An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

• Main Authors: Nataliia Kinash, Jaan Janno
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Mathematics
• Subjects: inverse problem; source reconstruction; final overdetermination; subdiffusion; tempered subdiffusion; fractional wave equation; generalized fractional derivative; atangana–baleanu derivative
• Online Access: https://www.mdpi.com/2227-7390/7/12/1138

In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function <i>u</i> based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.