Journal of the Ramanujan Mathematical Society

Volume 41, Issue 2, June 2026 pp. 113–123.

Inverse problem of determining the source function in a fractional diffusion-wave equation with variable coefficients

Authors: D. K. Durdiev
Author institution:Bukhara Branch of Romanovskii Institute of Mathematics, Uzbekistan Academy of Sciences, Bukhara, Uzbekistan. Bukhara State University, Bukhara, 705018 Uzbekistan.

Summary: In this paper, we study the inverse problem of finding a time-dependent multiplier of the right-hand side of a one-dimensional fractional time diffusion-wave equation with variable coefficients. The direct problem is an initial-boundary value one with the usual Cauchy conditions, homogeneous Dirichlet boundary conditions for this equation. The overdetermination condition has the form of an integral over a spatial segment from the solution of the direct problem, in which the weight function is a spatially dependent known factor of the right-hand side of the equation. This made it possible to construct a solution to the inverse problem in explicit form and prove its correctness in the class of regular solutions. Although we consider only the one-dimensional model, the analysis and computation in this part can be extended into the general multi-dimensional case, upon suitable modifications.

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