Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind
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, Xn,Yn, which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvabil...
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| Format: | Article |
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MDPI AG
2018-11-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/23/4/73 |
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| author | Silvia Alejandra Seminara María Inés Troparevsky |
| author_facet | Silvia Alejandra Seminara María Inés Troparevsky |
| author_sort | Silvia Alejandra Seminara |
| collection | DOAJ |
| description | 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, Xn,Yn, 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 un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. 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. |
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| id | doaj.art-e6f341bb10d447f7a55a609e021c9af7 |
| institution | Directory Open Access Journal |
| issn | 2297-8747 |
| language | English |
| last_indexed | 2024-12-13T21:29:27Z |
| publishDate | 2018-11-01 |
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| series | Mathematical and Computational Applications |
| spelling | doaj.art-e6f341bb10d447f7a55a609e021c9af72022-12-21T23:30:52ZengMDPI AGMathematical and Computational Applications2297-87472018-11-012347310.3390/mca23040073mca23040073Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second KindSilvia Alejandra Seminara0María Inés Troparevsky1Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, C1063ACV Buenos Aires, ArgentinaDepartamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, C1063ACV Buenos Aires, ArgentinaIn 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, Xn,Yn, 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 un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. 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.https://www.mdpi.com/2297-8747/23/4/73Fredholm integral equationsnumerical solutionsPetrov–Galerkin methodregular pairsiterated methods |
| spellingShingle | Silvia Alejandra Seminara María Inés Troparevsky Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind Mathematical and Computational Applications Fredholm integral equations numerical solutions Petrov–Galerkin method regular pairs iterated methods |
| title | Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind |
| title_full | Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind |
| title_fullStr | Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind |
| title_full_unstemmed | Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind |
| title_short | Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind |
| title_sort | iterated petrov galerkin method with regular pairs for solving fredholm integral equations of the second kind |
| topic | Fredholm integral equations numerical solutions Petrov–Galerkin method regular pairs iterated methods |
| url | https://www.mdpi.com/2297-8747/23/4/73 |
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