A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations

Abstract

In this paper, we have considered the fractional diffusion and fractional diffusion-wave equations in which the time derivative is a fractional derivative in the Caputo form and have obtained their numerical solutions by collocation method using cubic B-spline base functions. In the solution process, for the fractional diffusion equation $L1$ discretizaton formula of the fractional derivative is applied, and for the fractional diffusion-wave equation $L2$ discretizaton formula of the fractional derivative is applied. Accuracy of the proposed method is discussed by computing the error norms $L_{2}$ and $L_{\infty}$ . A stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.