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A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration. It is demonstrated that the signal builds up gradually from a zero value after the passage at the wavefront, resulting in signal delay. Attenuation splits into two parts: a frequency-independent amplitude modulating factor and a frequency-dependent part implicit in the basis functions. Explicit basis functions are obtained for the £-1/2 and t~n/3 singularities. Asymptotic solutions are derived for systems of PDEs with the t-1/2 singularity and for scalar equations with the t~n/3 singularities. Asymptotic fundamental solutions are constructed for scalar equations with singular kernels. For two families of scalar equations with singular kernels, asymptotic fundamental solutions are shown to be exact.

Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels