Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

In this paper, we propose an efficient numerical algorithm for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel. In time direction, we use second-order finite difference schemes to discretize the Caputo fractional derivative and also singular integral term. To achieve fully discrete scheme, we apply Galerkin method using generalized Jacobi polynomials as basis, which satisfy essentially all the underlying homogeneous boundary conditions. The proposed method is fast and efficient due to the resulting sparse coefficient matrices. We investigate the error estimate and prove that the method is convergent. Numerical results show the high accuracy and low CPU time of proposed method and confirmed the theoretical ones. Second-order accuracy in time direction and spectral accuracy in space component are also numerically demonstrated by some test problems. Finally we compare the numerical results with the results of other recently methods developed in literature.