Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations

This chapter is concerned with analyzing and presenting some algorithms for treating some kinds of fractional differential equations (FDEs) based on utilizing some suitable spectral methods. A Galerkin method is employed for solving time fractional telegraph equation. This method depends on choosing an appropriate basis functions satisfying the underlying boundary conditions. A double shifted Legendre expansion is proposed as an approximating polynomial. In addition, the two spectral methods, namely, Petrov–Galerkin and collocation methods are applied for obtaining spectral solutions of space fractional linear diffusion problem. The two suggested algorithms are built on using a certain double shifted Legendre basis. The philosophy of the application of spectral methods depends on transforming the problem with its boundary/initial conditions into a system of equations, which can be solved with suitable solvers. Investigation for the convergence and error analysis of the two suggested approximate double expansions are performed. Some numerical results are provided aiming to ensure the efficiency and applicability of the proposed algorithms.