An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

Abstract

In this work, we investigate unique solvability of inverse source problem (ISP) of determining a time‐dependent source term along with solution for a multi‐term fractional differential equation involving two‐parameter fractional derivative in time usually known as Hilfer fractional derivative. We applied the eigenfunction expansion method, and the corresponding spectral problem consists of Bessel differential equation in space variable. The multi‐term fractional order differential equations are solved by Laplace transform and solution involve multinomial Mittag–Leffler type functions. Banach fixed‐point theorem is used to prove unique existence of time‐dependent source term whenever integral type over‐determination condition is given. Some examples are provided to support our analysis.

On the solution of multi-term time fractional diffusion-wave equation involving ultra-hyperbolic operator

“…Single term fractional derivative (FD) has been considered by many researchers. Multi-term fractional derivatives have been considered in [11][12][13], because it is widely believed to violate the causality and lead to the instability yielding that the phenomena under consideration are not deterministic in a usual physical sense. In theoretical physics Bars [14], Sparling [15] and Tegmark [16] studied the string theory require additional dimensions for the time.…”