The Salzer Summation and the numerical inversion of the Laplace Transform: performance analysis for oscillatory, exponential and logarithmic functions

Authors

DOI:

https://doi.org/10.5902/2179460X87225

Keywords:

Laplace Transform, Gaver Functionals, Laplace Inverse Transform, Salzer Summation, Numerical Methods

Abstract

This article presents a study of the Salzer Summation, a technique for the numerical inversion of the Laplace Transform, applied to the inversion of five elementary functions with different behaviors: two oscillatory, two exponential and one logarithmic. Three of the functions studied have a variable parameter a (factor incorporated to investigate the efficiency of the method in dealing with functions of the same class). The algorithm's performance was analyzed for each value of M (number of terms in the sum) and parameter a chosen, through the Mean Absolute Error, graphical representation and execution times approximate. For the set of five functions presented (and for each a), the optimal value of M was determined. It was found
that a does not significantly influence the execution time, unlike the parameter M, which directly interferes. Also, it was concluded that for oscillatory functions, the method presents convergence difficulties as the frequency increases

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Author Biographies

Renan de Almeida Schmidt, Universidade Federal do Rio Grande

Master's student in Computational Modeling at the Graduate Program in Computational Modeling (PPGMC) at the Federal University of Rio Grande (FURG). Graduated in Applied Mathematics from the same university in 2019. He is interested in Pure Mathematics, Differential Equations and Numerical Methods.

Murilo da Cunha Paz, Universidade Federal do Rio Grande

Master's student in Computational Modeling at the Graduate Program in Computational Modeling (PPGMC) at the Federal University of Rio Grande (FURG). Graduated in Mathematics degree, from the same university, in 2022.  He is interested in the areas of Pure Mathematics, Mathematics Education and Statistics, with an emphasis on their practical applications.

Bárbara Denicol do Amaral Rodriguez, Universidade Federal do Rio Grande

Bachelor's degree in Applied and Computational Mathematics (2000), Master's degree in Applied and Computational Mathematics (2003) and PhD in Mechanical Engineering (2007), all from the Universidade Federal do Rio Grande do Sul– UFRGS. She is currently Associate Professor IV – DE and permanent professor of the Postgraduate Program in Computational Modeling and the Professional Master's Degree in Mathematics on a National Network at the Universidade Federal do Rio Grande – FURG. Works in Applied Mathematics, Engineering and Mathematics Teaching: in analytical methods development for solving particle transport problems, techniques and problem solving based on the Laplace transform and in the study of alternative teaching and learning methodologies in Higher Education.

João Francisco Prolo Filho, Universidade Federal do Rio Grande

Bachelor's degree in Applied and Computational Mathematics from the Universidade Federal do Rio Grande do Sul - UFRGS (2004), with master's degree (2007) and doctoral degree (2012) in Applied Mathematics from the Universidade Federal do Rio Grande do Sul - UFRGS. He is currently Associate Professor 2 - DE at the Universidade Federal do Rio Grande, collaborator professor of the Graduate Program in Ocean Engineering (PPGEO-FURG) and has co-supervised work in the Graduate Program in Computational Modeling (PPGMC-FURG) and Graduate Program in National Master's Degree in Physics Education (MNPEF/FURG). He works in the areas of Applied Mathematics and Engineering: i) in the development of analytical and spectral methods associated with the Boltzmann Equation and its applications in neutron transport, radioactive transfer and rarefied gas dynamics; ii) study of diffusive phenomena through the Laplace Transform and numerical inversion approaches, iii) theoretical and experimental study of alternative energy generators.

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Published

2024-11-07

How to Cite

Schmidt, R. de A., Paz, M. da C., Rodriguez, B. D. do A., & Prolo Filho, J. F. (2024). The Salzer Summation and the numerical inversion of the Laplace Transform: performance analysis for oscillatory, exponential and logarithmic functions. Ciência E Natura, 46(esp. 1), e87225. https://doi.org/10.5902/2179460X87225

Issue

Vol. 46 No. esp. 1 (2024): ERMAC e ENMC

Section

Special Edition 1

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