NUMERICAL SOLUTION OF TWO-DIMENSIONAL NONLINEAR FRACTIONAL DIFFUSION TYPE PROBLEMS WITH ATANGANA–BALEANU–CAPUTO TIME-FRACTIONAL DERIVATIVE

An efficient numerical scheme is presented to simulate the two-dimensional time fractional diffusion type equations with Atangana–Baleanu–Caputo (ABC) derivative. In this methodology, the Haar wavelets (HWs) are coupled with a numerical quadrature formula for time fractional ABC derivative. In the first step, time fractional ABC derivative is estimated with quadrature rule and then, an implicit formulation is adopted. Thereafter, the derivative is estimated by truncated HW series which on integration leads to the lower-order derivatives and solution. Afterward, the collocation strategy is implemented to transmute the fractional equation to the set of linear algebraic equations. Solving the resultant system gives the wavelet coefficients which refine the solution and derivatives for the iterative process. Moreover, the numerical stability of the proposed algorithm is presented theoretically, and verified computationally. The scheme is tested for five test problems which include, three linear, and two nonlinear. The accuracy of the scheme is elucidated via [Formula: see text], and [Formula: see text] error norms. Computations disclose that the computed results are comparable to the closed-form solution of the problems.