Taiwanese Journal of Mathematics

A Fully Discrete Spectral Method for the Nonlinear Time Fractional Klein-Gordon Equation

Hu Chen, Shujuan Lü, and Wenping Chen

Full-text: Open access

Abstract

The numerical approximation of the nonlinear time fractional Klein-Gordon equation in a bounded domain is considered. The time fractional derivative is described in the Caputo sense with the order $\gamma$ ($1 \lt \gamma \lt 2$). A fully discrete spectral scheme is proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in $H^1$ norm is $\mathrm{O}(\tau^{3-\gamma} + N^{1-m})$, where $\tau$, $N$ and $m$ are the time-step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to support the theoretical results.

Article information

Source
Taiwanese J. Math., Volume 21, Number 1 (2017), 231-251.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874565

Digital Object Identifier
doi:10.11650/tjm.21.2017.7357

Mathematical Reviews number (MathSciNet)
MR3613982

Zentralblatt MATH identifier
1357.35281

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

Keywords
fractional Klein-Gordon equation fully discrete spectral method stability convergence

Citation

Chen, Hu; Lü, Shujuan; Chen, Wenping. A Fully Discrete Spectral Method for the Nonlinear Time Fractional Klein-Gordon Equation. Taiwanese J. Math. 21 (2017), no. 1, 231--251. doi:10.11650/tjm.21.2017.7357. https://projecteuclid.org/euclid.twjm/1498874565


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