## Taiwanese Journal of Mathematics

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

#### Abstract

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

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

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

https://projecteuclid.org/euclid.twjm/1563436877

Digital Object Identifier
doi:10.11650/tjm/180901

Mathematical Reviews number (MathSciNet)
MR3982070

Zentralblatt MATH identifier
07088956

#### Citation

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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