Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

TL;DR

This paper establishes the well-posedness and regularity of solutions for a hyperbolic integro-differential equation with singular kernels modeling fractional viscoelasticity, using Galerkin's method and regularity analysis.

Contribution

It proves existence, uniqueness, and regularity of solutions for a class of integro-differential equations with singular kernels, extending to smooth kernels in viscoelastic models.

Findings

Existence and uniqueness of solutions are established.

Regularity of solutions is proved and discussed.

The approach applies to models with smooth kernels in viscoelasticity.

Abstract

A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions. Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity, under the appropriate assumptions on data.