Adv. Appl. Math. Mech., 10 (2018), pp. 1440-1458.
Published online: 2018-09
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In this paper, we study a class of nonlinear fractional integro-differential equations. The fractional derivative is described in the Caputo sense. Using the properties of the Caputo derivative, we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel, then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra type. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in $L^∞$-norm and weighted $L^2$-norm.
}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0038}, url = {http://global-sci.org/intro/article_detail/aamm/12718.html} }In this paper, we study a class of nonlinear fractional integro-differential equations. The fractional derivative is described in the Caputo sense. Using the properties of the Caputo derivative, we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel, then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra type. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in $L^∞$-norm and weighted $L^2$-norm.