Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Abstract In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α ( x , t ) ∈ ( 0 , 1 ) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model.