Unconditionally convergent and superconvergent finite element method for nonlinear time-fractional parabolic equations with distributed delay

In this paper, we propose an efficient Newton linearized numerical method for the nonlinear time-fractional parabolic equations with distributed delay based on the Galerkin finite element method in space and the nonuniform L1 scheme in time. The term of distributed delay is approximated by using the compound trapezoidal formula. For the constructed numerical scheme, we mainly focus on the unconditional convergence and superconvergence without any time-space ratio restrictions, the key of which is the use of fractional discrete Gronwall inequality and time-space error splitting technique. Numerical tests for several biological models, including the fractional single-species population model with distributed delay, the fractional diffusive Nicholson's blowflies equation with distributed delay, and the fractional diffusive Mackey-Glass equation with distributed delay, are conducted to confirm the theoretical results. Finally, combined with the nonunifom Alikhanov scheme in time and the FEM in space, we extend a higher-order Newton linearized numerical scheme for the nonlinear time-fractional parabolic equations with distributed delay and give some numerical tests for some biological models.