A numerical approach for nonlinear time-fractional diffusion equation with generalized memory kernel

In this manuscript, a nonlinear time-fractional diffusion equation with a generalized memory kernel is studied. Initially, the original model problem is linearized by implementing the Newton's quasilinearization technique. In the time-fractional term, a generalized Caputo derivative is considered and approximated using the non-uniform L1-scheme as the solution has a singularity at t = 0. The main contribution of this work is to develop a generalized discrete fractional Gr & ouml;nwall inequality. Thereafter, permitting its use to establish the stability and analyze the error estimate, under a proper regularity condition in the L-2-norm, and an optimal convergence order O(N-(2-zeta)) is obtained for the L1-scheme with respect to the graded mesh. Numerical results are inserted to corroborate the effectiveness of the theoretical analysis.