Inverse problems for a multi-term time fractional evolution equation with an involution
Asim Ilyas, Salman A. Malik, Summaya Saif
Abstract
This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.