On solutions of a class of matrix-valued convolution equations

TL;DR

This paper explores matrix-valued convolution equations, establishing unique solutions using advanced function classes, with applications in viscoelasticity and fractional calculus.

Contribution

It introduces a novel approach linking Bernstein and Stieltjes functions to solve matrix-valued convolution equations.

Findings

Proved uniqueness of solutions for certain matrix-valued convolution equations.

Applied results to viscoelastic duality and Sonine equations.

Extended fractional calculus to matrix-valued functions.

Abstract

We apply a relation between matrix-valued complete Bernstein functions and matrix-valued Stieltjes functions to prove that certain convolution equations for matrix-valued functions have unique solutions in a special class of functions. In particular the cases of the viscoelastic duality theorem and the Sonine equation are discussed, with applications in anisotropic linear viscoelasticity and a generalization of fractional calculus.