Solving fractional diffusion and fractional diffusion-wave equations by Petrov-Galerkin finite element method

In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms [L.sub.2] and [L.sub.∞] have been calculated for testing the accuracy of the proposed scheme. Keywords: Finite element method, Petrov-Galerkin method, Fractional diffusion equation, Fractional diffusion-wave equation, Quadratic B-Spline, Linear B-Spline. AMS Subject Classification: 97N40, 65D07, 74S05, 26A33, 34A08, 65L60