Abstract
In this work, we consider two-dimensional linear and nonlinear Fredholm integral equations of the first kind. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method is used to find a solution to the equation. The application of this method is based upon converting the first kind of equation to the second kind by applying the regularization method. Then the homotopy perturbation method is employed to the resulting second kind of equation to obtain a solution. A few examples including linear and nonlinear equations are provided to show the validity and applicability of this approach.
Highlights
Integral equations appear in many scientific applications with a very wide range from physical sciences to engineering
Like in one dimensional case, the regularization method transforms the first kind of equation: f ( x, t) =
This is expected because Fredholm integral equations of the first kind are often ill-posed problems
Summary
Integral equations appear in many scientific applications with a very wide range from physical sciences to engineering. There are numerous articles and books on the investigation of analytical and numerical solutions of one dimensional Fredholm integral equations of the first kind [1,2,3]. The main goal in this work is to extend the regularization-homotopy method introduced in [7] for one dimensional Fredholm integral equations of the first kind to two-dimensional Fredholm integral equations of the first kind. This method can applied for obtaining numerical solutions of the Fredholm integral equations of the first kind. Motivated by [14,15,16], one possible application area could be image restoration and denoising
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