Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

Abstract

In this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.MSC 2010 : Primary 26A33; Secondary 34A08.

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

“…Next, numerical results with different scale functions z (t) = t 0.5 , t , t 2 and t 5 are shown in Figure 3 when T = 1. Denoting z (t) = t s , 0 ≤ t ≤ 1, scale function z (t) behaves like a "contracting function" for the time variable over domain [0, 1] when s > 1, while behaving like a "stretching function" as s < 1 (see [25]). From Figure 3, when s = 0.5, decomposing and diffusion phenomena appeared earlier in contrast with the case when s = 1, just like being contracted.…”

“…Similarly, for one function containing two variables, that is, u(x, t), its (left) Caputo-type generalized fractional partial derivative with respective to t is defined as [25] …”

“…Taking a scale function and a weight function into consideration, fractional integrals and derivatives were essentially generalized. Partial differential equations with such generalized fractional derivatives are not studied much [25][26][27][28]. Therefore, in this paper, we choose the KdV equation to further study this topic.…”

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

“…Other properties, such as integration by parts formulae, and the application to Burgers equation of these generalized fractional integrals and generalized fractional derivatives could be found in [25,30,31]. Note that the generalized fractional integrals include both the left-and the right-sided fractional integrals, and the generalized fractional derivatives include the left-and the right-sided Riemann-Liouville and Caputo fractional derivatives.…”

Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow

“…The solution strategies employed homotopy analysis [27] and Adomian decomposition methods [28] to solve the space-and time-fractional versions of the fractional BE. In this context, the clas-sic finite difference method was proposed to solve the generalized FBE [29]. Furthermore, the variational iter-ation method (VIM) was successfully applied for tak-ing the Burgers' flows with fractional derivatives [30].…”