An efficient difference scheme based on cubic B-spline Quasi-interpolation for a time fractional partial integro-differential equation with a weakly singular kernel

In this paper, a difference scheme based on cubic B-spline Quasi--interpolation has been derived for the solution of a time fractional partial integro-differential equation with a weakly singular kernel. The fractional derivative of the mentioned equation has been described in the Riemann-Liouville sense. Time fractional derivative is approximated by a scheme of order $O(\tau^{2})$. The spatial second derivative has been approximated using the second derivative of the cubic B-spline Quasi--interpolation. We show that the proposed scheme is uniquely solvable, stable and convergent. It has also been shown that the method is the convergence of order $O(\tau^{\frac{3}{2}}+h^{2})$. Numerical experiments are provided to verify the effectiveness of the proposed method.