Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we study a class of nonlinear fractional integro-differentialequations. The fractional derivative is described in the Caputo sense. Using the propertiesof the Caputo derivative, we convert the fractional integro-differential equationsinto equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinearintegro-differential equations of Volterra type. We provide a rigorous error analysis forthe spectral methods, which shows that both the errors of approximate solutions andthe errors of approximate fractional derivatives of the solutions decay exponentially in $L^∞$-norm and weighted $L^2$-norm.