Eliminating solution singularity of variably distributed-order time-fractional diffusion equation via strongly singular initial distribution

We investigate a distributed-order time-fractional diffusion equation with a time-dependent density function and its support. The well-posedness and regularity of the equation are analyzed. In particular, by proposing appropriate assumptions on the density function, which may lead to a strongly singular initial distribution instead of smooth distributions that are usually imposed in the literature, we prove smoothing properties of the solutions and eliminate their nonphysical initial singularities without affecting the memory and hereditary properties of the model, i.e. the current state of the model depends on its states at previous time instants, away from the initial time. The results generalize the solution theory of distributed-order equations and provide a model correction for problems that do not exhibit initial singularities.