Shifted Jacobi spectral-Galerkin method for solving fractional order initial value problems

Abstract

In this paper, the shifted Jacobi spectral-Galerkin method is introduced to deal with fractional order initial value problems (FIVPs). In the proposed method, the exact solution of the FIVP is approximated by using shifted Jacobi polynomials on each subinterval of the total time. The main advantage of the proposed method is that the rate of convergence in L2-norm depends on the local smoothness of solution. This enables the proposed method to work for nonsmooth solutions. The second merit is that the flexibility between the length of sub-interval and the degree of polynomial significantly enhances the numerical accuracy. We derive the spectral-Galerkin approximation formula for the Volterra integral equation of the second kind which is equivalent to the FIVP. Error analysis in L2-norm for the case α=β=0 in shifted Jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is Lipschitz continuous in the second argument. Numerical illustrations for multi-order, linear, and nonlinear FIVPs are demonstrated to confirm the convergence rate.