An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

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.

The current work analyzes the transmission behavior of plane harmonic waves in an isotropic medium. The observation is made for homogeneous type solid in the context of generalized dual phase lag model of thermoelasticity. Concept micro-temperature, where the microelements have different temperatures has also been considered. The basic focus of thework is to predict the influence of initially applied magnetic field on plane waves through the elastic solid. We have made an attempt to find exact solution of the problem using an analytical technique of a normal mode analysis method. The theoretical results are obtained for a generalized solid in order to test the numerical calculation of a magnesium crystal. It is found that the magnetic field reduces the strength of the attenuation factor.