An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation

Abdallah El Hamidi, Mokhtar Kirane and Ali Tfayli
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Abdallah El Hamidi: Laboratoire LaSIE, La Rochelle Université, 17042 La Rochelle, France
Mokhtar Kirane: Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi P.O. Box 127788, United Arab Emirates
Ali Tfayli: Laboratoire LaSIE, La Rochelle Université, 17042 La Rochelle, France

Mathematics, 2022, vol. 10, issue 15, 1-17

Abstract: 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 t ∈ ( 0 , T ] , 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.

Keywords: inverse problem; fractional derivative; time-space fractional equation; integral equations; biorthogonal system of functions; Fourier series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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