A numerical method for nonlinear fractional reaction–advection–diffusion equation with piecewise fractional derivative

This study introduces a new fractional version of the nonlinear reaction–advection–diffusion equation using a kind of piecewise fractional derivatives defined by Atangana and Araz. A hybrid approach using the Chebyshev cardinal functions and piecewise Chebyshev cardinal functions is established for finding a solution to this equation. The presented method transforms solving the generated fractional problem into finding the solution of a nonlinear algebraic system by expanding the solution of the problem in terms of the mentioned basis functions and employing the piecewise fractional derivative matrix of the piecewise Chebyshev cardinal functions (which is derived in this study). The accuracy of the constructed algorithm is checked in some illustrative examples.