A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time–space fractional diffusion equation

Abstract

In this analysis, a numerical algorithm in the RKHS approach is utilized to the inverse source problem for the diffusion equation in a time–space fractional sense, where determinations of state variable and source parameter subject to initial–boundary and overdetermination conditions are the main goal. Consequently, specifics theoretical demonstrations are presented to interpret the NPSs to such a fractional problem. In this direction, convergence analysis and error estimates of the developed approach are studied and analyzed as well. Concerning the considered equation, specific unidirectional physical experiments are given in a finite compact regime to confirm the theoretical aspects and to demonstrate the superiority of the utilized approach. Some representative results are presented in two-dimensional graphs, whilst dynamic behaviors of fractional parameters are reported for several fixed α,βvalues. From the practical viewpoint, the archived simulations and consequences justify that the iterative approach is a straightforward and appropriate tool with computational efficiency for numeric solutions of the inverse source problem.

Keywords

• Inverse source problem
• Reproducing kernel Hilbert space
• Fractional diffusion equation
• Fractional partial differential equations
• State variable
• Source parameter