the solution

Document Type : Original Paper

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Department of Mathematics, University of Saravan, Saravan, Iran

Abstract

Since most integral equations of the first kind are ill-posed, in this article we tried to present a method based on Legendre wavelets to solve some of them. The proposed method is that at first all the components of the equation are expressed in matrix form based on Legendre wavelet functions, then these matrices are replaced in the main equation. This method, along with the collocation method, adjusts the equation to a system of algebraic equations, which can be solved by usual numerical or analytical methods. To show the convergence of the method, some computable error bounds are obtained by preparing some theorems and lemmas. Moreover, the stability analysis, the existence, and uniqueness of the solution are investigated. Numerical results with comparisons are given to confirm the reliability of the proposed method. In practice, in order to check the stability, we calculated the condition number in the examples, and also the convergence rate was calculated and the convergence was displayed in graphs by the ratio test. The outcomes reveal that this method is very effective and more accurate than those of the literature. Also, in many cases, one can get the exact solution to the problem.

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History

  • Receive Date: 21 April 2024
  • Revise Date: 09 September 2024
  • Accept Date: 08 November 2024

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