Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional order integro-differential equation

Abstract

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [−1,1] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.