We solve the one- and two-dimensional fractional coupled Burgers’ equations (FCBEs) by three different methods. The proposed methods are the Laplace–Adomian decomposition method (LADM), the Laplace-variational iteration method (LVIM) and the reduced differential transform method (RDTM). The solutions are obtained as rapidly convergent series with simply calculable terms. Numerical studies of the application of theseapproaches for a number of sample problems are given and are illustrated graphically. With these methods, it is possible to investigate the nature of solutions when the fractional derivative parameters are changed. The numerical results reveal the effectiveness and the correctness of the proposed methods.

This paper declares a new spectral collocation technique to provide accurate approximate solutions of the one- and two-dimensional time and space fractional coupled Burgers’ equations (TSFCBEs) in which the fractional derivatives are defined according to Caputo’s definition. The suggested method is based on the shifted Gegenbauer polynomials (SGPs) for approximating the solution of TSFCBEs. The suggested technique reduces the considered problems to the solution of nonlinear algebraic equations (NLAEs). Moreover, the accuracy and reliability of the proposed method are confirmed through numerical examples. Finally, the obtained numerical results are compared with those previously reported in the literature.