Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems

Mikusinski's operational calculus has proven to be a powerful tool for tackling various fractional order differential equations. Notably, it has been extended to encompass the Hilfer-Prabhakar fractional order operator. By employing this operational calculus, we have been able to recover and connect previously derived results pertaining to the Hilfer, Riemann-Liouville, and Caputo operators. Furthermore, we address two inverse problems that involve determining space and time dependent source terms and diffusion concentration. Using our operational calculus results, we obtain series solutions for these inverse problems. It is essential to note that the inverse problems belong to the class of ill-posed problems according to Hadamard's definition. To establish the validity and uniqueness of the solutions to both inverse problems, we rigorously prove the existence and uniqueness results. Additionally, we provide specific examples of inverse problems, thereby illustrating the practical applications and versatility of the operational calculus approach.