Application Jacobi spectral method for solving the time-fractional differential equation

Our aim from this article is to propose a spectral method for numerically solving a general form of time-fractional differential equation with boundary conditions. Our method is based on the shifted Jacobi polynomials and has four steps as follows. At first, the problem is converted to a new problem with homogeneous conditions (step 1). Then, an integro-differential equation is obtained which is equivalent with the new problem (step 2). Next, all the known and unknown functions which are existing in the integro-differential equation are approximated via the shifted Jacobi polynomials (step 3). Utilizing operational matrices of shifted Jacobi polynomials and collocation method, a system of nonlinear algebraic equations is achieved which is solved by Newton's iterative method (step 4). In order to show and verify the accuracy and performance of the proposed method, we examine our method on some illustrative examples.