An indirect collocation method for variable-order fractional wave equations on uniform or graded meshes and its optimal error estimates

Abstract

We develop an indirect collocation method for a variable-order fractional wave equation. Fractional differential equations are well known to exhibit initial weak singularity, which makes it unrealistic to carry out error estimates of their numerical approximations based on the smoothness assumptions of the true solutions. In this paper, we analyze the convergence behaviour of the method without artificially assuming the (often untrue) full regularity of the true solution of the problem, but only based on the behaviour of the coefficients and variable order of the problem. More precisely, we prove the following results: (i) If the variable order has an integer initial value, the method discretized on a uniform partition has an optimal-order convergence rate in the $L_{\mathrm{\infty }}$ norm. (ii) Otherwise, the method discretized on a uniform mesh has only a sub-optimal order convergence rate. The method discretized on a graded mesh with the mesh grading parameter determined by the initial value of the variable order has an optimal-order convergence rate in the $L_{\mathrm{\infty }}$ norm. Numerical experiments are performed to substantiate the theoretical results.

Keywords:

• Variable order
• fractional wave equation
• collocation method
• graded mesh
• optimal-order error estimate

2010 Mathematics Subject Classifications:

• 65L20
• 65L60

Acknowledgments

The authors would like to express their most sincere thanks to the editors and the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was partially funded by the ARO MURI Grant W911NF15-1-0562, by the National Science Foundation under Grant DMS-2012291, and by the National Natural Science Foundation of China under Grant 12071262.

Disclosure statement

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

Additional information

Funding

This work was supported by Army Research Office [ARO MURI Grant W911NF-15-1-0562] and National Natural Science Foundation of China [12071262] and National Science Foundation [DMS-2012291].