Taiwanese Journal of Mathematics

Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions

Haiyu Liu, Shujuan Lü, and Hu Chen

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A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 981-1000.

Received: 9 February 2018
Revised: 3 September 2018
Accepted: 9 September 2018
First available in Project Euclid: 18 July 2019

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Zentralblatt MATH identifier

Primary: 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods 65M70: Spectral, collocation and related methods 35R11: Fractional partial differential equations

nonlinear fractional delay equations spectral method stability and convergence nonsmooth solutions correction terms


Liu, Haiyu; Lü, Shujuan; Chen, Hu. Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions. Taiwanese J. Math. 23 (2019), no. 4, 981--1000. doi:10.11650/tjm/180901. https://projecteuclid.org/euclid.twjm/1563436877

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