Abstract
We prove the wellposedness of a nonlinear variable-order fractional differential equation and the regularity of its solutions. The regularity of the solutions is determined solely by the values of the variable order and its high-order derivatives at time t = 0 (in addition to the usual regularity assumptions on the variable order and the coefficients). If the variable-order reduces to an integer order at t = 0, then the solution has full regularity as the solution to a first-order ordinary differential equation. In this case, we prove that the corresponding finite difference scheme discretized on a uniform mesh has an optimal-order convergence rate. However, if the variable order does not reduce to an integer order at t = 0, then the solution has a singularity at time t = 0, as Stynes et al. proved in [15] for the constant-order time-fractional diffusion equations. The corresponding finite difference scheme discretized on a uniform mesh has only a suboptimal-order convergence rate. Instead, we prove that the finite difference scheme discretized on a graded mesh determined by the value of the variable order at time t = 0 has an optimal-order convergence rate in terms of the number of the time steps. Numerical experiments substantiate these theoretical results.
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The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.
Funding
This work was funded by the OSD/ARO under MURI Grant W911NF-15-1-0562 and the National Science Foundation under Grant DMS-1620194.
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Communicated by: Martin Stynes
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Wang, H., Zheng, X. Analysis and numerical solution of a nonlinear variable-order fractional differential equation. Adv Comput Math 45, 2647–2675 (2019). https://doi.org/10.1007/s10444-019-09690-0
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DOI: https://doi.org/10.1007/s10444-019-09690-0