Numerical modeling of a time-fractional Burgers equation

Abstract

A fractional time derivative is introduced into Burgers equation to model losses of nonlinear waves arising in acoustics. A diffusive representation of the fractional derivative replaces the non-local operator by a continuum of memory variables that satisfy local ordinary differential equations. A quadrature formula yields a system of local partial differential equations. The quadrature coefficients are computed by optimization with a positivity constraint. One resolves the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Extensive details can be found in [3].