Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations
In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2025-01-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://gc.vgtu.lt/index.php/MMA/article/view/19339 |
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| author | Zewen Wang Shufang Qiu Xiuxing Rui Wen Zhang |
| author_facet | Zewen Wang Shufang Qiu Xiuxing Rui Wen Zhang |
| author_sort | Zewen Wang |
| collection | DOAJ |
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In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations.
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| format | Article |
| id | doaj-art-08ac9457ad264bc0aa4c3e0a222268f3 |
| institution | Kabale University |
| issn | 1392-6292 1648-3510 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj-art-08ac9457ad264bc0aa4c3e0a222268f32025-01-27T16:30:20ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102025-01-0130110.3846/mma.2025.19339Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equationsZewen Wang0Shufang Qiu1Xiuxing Rui2Wen Zhang3School of Arts and Sciences, Guangzhou Maritime University, 510725 Guangzhou, ChinaSchool of Arts and Sciences, Guangzhou Maritime University, 510725 Guangzhou, China; School of Science, East China University of Technology, 330013 Nanchang, ChinaSchool of Science, East China University of Technology, 330013 Nanchang, ChinaSchool of Science, East China University of Technology, 330013 Nanchang, China In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations. https://gc.vgtu.lt/index.php/MMA/article/view/19339numerical differentiationfractional derivativesource inversionhyperbolic equationill-posed problem |
| spellingShingle | Zewen Wang Shufang Qiu Xiuxing Rui Wen Zhang Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations Mathematical Modelling and Analysis numerical differentiation fractional derivative source inversion hyperbolic equation ill-posed problem |
| title | Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| title_full | Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| title_fullStr | Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| title_full_unstemmed | Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| title_short | Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| title_sort | numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations |
| topic | numerical differentiation fractional derivative source inversion hyperbolic equation ill-posed problem |
| url | https://gc.vgtu.lt/index.php/MMA/article/view/19339 |
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