A novel finite difference based numerical approach for Modified Atangana-Baleanu Caputo derivative

In this paper, a new approach is presented to investigate the time-fractional advection-dispersion equation that is extensively used to study transport processes. The present modified fractional derivative operator based on Atangana-Baleanu's definition of a derivative in the Caputo sense involves singular and non-local kernels. A numerical approximation of this new modified fractional operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it has been proved that the proposed scheme is unconditionally stable. Numerical examples are solved that validate the theoretical results presented in this paper and ensure the proficiency of the numerical scheme.

Automated summary

This paper presents a new approach to investigate the time-fractional advection-dispersion equation extensively used in studying transport processes. A new modified fractional derivative operator, based on Atangana-Baleanu's definition of a derivative in the Caputo sense, is proposed, involving singular and non-local kernels. A numerical approximation of this new operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it is proved that the proposed scheme is unconditionally stable. Numerical examples validate the theoretical results and demonstrate the proficiency of the numerical scheme.