Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity
Provides rigorous mathematical analysis for a class of viscoelastic models, but the result is incremental as it extends known techniques to specific kernel types.
The paper proves existence, uniqueness, and regularity of solutions for a hyperbolic integro-differential equation with weakly singular kernels modeling fractional viscoelasticity, using Galerkin's method. Regularity limitations are discussed, and the approach extends to smooth kernel cases.
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.