Efficient numerical simulations of singular kernel fractional diffusion equations with nonlinear sources

In recent years, a novel type of fractional derivative operator associated with a singular kernel has been proposed as a consistent inverse of an integral operator incorporating a Prabhakar kernel. This operator retains key features of the Caputo derivative, particularly its nonlocal structure and the singular nature of its kernel. The present study aims to provide accurate numerical solutions for singular kernel fractional diffusion equations coupled with nonlinear source terms, such as fractional forms of the Fisher and Newell-Whitehead equations. The considered derivatives include both the newly proposed singular kernel derivative and the classical Caputo derivative. To this end, efficient numerical schemes are constructed based on finite difference discretization and predictor–corrector techniques to simulate the behavior of the proposed models. The schemes are formulated over uniform spatio-temporal grids, and their stability properties are carefully analyzed. Numerical approximations and three-dimensional visualizations are presented to confirm the reliability and performance of the developed approaches. In addition, a numerical comparison is performed between the solutions obtained using the newly formulated singular kernel fractional derivative and those derived from the classical Caputo derivative, highlighting the distinct effects of each operator on the dynamic behavior of the models.