Approximate analytical solutions of time-fractional advection-diffusion equations using singular kernel derivatives: a comparative study

A new singular kernel fractional derivative operator has recently been proposed as a convenient inverse of a fractional integral operator whose kernel is a combination of the power law function and the Mittag-Leffler function. This new operator has properties similar to the Caputo operator in terms of non-locality and singularity of the kernel, while having other features that differ from those of the Caputo operator. This paper is concerned with providing approximate analytical solutions for time-fractional advection-diffusion equations associated with singular kernel fractional derivatives. We expand the scope of application of the homotopy analysis method and then develop an optimal approach of this method to deal analytically with the studied problems. We also introduce some results regarding convergence and error estimates of the used approaches. Then, we present approximate solutions to fractional extensions of some advection-diffusion equations, where the used fractional derivatives include the recently developed singular kernel derivative and the Caputo derivative. Furthermore, we provide a comparison between the behavior of the solutions for the studied problems when using the newly developed derivative and the Caputo derivative.