Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics

Physical applications involving time-fractional derivatives are reflecting some memory characteristics. These inherited memories have been identified as a homotopy mapping of the fractional-solution into the integer-solution preserving its physical shapes. The aim of the current work is threefold. First, we present a new technique which is constructed by combining the Laplace transform tool with the residual power series method. Precisely, we provide the details of implementing the proposed method to treat time-fractional nonlinear problems. Second, we test the validity and the efficiency of the method on the temporal-fractional Newell–Whitehead–Segel model. Then, we implement this new methodology to study the temporal-fractional (1+1)-dimensional Burger’s equation and the Drinfeld–Sokolov–Wilson system. Further, for accuracy and reliability purposes, we compare our findings with other methods being used in the literature. Finally, we provide 2-D and 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.