Approximate analytical solution of the nonlinear Bethe equation
The authors consider the Bethe equation, which is a nonlinear differential equation, describing the energy loss of electrons when they penetrate a material. They have used the Laplace Adomian decomposition method (LADM) to solve this equation approximated by \[ \frac{du}{dx}+\frac{ln(u+1)}{u}=0,\quad u(0)=u_0, \] where \(u\) is a dimensionless measure of the kinetic energy with \(u \geq 1 \), and \(x\) is a dimensionless measure of the distance pernetrated by the particle into the medium. The LADM has been applied to this method by considering \(\frac{d}{dx}\) as the linear operator \(L_x\) and the nonlinear operator \(Nu=\frac{ln(u+1)}{u}\). They have worked out several examples and proved the accuracy of the solution of the Bethe equation obtained through LADM.
scientific article; zbMATH DE number 7046074