Front Propagation in a Fractional Viscoelastic Rod Model: Control of the Minimal Speed by the Rouse Exponent and Its Slow Approach

Abstract

We consider a Caputo-fractional nonlinear telegraph equation with a Rouse memory exponent  as a model of strain fronts in a viscoelastic rod. A closed-form hyperbolic-tangent kink of this model, obtained with the -expansion method, predicts a front speed of 451 m/s. We study front propagation in this model by direct numerical solution of the Caputo equation. From localised (step) data the front speed rises towards a minimal speed  given in closed form by a leading-edge argument; at  it is  in units of the elastic wave speed  m/s, i.e. 12.5 m/s. This is about 36 times lower than the kink value and 34 times lower than the classical ( ) minimal speed of 427 m/s. The approach to  is slow: the speed is 42% of  at , 90% at  and 97% at , with a shortfall of the order  (within 20% of it for ). The front widens to about 90 length units. The speed does not depend on the cubic coefficient and so not on the strain amplitude, and its elasticity with respect to  is 7.4 at : a 1% error in the memory exponent changes the speed by about 7%. Measurements over times much shorter than  in these units would therefore report a speed far below the asymptotic one.