Modeling nonlinear fractional diffusion-wave equations with Burgers-type evolution equations

Abstract

Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions under which the FPDE can be modeled accurately with a Burgers-type evolution equation for progressive wave motion. The method of lines in combination with a general Runge-Kutta method for forward integration is used for numerical analysis. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions ...