Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations |
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Authors: | Pouria Assari |
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Institution: | 1. Faculty of Sciences, Department of Mathematics, Bu-Ali Sina University , Hamedan, Iran.passari@basu.ac.ir |
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Abstract: | The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates. |
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Keywords: | Non-linear integral equation thin-plate spline weakly singular kernel discrete Galerkin method error analysis |
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