# On solutions of matrix-valued convolution equations, CM-derivatives and their applications in linear and nonlinear anisotropic viscoelasticity

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## 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 also show that a new class of “fractional derivatives” called here CM-derivatives can be defined by replacing the kernel \(t^{-\alpha }/\Gamma (1-\alpha )\) of the Caputo derivatives with completely monotone kernels which are weakly singular at 0.

## Keywords

Viscoelasticity Completely monotone Bernstein Complete Bernstein Stieltjes Sonine equation Fractional calculus## List of symbols

*D*\(= \mathrm {d}/\mathrm {d}t\)

- \(\mathbb {R}\)
The set of real numbers

- \(\mathbb {N}\)
The set of nonnegative integers

- [
*a*,*b*[ The set of

*x*such that \(a \le x < b\)

## Mathematics Subject Classification

74D05 26A33## Notes

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