Abstract
Inverse problems to reconstruct a solution of a time fractional diffusion-wave equation in a cylindrical domain are studied. The equation is complemented by initial and final conditions and partly given boundary conditions. Two cases are considered: (1) full boundary data on a lateral hypersurface of the cylinder are given, but the boundary data on bases of the cylinder are specified in a neighborhood of a final time; (2) boundary data on the whole boundary of the cylinder are specified in a neighborhood of the final time, but the cylinder is either a cube or a circular cylinder. Uniqueness of solutions of the inverse problems is proved. Uniqueness for similar problems in an interval and a disk is established, too.
Highlights
Theoretical study of fractional differential equations that generalize mathematical models of various anomalous processes have been a focus of mathematicians in the last few decades
We have proven the uniqueness for inverse problems to reconstruct solutions of the fractional diffusion-wave equation in a cylinder ω × (0, 1) in two cases: (1)
The initial and final data and full boundary data on lateral hypersurface of the cylinder are given, but the boundary data on the bases of the cylinder are specified in a partial time interval ( T − δ, T ) (IP1); the initial and final data are given and Dirichlet boundary data on the whole boundary of the cylinder are specified in ( T − δ, T ) (IP2)
Summary
Theoretical study of fractional differential equations that generalize mathematical models of various anomalous processes have been a focus of mathematicians in the last few decades. Inverse problems to reconstruct space-dependent parameters of fractional diffusion equations by means of final measurements have been studied in several papers. The author of the present paper proved that a time-dependent factor f (t) of a source term in the fractional diffusion-wave equation is uniquely recovered by the final data [22]. We continue the study of inverse problems with time-dependent unknowns and over-specified final data. We consider two problems for the time fractional diffusion-wave equation in a cylindrical domain. In both problems, the initial and final conditions are given. Problems with final overdetermination and unknown boundary conditions are completely new and have not yet been studied neither in the fractional nor in the classical integer case
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