Solution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method

TL;DR

This paper introduces a novel combined Laplace-Adomian Decomposition Method to solve the nonlinear relativistic harmonic oscillator equation, providing highly accurate series solutions with less computational effort compared to existing methods.

Contribution

The paper presents the first application of the combined Laplace-Adomian Decomposition Method to this nonlinear relativistic oscillator, yielding new series solutions not previously reported.

Findings

Series solutions match well with other approximate methods

Method achieves high accuracy with less computational effort

First reported series solutions for this specific nonlinear oscillator

Abstract

Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by means of a combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). The results that we have obtained, a series of powers of functions, have never been reported and show a very good match when compared with other approximate solutions, obtained by different methods. The method here proposed works with high degree of accuracy and because it requires less computational effort, it is very convenient to solve this kind of nonlinear differential equations.