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

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.

Appendices

Appendix A. A remark on the convolution algebra

For our purposes it is important that the convolution algebra has a unity. The unity is not a function. Hence, the convolution algebra must include Borel measures. The convolution $\rho *\nu$ of two measures $\rho$ and $\nu$ defined on $[0,\infty [$ is defined as the Borel measure $\lambda$ satisfying the identity

$$\begin{aligned} \int _{[0,\infty [} f(r)\, \mathrm {d}\lambda (\mathrm {d}r) = \int _{[0,\infty [} \int _{[0,\infty [} f(r + s)\,\rho (\mathrm {d}r)\,\nu (\mathrm {d}s) \end{aligned}$$

for every continuous function f with compact support.

This definition is easily extended to matrix-valued measures.

For our purposes the convolution algebra has to involve only Borel measures of the form $u \, \mathbf {C}+ \mathbf {F}(t) \, \mathrm {d}t$, where the unity is a measure defined in Sect. 2.

Appendix B. Matrix-valued Stieltjes functions and complete Bernstein functions

We shall now use some results from Appendix B of [4].

A matrix-valued Stieltjes function $\mathbf {Y}(p)$ has the following integral representation:

$$\begin{aligned} \mathbf {Y}(p) = \mathbf {B} + \int _{[0,\infty [} (p + r)^{-1} \, \mathbf {H}(r) \, \mu (\mathrm {d}r) \end{aligned}$$

(38)

where $\mathbf {B} \in \mathcal {S}_+$, $\mu$ is a Borel measure on $[0,\infty [$ satisfying (9), and $\mathbf {H}(r)$ is an $\mathcal {S}_+$-valued function defined and bounded $\mu$-almost everywhere on $[0,\infty [$.

Conversely, any matrix-valued function with the integral representation (38) is an $\mathcal {S}_+$-valued Stieltjes function.

Theorem B.1

Every matrix-valued Stieltjes function is the Laplace transform of a matrix-valued e-LICM.

Proof

The Laplace transform of the $\mathcal {S}_+$-valued LICM function $\mathbf {A}(t)$ (Eq. (8)) is given by the equation

$$\begin{aligned} \tilde{\mathbf {A}}(p) = \int _{[0,\infty [} (p + r)^{-1} \, \mathbf {H}(r) \, \mu (\mathrm {d}r) \end{aligned}$$

(39)

where $\mu$, $\mathbf {H}$ satisfy the same conditions as in (38).

The second term on the right-hand side of Eq. (38) involves a double Laplace transform; hence, it equals

$$\begin{aligned} \mathbf {V}(p) := \int _0^\infty \mathrm {e}^{-p t} \left[ \int _0^\infty \mathrm {e}^{-r t} \, \mathbf {H}(r) \, \mu (\mathrm {d}r) \right] \mathrm {d}t \end{aligned}$$

where the Borel measure $\mu$ satisfies inequality (9). The inner integral represents a general matrix-valued LICM $\mathbf {F}(t)$. Thus, $\mathbf {V}(p)$ is the Laplace transform of a general matrix-valued LICM and $\mathbf {Y}(p)$ is the Laplace transform of a general matrix-valued e-LICM. $\square$

An $\mathcal {S}_+$-valued CBF $\mathbf {Z}(p)$ has the following integral representation:

$$\begin{aligned} \mathbf {Z}(p) = p\, \mathbf {B} + p \int _{[0,\infty [} (p + r)^{-1} \, \mathbf {H}(r) \, \nu (\mathrm {d}r) \end{aligned}$$

(40)

where $\mathbf {B} \in \mathcal {S}_+$, $\nu$ is a Borel measure on $[0,\infty [$ satisfying (9), and $\mathbf {H}(r)$ is an $\mathcal {S}_+$-valued function defined $\nu$-almost everywhere on $[0,\infty [$.

Conversely, any $\mathcal {S}_+$-valued function with the integral representation (40) is a $\mathcal {S}_+$-valued CBF.

It follows immediately that the function $p^{-1} \, \mathbf {Z}(p)$, where $\mathbf {Z}$ is an $\mathcal {S}_+$-valued CBF function, is an $\mathcal {S}_+$-valued Stieltjes function.

We quote Lemma 1 in Appendix B of op. cit. in the form of the following theorem

Theorem B.2

If $\mathbf {Z}(p)$ is an $\mathcal {S}_+$-valued CBF and does not vanish identically, then $\mathbf {Z}(p)^{-1}$ is an $\mathcal {S}_+$-valued Stieltjes function.

Conversely, if $\mathbf {Y}(p)$ is an $\mathcal {S}_+$-valued function does not vanish identically then $\mathbf {Y}(p)^{-1}$ is a CBF.