Time Fractional Burgers’ Equation

Abstract

The Burgers' equation is a time-dependent, nonlinear partial differential equation that finds applications in various scientific and technical domains, such as heat conduction, traffic flow, gas dynamics, and non-linear acoustics. This research article provides an overview of Burgers' equations, including their basic form and the time fractional form. The conditions under which exact solutions are available are discussed, and an approximation analytic Adomian approach is introduced to address the standard Burgers' equation.

The paper highlights the challenges in numerically solving the Burgers' equations due to their nonlinear behavior and low viscosity. Analytical solutions require infinite series with slow convergence for low viscosity coefficients. Consequently, numerical approaches are of significant interest for obtaining solutions to the Burgers' equation.

Furthermore, the article discusses how Burgers' equations can be derived from more complex models, making them commonly referred to as toy models. Examples of such simplifications are presented, and the concept of the outer product of vectors is introduced to aid in the understanding of Burgers' equations.