The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Abstract

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.

Modelling the Frequency of Interarrival Times and Rainfall Depths with the Poisson Hurwitz-Lerch Zeta Distribution

“…is log-concave with p 0 = P(X = 0) = G Z [G Y (0)] and qx = P(Z = x),q x = P(Y = x). Equation ( 30) follows from Equation (2.3) in [23] using the general partition polynomials (8). When Z d = PS(λ) a necessary and sufficient condition to recover strong unimodality is related to the magnitude of q 1 and q 2 , as the following theorem shows.…”

“…In Section 4, we summarize its main properties using the combinatorics of exponential Bell polynomials. It is noteworthy to mention that Bell polynomials are used within fractional calculus, see for example [7,8] and within fractal models [9]. Moreover, new results are added on the the PSHLZD, as for example on log-concavity.…”