Solution Dynamics of the Time-Tempered Fractional Burgers-Like Equation via an Analytical Tempered Laplace Residual Power Series Method

This work discusses the time-tempered fractional Burgers’ equation in the context of Caputo tempered fractional derivative. This equation is crucial for modeling non-Newtonian fluid flow and long-memory phenomena that involves a time-tempering effect. In this paper, we employ a novel approach that combines Laplace transformation and the residual power series method to obtain approximate analytical solutions for the time-tempered fractional Burgers’ equation, the tempered Laplace residual power series method. The proposed method is not an incremental extension of existing Laplace-based approaches; it requires a fundamentally new expansion structure and original convergence theory and yields solutions that encode the tempering dissipation effect intrinsically through a closed-form exponential factor. The results obtained during the work are presented in two- and three-dimensional figures, in addition to tables showing the exact numerical results for comparison and a study of the relative and absolute errors of these solutions. In this paper, we explore the impact of the fractional derivative on the solutions obtained through our method and provide an explanation for these effects. We present the results at various parameter values in the governing equation to examine the resulting changes in these parameters. We also show real-world examples related to the time-tempered fractional Burgers’ equation to demonstrate how useful this study is for modeling complex patterns. The modeling of intricate fluid mechanics phenomena and the creation of novel numerical algorithms to control these models are made possible by this work. Additionally, it illustrates the significance of fractional differential equations in several scientific domains, including engineering and physics.