Exact analytical solutions to the geodesic equations in general relativity via (G′/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G'/G)$$\end{document} - expansion method

In this paper, the (G′/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G'/G)$$\end{document} -expansion method is used to construct two certain classes of one-parameter exact solutions to the geodesic equations in General Relativity (GR). This method has been developed as an effective technique to construct exact solutions for nonlinear evolution equations or nonlinear partial derivative equations (PDE). At the first stage of this method, a nonlinear PDE is transformed into nonlinear ordinary derivative equation (ODE) of a polynomial form. Therefore, if we have some nonlinear ODE of a polynomial form, say, the geodesic equation, then sometimes its solutions can be obtained following to the procedure of (G′/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G'/G)$$\end{document} -expansion method. Here, this method allows us to obtain two classes of exact analytical solutions for the isotropic and time-like geodesic equations in the spherically symmetric spacetime metrics of the Schwarzschild and Kiselev black holes.