An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation.

• Mittag-Leffler kernel derivative. • Advection-reaction-diffusion equation with fractional order. • fractional differentiation. • Legendre operational matrix computations. This paper presents the fractional formulation and numerical solution of a non-linear fractional diffusion equation with advection and reaction terms. The fractional derivative employed in this equation is of non-singular type with Mittag-Leffler kernel. To investigate this fractional equation numerically, we adopted a collocation method with the Legendre operational matrix. The numerical value of Atangana-Baleanu Caputo (ABC) fractional differentiation of polynomial function f (ϑ) = ϑ k is determined by using the definition of Mittag-Leffler kernel fractional differentiation. The operational matrix of fractional differentiation of Mittag-Leffler kernel fractional derivative is determined numerically employing a polynomial function. Within the above context, we solve the fractional equation in-hand by using this matrix and using the collocation method based upon Legendre polynomial. The effectiveness of these techniques is illustrated through different cases that include fractional diffusion, advection-diffusion, and reaction-diffusion equation. The feasibility of the proposed derived ABC fractional differentiation operational matrix is clearly demonstrated by validating the numerical resolutions against exact solutions. The results demonstrated that the method is capable of resolving the non-linear fractional diffusion equation with different fractional order. [ABSTRACT FROM AUTHOR]