On evolutionary equations with material laws containing fractional integrals

Abstract

Abstract. A well-posedness result for a time-shift invariant class of evolutionary operator equations involving material laws with fractional time-integrals of order α ∈ ]0, 1[ is considered and exemplified by an application to a Kelvin-Voigt type model.

Homogenization in Fractional Elasticity

“…According to [3] material models of fractionl type are used to describe the 1 The condition that α only ranges from 1{2 to 1 is rather technical. In fact, using other methods andat the same time -imposing more conditions on the operator C namely, besides its boundedness, one has to assume that it is selfadjoint and non-negative definite, we have shown in [11,Theorem 4.1], that the range of α may vary in p0, 1s.…”

A Discontinuous Galerkin Method for Fractional Differential Equations

“…Hence a more natural approach is to consider initial function conditions. This can be realised, as suggested by Picard [1], by using the decomposition of the solution u = u hist + v into the history term u hist = u1 (−∞,0] and the future term v = u1 (0,∞) , assuming (0, ∞) as the domain of interest. This changes the inital value problem to:…”

“…However this there are still many different definitions used for fractional derivatives. Based on the definition given by Picard in [1] we will use the following definition on the weighted L 2 space H ρ,0 (R; R) := {f ∈ L 2 (R; R); f exp(−ρ ·) 2 L2 < ∞}: Definition 1.1 For f ∈ H ρ,0 (R; R) and α ∈ [0, 1) the fractional integral of f of order α is given by…”

On abstract grad–div systems

“…In a number of studies [21,20,14,33,29,22] it has been illustrated, that typical initial boundary value problems of mathematical physics can be represented in the general form…”

“…[15,16]. The operator M is referred to as the material law operator, which in the situation discussed here is a linear operator acting on a Hilbert space realizing the space-time the problems are formulated in, [24,34,21,15].…”