Inverse Problems for a Time-Fractional Diffusion Equation with Unknown Right-Hand Side

Durdiev D.K. Sultanov M.A. Rahmonov A.A. Nurlanuly Y.
2023 Natural Sciences Publishing

Progress in Fractional Differentiation and Applications
2023 #9 Issue 4 639 - 653 pp.

This paper is devoted to forward and inverse source problems for a 2D in space variables time-fractional diffusion equation. The forward problem is an initial-boundary/value problem for given equation in a rectangular area. In inverse problem the unknown right side of the equation is assumed to have the form of a product of two functions: one of which depends only on the time variable, while the other one - depends on the spatial variables. Two inverse problems of finding these functions separately under the condition that the other function is known are investigated. For the inverse problem for determining a time-dependence function, Abel’s integral equation of the first kind is obtained, which is further reduced to an integral equation of the second kind with the application of fractional differentiation to it. To solve the direct problem and inverse problem of determining a spatial-dependence function, the Fourier spectral method is used. Theorems of unique solvability of the formulated problems are proved. The existence and uniqueness results are based on the Fourier method, fractional calculus and properties of Mittag-Lefller function.

Bessel inequality , Fourier coefficient , Fourier spectral method , Gerasimov-Caputo fractional derivative , Inverse problem

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Bukhara Branch of the Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, Bukhara, 100170, Uzbekistan
Department of Mathematics, Faculty of Natural Science, Khoja Akhmet Yassawi International Kazakh, Turkish University, Turkestan, 161200, Kazakhstan
V.I. Romanovskii Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100174, Uzbekistan
Khoja Akhmet Yassawi International Kazakh, Turkish University, Turkestan, 161200, Kazakhstan

Bukhara Branch of the Institute of Mathematics
Department of Mathematics
V.I. Romanovskii Institute of Mathematics
Khoja Akhmet Yassawi International Kazakh

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