Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method

In this chapter, we present the time-space-fractional Cauchy equation with constant coefficients, the space and time-fractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. The implicit scheme is introduced to solve time-space-fractional Cauchy problem in a matrix form by utilising fractionally Grunwald formulas for discretization of Riemann-Liouville fractional integral, and L1-algorithm for the discretization of time-Caputo fractional derivative, additionally, we provided a proof of the von Neuman type stability analysis for the fractional Cauchy equation of fractional order. Several numerical examples are introduced to illustrate the behaviour of approximate solution for various values of fractional order.