A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems
In this paper, two inverse problems of determining an unknown source term in a parabolic equation are considered. First, the unknown source term is estimated in the form of a combination of Chebyshev functions. Then, a numerical algorithm based on Chebyshev polynomials is presented for ob...
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| Format: | Article |
| Language: | fas |
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Kharazmi University
2016-09-01
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| Series: | پژوهشهای ریاضی |
| Subjects: | |
| Online Access: | http://mmr.khu.ac.ir/article-1-2577-en.html |
| _version_ | 1797871734489088000 |
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| author | S Nemati A Babaei S Sedaghat |
| author_facet | S Nemati A Babaei S Sedaghat |
| author_sort | S Nemati |
| collection | DOAJ |
| description | In this paper, two inverse problems of determining an unknown source term in a parabolic equation are considered. First, the unknown source term is estimated in the form of a combination of Chebyshev functions. Then, a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem. For solving the problem, the operational matrices of integration and derivation are introduced and utilized to reduce the mentioned problem into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. Due to ill-posedness of these inverse problems, the Tikhonov regularization method with generalized cross validation (GCV) criterion is applied to find stable solutions. Finally, some examples are presented to illustrate the efficiency of this numerical method. The numerical results show that the proposed method is a reliable method and can give high accuracy approximate solutions. |
| first_indexed | 2024-04-10T00:47:20Z |
| format | Article |
| id | doaj.art-80c21ddc8a874cb8b6a9a8397e3f55b3 |
| institution | Directory Open Access Journal |
| issn | 2588-2546 2588-2554 |
| language | fas |
| last_indexed | 2024-04-10T00:47:20Z |
| publishDate | 2016-09-01 |
| publisher | Kharazmi University |
| record_format | Article |
| series | پژوهشهای ریاضی |
| spelling | doaj.art-80c21ddc8a874cb8b6a9a8397e3f55b32023-03-13T19:18:12ZfasKharazmi Universityپژوهشهای ریاضی2588-25462588-25542016-09-01214768A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source ProblemsS Nemati0A Babaei1S Sedaghat2 In this paper, two inverse problems of determining an unknown source term in a parabolic equation are considered. First, the unknown source term is estimated in the form of a combination of Chebyshev functions. Then, a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem. For solving the problem, the operational matrices of integration and derivation are introduced and utilized to reduce the mentioned problem into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. Due to ill-posedness of these inverse problems, the Tikhonov regularization method with generalized cross validation (GCV) criterion is applied to find stable solutions. Finally, some examples are presented to illustrate the efficiency of this numerical method. The numerical results show that the proposed method is a reliable method and can give high accuracy approximate solutions.http://mmr.khu.ac.ir/article-1-2577-en.htmlparabolic equationinverse problemunknown source termtikhonov regularizationchebyshev polynomialsoperational matrix |
| spellingShingle | S Nemati A Babaei S Sedaghat A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems پژوهشهای ریاضی parabolic equation inverse problem unknown source term tikhonov regularization chebyshev polynomials operational matrix |
| title | A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems |
| title_full | A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems |
| title_fullStr | A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems |
| title_full_unstemmed | A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems |
| title_short | A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems |
| title_sort | numerical algorithm based on chebyshev polynomials for solving some inverse source problems |
| topic | parabolic equation inverse problem unknown source term tikhonov regularization chebyshev polynomials operational matrix |
| url | http://mmr.khu.ac.ir/article-1-2577-en.html |
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