Fractional Time–Space Viscoelastic Wave Equation: Analysis and Solution Framework

We investigate a fractional generalization of a one-dimensional scalar wave equation with a singular memory kernel, a model originally introduced to describe single-mode propagation in viscoelastic media exhibiting weakly singular hereditary effects. The fractional formulation is constructed by replacing the standard second-order time derivative with the Caputo fractional derivative of order α, where the order ranges from one to two, and by replacing the spatial Laplacian with the Riesz fractional pseudo-differential operator of order β, with the order likewise ranging from one to two.

An explicit solution is obtained in the form of a series involving Fox H-functions, which arise naturally from the combined influence of temporal and spatial fractional operators. This series representation provides a mathematically tractable structure that facilitates asymptotic analysis and numerical approximation. Furthermore, in the limiting cases α=2 and β=2, the formulation reduces to the classical viscoelastic wave equation, thereby illustrating the consistency of the fractional model with established theories of wave propagation in media with memory.

The results demonstrate that the fractional extension captures a broadened spectrum of propagation behaviors characterized by smoother wave fronts, delayed response, and modified dispersion patterns that depend sensitively on the fractional orders. These findings underscore the capacity of fractional operators to extend classical viscoelastic models and to describe complex dynamical effects associated with memory and spatial nonlocality in a unified mathematical setting.