Chapter 5: Numerical Methods for Solving Fractional PDEs in the Book: Fractional Modelling of Fluid Flow and Transport Phenomena

Abstract

This chapter provides an extensive review of numerical methods tailored for solving fractional-order ordinary differential equations (ODEs), fractional partial differential equations (FPDEs), and systems of such equations. We discuss several numerical schemes, including explicit and implicit finite difference, Galerkin and mixed finite element, spectral element methods, and meshless methods, highlighting their application to both time and space fractional-order PDEs. Starting with the adaptation of classical numerical techniques, such as the Euler method and Runge-Kutta methods, to the fractional calculus framework, demonstrating their effectiveness through the introduction of fractional versions like the Fractional Forward Euler Method and Explicit Fractional Order Runge-Kutta (EFORK) methods, through examples and pseudocodes.