Analysis of Legendre pseudospectral approximations for nonlinear time fractional diffusion-wave equations

Abstract

A finite difference/pseudospectral scheme is developed for solving nonlinear time fractional parabolic equations with Caputo fractional derivative of order $1<\alpha <2$. The boundedness and unique solvability of numerical solution are given. Then we prove rigorously the unconditional stability and convergence of the fully discrete scheme, where the optimal error estimate in $H^1$ norm is obtained. Furthermore, an improved scheme by adding correction terms is proposed to deal with the weak singularity, which makes the approximations of fractional derivative and nonlinear term exact or sufficiently accurate for the weak singular parts of solutions. Numerical experiments are provided to show the sharpness of the error analysis.

Keywords:

• Nonlinear fractional diffusion-wave equation
• pseudospectral method
• unique solvability
• stability and convergence
• weak singularity

2010 AMS Subject Classifications:

• 35R11
• 65M70
• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11672011, 11272024] and Scientific Research Funds of Beijing Information Science and Technology University [grant number 5222010952].