Approximate Analytical Solution of the Nonlinear Bethe Equation
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Bethe equation is a nonlinear differential equation that appears in nuclear physics and that plays an important role in various applications ranging from basic science to engineering and medicine. Please confirm the corresponding author is correctly identified and amend if necessary. It describes the behavior of a charged particle when it enters a material medium. Despite its importance, it is unusual to find exact solutions for this non-linear equation in the literature and practically all the related studies focus on contrasting the predictions given by approximate solutions with the corresponding experiments. In this work, we solve approximately 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). Additionally, we illustrate our approach by solving some examples in which the initial conditions are considered within the numerical ranges typical of experimental situations. Our results indicate that LADM is highly accurate and can be considered as a very useful and valuable method in the study of this equation.
KeywordsBethe equation Stopping power Nonlinear equations Analytical approximate solution Adomian decomposition method
Mathematics Subject Classification34L30 34L25 65D15
We are very grateful to the anonymous referees for their invaluable comments and suggestions which helped much to improve the quality of the paper.