On solutions of matrix-valued convolution equations, anisotropic fractional derivatives and their applications in linear and non-linear anisotropic viscoelasticity

TL;DR

This paper explores matrix-valued convolution equations and introduces anisotropic generalized fractional derivatives, with applications in linear and nonlinear anisotropic viscoelasticity, extending fractional calculus and analyzing solution classes.

Contribution

It establishes a link between matrix-valued Bernstein and Stieltjes functions, and defines anisotropic GFDs using matrix-valued kernels, advancing the mathematical framework for anisotropic viscoelasticity.

Findings

Solutions belong to special function classes

Duality requires Newtonian viscosity inclusion

Introduces anisotropic GFDs with matrix kernels

Abstract

A relation between matrix-valued complete Bernstein functions and matrix-valued Stieltjes functions is applied to prove that the solutions of matricial convolution equations with extended LICM kernels belong to special classes of functions. In particular the cases of the solutions of the viscoelastic duality relation and the solutions of the matricial Sonine equation are discussed, with applications in anisotropic linear viscoelasticity and a generalization of fractional calculus. In the first case it is in particular shown that duality of completely monotone relaxation functions and Bernstein creep functions in general requires inclusion in the relaxation function of a Newtonian viscosity term in addition to the memory effects represented by the completely monotone kernel. We define anisotropic generalized fractional derivatives (GFD) by replacing the kernel $t^{-\alpha}/\Gamma(1-\alpha)$ of the Caputo derivatives with completely monotone matrix-valued kernels which are weakly singular at 0.