Green’s Function for a One-Dimensional Fractional Viscoelastic Wave Equation

We investigate the Green’s function associated with a one-dimensional fractional viscoelastic wave equation driven by a Dirac delta source. The starting point of the formulation is the classical one-dimensional viscoelastic wave equation posed with a Dirac delta driving term, which defines the fundamental solution of the corresponding wave operator. Fractional effects are incorporated directly at the level of this forced equation by generalizing the time and space operators: the second-order time derivative is replaced by the Caputo fractional derivative of order α∈(1,2], while the spatial Laplacian is replaced by the Riesz fractional operator of order β∈(1,2]. This construction leads to a fractional wave equation with memory in time and nonlocality in space, formulated explicitly as a Green’s function problem.

The Green’s function is obtained by applying the Laplace transform in time and the Fourier transform in space to the fractional wave equation, reducing the problem to an algebraic equation in the transform domain. The inverse transforms yield an explicit representation of the fundamental solution in terms of Fox H-function series, which naturally arise from the combined action of the fractional temporal and spatial operators. The resulting expression provides an explicit form of the Green’s function for the fractional viscoelastic wave equation with singular forcing.