On the shock wave approximation to fractional generalized Burger–Fisher equations using the residual power series transform method

Abstract

The time-fractional generalized Burger–Fisher equation (TF-GBFE) has various applications across various scientific and engineering disciplines. It is used for investigating various phenomena, including the dynamics of fluid flow, gas dynamics, shock-wave formation, heat transfer, population dynamics, and diffusion transport, among other areas of research. By incorporating fractional calculus into these models, researchers can more effectively represent the non-local and memory-dependent effects frequently observed in natural phenomena. Due to the importance of the family of TF-GBFEs, this work introduces a changed iterative method for analyzing this family analytically to gain a deep understanding of many nonlinear phenomena described by this family (e.g., shock waves). The proposed approach combines two algorithms: the Laplace transform and the residual power series method. The suggested technique is thoroughly discussed. Two numerical problems are discussed to check the effectiveness and accuracy of the proposed method. The approximations for integer and fractional orders are compared with the exact solution for integer-order problems. Finally, to investigate how the fractional order affects these problems, the obtained results are discussed graphically and numerically in the tables.