An efficient parametric finite difference and orthogonal spline approximation for solving the weakly singular nonlinear time-fractional partial integro-differential equation

In this paper, a numerical method is presented to solve a nonlinear weakly singular time-fractional partial integro-differential equation with Caputo fractional derivative. An orthogonal basis of spline space called O-spline is introduced, which is used for spatial approximations. Also, an approximation for the temporal Riemann-Liouville integral of the function is presented. A new parametric approximation is developed for the temporal Caputo fractional derivative of the function. Finally, using the weighted finite difference method, a numerical time scheme is obtained to approximate the equation. The convergence of this numerical scheme is investigated and some numerical examples are provided to illustrate the accuracy and efficiency of the method.