Sums of exponentials approximations for the Kohlrausch function

Abstract

The mathematical foundation of many real-world problems can be quite deep. Such a situation arises in the study of the flow and deformation (rheology) of viscoelastic materials such as naturally occurring and synthetic polymers. In order to advance polymer science and the efficient manufacture of synthetic polymers, it is necessary to recover information about the molecular structure within such materials. For the recovery of such information about a specific polymer, it is necessary to determine its relaxation modulus G(t) and its creep modulus J(t). They correspond to the kernels of the Boltzmann causal integral equation models of stress relaxation and strain accumulation experiments performed on viscoelastic solids and fluids. In order to guarantee that the structure of such models is consistent with the conservation of energy, both the relaxation modulus and the derivative of the creep modulus must be completely monotone (CM) functions. As well as the exponential function exp(-αt) and Dirichlet series 1 ∑ n=1∞ a i exp(-λ i), 0 < λ 1 < λ 2 < β < 1. Such functions arise in a broad spectrum of applications and have important mathematical properties Anderssen et al. (2004) (e.g. the Weibull cumulative probability distribution). Because of their importance in applications, there is a need to have accurate approximations for it, so that its properties can be approximated. For example, even though the relaxation and creep modulii G(t) and J(t) of linear viscoelasticity are known to satisfy the interconversion equation ∫ 0t G(t - τ )J(τ )dτ = ∫ 0t J(t - τ )G(τ )dτ = t, the specific form of J(t) is unknown when G(t) is a Kohlrausch function. Theoretically, it is known that J(t) must be a strictly monotonically increasing function. Recently, by taking advantage of the Pollard (1946) result that a Kohlrausch function is the Laplace transform of a known stable distribution, Anderssen and Loy (2011) have shown how uniformly convergent sums of exponentials approximations for the Kohlrausch function can be constructed. In this paper, the derivation, significance and validation of such approximations are examined.