On the generalization of Calogero-Ahmed summation formulas

Abstract

The use of the Laplace transform gives the solution of water hammer equations in the frequency domain. The inverse transform of this solution over the years seemed impossible to derive, due to the significant complexity and the fact that the square root of the Bessel function was embedded in the argument of the resulting hyperbolic functions. In this work, we consider some generalizations that enable the determination of the modified Calogero-Ahmed infinite series. These generalizations will allow us in the near future (using the machine learning and artificial intelligence algorithms) a return to the time domain in a very wide range of the dynamic viscosity function, which plays the most important role in this complex fluid dynamic problem.

Infinite Series Based on Bessel Zeros

“…After developing the basic solutions (recalled in this work) in the following work [8], Calogero, supported by Ahmed, developed knowledge about these series while also analyzing solutions for their modified forms. As shown in [9], the modified Calogero-Ahmed series can always be written in a form depending on the standard Rayleigh series and the standard Calogero series. Standard Calogero series is characterized by the fact that they are based on differences computed between zeros of Bessel functions of the same order as follows:…”