Diffusion models of heat and momentum with weakly singular kernels in the fading memories: How the integral-balance method can be applied?

This work presents an attempt to apply the heat-balance integral approach to diffusion models with fading memories with weakly singular kernels resulting in closed-form solutions. The main steps are exemplified by solutions where the fading memory is represented by Volterra integrals and by a time-fractional Riemann-Liouville derivative. The examples address sole elastic (damping) effects and cases where the viscous diffusivity should be taken into account. As examples polynomial approximation is applied, demonstrating how to avoid problems in determination of the exponent of the general parabolic profile, but without freedom to optimize the final closed-form solution. In general, this is a new implementation of an old idea and related methods to new models and we hope the demonstrated technique could be useful in solutions of practical problems