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Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, C1063ACV Buenos Aires, Argentina
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Math. Comput. Appl. 2018, 23(4), 73; https://doi.org/10.3390/mca23040073
Received: 22 October 2018 / Revised: 9 November 2018 / Accepted: 12 November 2018 / Published: 13 November 2018
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Abstract

In this work we obtain approximate solutions for Fredholm integral equations of the second kind by means of Petrov–Galerkin method, choosing “regular pairs” of subspaces, { X n , Y n } , which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvability and numerical stability of the approximation scheme in an easy way, and the selection of orthogonal basis for the subspaces make the calculations quite simple. Afterwards, we explore an interesting phenomenon called “superconvergence”, observed in the 1970s by Sloan: once the approximations u n X n to the solution of the operator equation u K u = g are obtained, the convergence can be notably improved by means of an iteration of the method, u n * = g + K u n . We illustrate both procedures of approximation by means of two numerical examples: one for a continuous kernel, and the other for a weakly singular one. View Full-Text
Keywords: Fredholm integral equations; numerical solutions; Petrov–Galerkin method; regular pairs; iterated methods Fredholm integral equations; numerical solutions; Petrov–Galerkin method; regular pairs; iterated methods
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Seminara, S.A.; Troparevsky, M.I. Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind. Math. Comput. Appl. 2018, 23, 73.

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