Lie Symmetry Analysis and Analytical Solutions of Klein-Gordon Equations on Spherical Coordinate

Abstract

In this paper, we discuss the Lie symmetry analysis of classical and fractional Klein-Gordon differential equations on spherical coordinates. We show that, the Klein-Gordon equations admits infinite dimensional algebras, and the admitted infinitesimal generators satisfied the Lie commutator relations. Lie symmetry reductions and analytical solutions of the classical and fractional-order equations are obtained by using the similarity variables of the corresponding generators. The solutions of the classical equation are expressed in the forms of trigonometric, hyperbolic and Bessel's functions, while the solution of the corresponding fractional equation is derive in the form of well-know Mittag-Leffler function. The graphical visualization of the solutions are presented.