Solution of the Bethe Equation Through the Laplace-Adomian Decomposition Method

TL;DR

This paper introduces a novel analytical method combining Adomian Decomposition and Laplace Transform to solve the nonlinear Bethe equation, providing accurate solutions relevant to nuclear physics applications.

Contribution

The paper presents a new combined approach (LADM) for solving the Bethe equation, which is difficult to solve exactly and mostly addressed experimentally.

Findings

LADM yields highly accurate solutions for the Bethe equation.

The method is effective across various initial conditions.

It offers a practical analytical tool for applications in nuclear physics.

Abstract

The Bethe equation is a nonlinear differential equation that plays an important role in nuclear physics and a variety of applications related to it, such as the description of the behavior of an energetic particle when it penetrates into matter. Despite its importance, its unusual to find the exact solution to this nonlinear equation in literature and practically all of them are of experimental nature. In this paper, we solve this equation and present a new approach to obtain the solution through the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). In addition, we illustrate our approach solving three examples, in which initial conditions are considered within the typical numerical ranges derived from the applications. Our results indicate that LADM is highly accurate and can be considered a very useful and valuable method.