An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation
We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><sema...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2586 |
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| author | Abdallah El Hamidi Mokhtar Kirane Ali Tfayli |
| author_facet | Abdallah El Hamidi Mokhtar Kirane Ali Tfayli |
| author_sort | Abdallah El Hamidi |
| collection | DOAJ |
| description | We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula> as well as the initial and the boundary conditions. The existence and the continuity with respect to the data of the solution for the direct and the inverse problem are proven by Fourier’s method and the Schauder fixed-point theorem in an adequate convex bounded subset. In the published articles on this topic, the incorrect use of the estimates in the generalized Mittag–Leffler functions is commonly performed. This leads to false proofs of the Fourier series’ convergence to recover the equation satisfied by the solution, the initial data or the boundary conditions. In the present work, the correct framework to recover the decay of fractional Fourier coefficients is established; this allows one to recover correctly the initial data, the boundary conditions and the partial differential equations within the space-time domain. |
| first_indexed | 2024-03-09T12:23:50Z |
| format | Article |
| id | doaj.art-8253b06883f349e5974e6d271c1f6cff |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T12:23:50Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-8253b06883f349e5974e6d271c1f6cff2023-11-30T22:37:28ZengMDPI AGMathematics2227-73902022-07-011015258610.3390/math10152586An Inverse Problem for a Non-Homogeneous Time-Space Fractional EquationAbdallah El Hamidi0Mokhtar Kirane1Ali Tfayli2Laboratoire LaSIE, La Rochelle Université, 17042 La Rochelle, France Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi P.O. Box 127788, United Arab EmiratesLaboratoire LaSIE, La Rochelle Université, 17042 La Rochelle, FranceWe consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>,</mo></mrow></semantics></math></inline-formula> as well as the initial and the boundary conditions. The existence and the continuity with respect to the data of the solution for the direct and the inverse problem are proven by Fourier’s method and the Schauder fixed-point theorem in an adequate convex bounded subset. In the published articles on this topic, the incorrect use of the estimates in the generalized Mittag–Leffler functions is commonly performed. This leads to false proofs of the Fourier series’ convergence to recover the equation satisfied by the solution, the initial data or the boundary conditions. In the present work, the correct framework to recover the decay of fractional Fourier coefficients is established; this allows one to recover correctly the initial data, the boundary conditions and the partial differential equations within the space-time domain.https://www.mdpi.com/2227-7390/10/15/2586inverse problemfractional derivativetime-space fractional equationintegral equationsbiorthogonal system of functionsFourier series |
| spellingShingle | Abdallah El Hamidi Mokhtar Kirane Ali Tfayli An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation Mathematics inverse problem fractional derivative time-space fractional equation integral equations biorthogonal system of functions Fourier series |
| title | An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation |
| title_full | An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation |
| title_fullStr | An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation |
| title_full_unstemmed | An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation |
| title_short | An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation |
| title_sort | inverse problem for a non homogeneous time space fractional equation |
| topic | inverse problem fractional derivative time-space fractional equation integral equations biorthogonal system of functions Fourier series |
| url | https://www.mdpi.com/2227-7390/10/15/2586 |
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