Nonlinear dynamic evolution of a novel normalized time-fractional Burgers equation

Abstract

In this paper, a novel normalized time-fractional Burgers equation is proposed to enable a fair computational comparison study of its nonlinear dynamic evolution for various fractional order values. The introduced equation is formulated using a recently developed normalized time-fractional derivative, defined by the unique property that the sum of its weighting coefficients is equal to one. This ensures a well-balanced contribution of fractional terms, resulting in a normalized formulation that allows fair comparison. The classical Burgers equation is a basic partial differential equation (PDE) applied to model many physical phenomena such as traffic flow, gas dynamics and fluid dynamics, while the time-fractional Burgers equation is a modified form incorporating a fractional derivative in time to model diffusion and non-linear wave phenomena with memory effects. These memory effects are essential in accurately representing processes where the current state depends on the entire history of the system. We present several characteristic computational tests to study the effects of the time-fractional order. It is noteworthy that when a small time-fractional order is applied to an oscillatory advection velocity, increasing local maximum values may be observed as time progresses. This observation highlights the impact of the time-fractional order on the progression of the system’s dynamic features and provides valuable insights into how fractional derivatives influence the propagation and interaction of nonlinear waves in systems with memory.