Chebyshev-Legendre Spectral Approach in Fractional Integro-Differential Equations

Objectives: In the current research we implement Chebyshev-Legendre spectral approach for a class of linear fractional integro-differential equations (FIDE) including those of Fredholm and Volterra types via initial conditions or two-point boundary value conditions. Methods: An equations derived that characterizes the approximation solution by using the shifted Legendre and shifted Chebyshev collocation points. By providing various numerical instances, the precision and efficiency of the developed algorithms were evaluated. We show that for a class of equations, we obtain the exact solutions. Results: Some particular linear solvers were presented for the linear and nonlinear FIDE with initial conditions by applying downshifted Legendre polynomials. The fractional derivatives were defined in the Caputo sense. Furthermore, a novel method developed by executing downshifted Legendre procedure in union with the downshifted Chebyshev collocation way for the solution of FIDE. Based on our research, the presented research is the first investigation regarding the Chebyshev-Legendre spectral approach for determining FIDE with initial conditions. Conclusion: In the current paper, A numerical algorithm was proposed to determine the overall equations, utilizing Gauss-collocation features and estimated the solution directly applying the downshifted Legendre polynomials.