Volume 10, Issue 6
Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations

Adv. Appl. Math. Mech., 10 (2018), pp. 1440-1458.

Published online: 2018-09

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• Abstract

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.

• Keywords

Spectral method, nonlinear, fractional derivative, Volterra integro-differential equations, Caputo derivative.

65R20, 45J05, 65N12

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@Article{AAMM-10-1440, author = {Xingfa and Yang and and 19056 and and Xingfa Yang and Yin and Yang and and 19057 and and Yin Yang and Yanping and Chen and and 19058 and and Yanping Chen and Jie and Liu and and 19059 and and Jie Liu}, title = {Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {1440--1458}, abstract = {

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} }
TY - JOUR T1 - Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations AU - Yang , Xingfa AU - Yang , Yin AU - Chen , Yanping AU - Liu , Jie JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1440 EP - 1458 PY - 2018 DA - 2018/09 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0038 UR - https://global-sci.org/intro/article_detail/aamm/12718.html KW - Spectral method, nonlinear, fractional derivative, Volterra integro-differential equations, Caputo derivative. AB -

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.

Xingfa Yang, Yin Yang, Yanping Chen & Jie Liu. (1970). Jacobi Spectral Collocation Method Based on Lagrange Interpolation Polynomials for Solving Nonlinear Fractional Integro-Differential Equations. Advances in Applied Mathematics and Mechanics. 10 (6). 1440-1458. doi:10.4208/aamm.OA-2018-0038
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