A Semi Analytical Investigation of the Paracial Ray Equation

A new common procedure for solving paraxial ray equation of a charge particle going through a pure magnetic field has been introduced. The presented approach is mainly concern with using power series method usually used to solve differential equations of special forms. In order to make the power series method applicable for solving equations of sort of paraxial ray equation, the domain of solution, the axial coordinate where the magnetic field exist, are descretized into several subdomains, such that magnetic field can consider to be fixed throughout each of these subdomains. Computer has been adopted to be a main tool for execution in order to make prt:sent technique of a practical impo1tance. The outputs of this procedure are compared with some of the available con-espondence analytical equations. The results clearly shown that the new technique have an accurate results and has the property of analytical proce9ure. Anyway, the algorithm of presented method are fom1Ulated to a subroutine written in Fortran-77 language •vhich may used to replace any of the subroutines that work with numerical methods in the well known software. However, one would not face the numerical errors any more