A model of diffusive waves in viscoelasticity based on fractional calculus

The partial differential equation of diffusion is generalized by replacing the first time derivative by a fractional derivative of order alpha. This generalized equation is shown to govern the propagation of stress waves in viscoelastic solids, which exhibit a power law creep of degree p with 0 < p < 1, provided that 1 < alpha = 2-p < 2. For the basic Cauchy and Signaling problems the corresponding Green functions are expressed in terms of an entire function for which integral and series representations are provided. Numerical results are presented which show the transition from a pure diffusion process (alpha = 1) to a pure wave process.