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.