The KWW Relaxation Spectrum Determination Using the Post-Widder Inversion Formula

The problem of recovering the relaxation spectrum of the process described by the stretched exponential Kohlrausch-Williams-Watts (KWW) model is considered using the Post-Widder Laplace transform inversion rule. Based on the specific properties of the stretched exponential function, an analytical formula was derived that describes an arbitrarily high-order Post-Widder approximation of the spectrum directly in terms of the relaxation modulus, without any-neither analytical nor numerical-differentiation of the modulus as a product of finite power series of the relaxation times and the relaxation modulus. An alternative recurrence formula defining a sequence of the Post-Widder approximate models of the relaxation spectrum was developed. The positive definiteness, multiple differentiability, and zero asymptotic properties of the spectrum model are demonstrated; its extreme properties are discussed, and a sensitivity analysis with respect to the model parameters is conducted. The developed algorithm ensures fast convergence of the generated model sequence by applying a simple adaptive rule to select subsequent model orders, which relates them to the discrepancy between successive models and results in high-order models in at most a dozen iterations. Detailed numerical studies performed for nine values of the stretching exponent for KWW spectra covering relaxation times from 3 to 41 decades show that the proposed approach allows for generating an excellent approximation of the KWW spectrum by using only the relaxation modulus values in simple algebraic calculations.