Numerical simulation of blow-up solution to the Caputo-Hadamard fractional differential equation

In this paper, the nonlinear initial-boundary value problem with Caputo-Hadamard fractional derivative is studied numerically, where the L1 formula is used in time direction and the moving mesh finite element method is applied to spatial direction. The fully discrete scheme for the nonlinear problem is obtained by selecting a suitable projection, which keeps the convergence accuracy of the numerical scheme well. Finally, several numerical examples are displayed to arrive at some numerical results such as errors and convergence order in different norms to verify the validity of the numerical scheme, and then the corresponding blow-up solutions for the nonlinear problem are simulated successfully.