Analysis and numerical solution of a nonlinear variable-order fractional differential equation

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.