Some new analytical solutions of the variant Boussinesq equations

In this manuscript, the exact analytical solutions for the variant Boussinesq equations are constructed by utilizing the improved tanh(ϕ(ξ)/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tanh (\phi (\xi )/2)$$\end{document}-expansion method (ITEM) and the improved (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 (IGEM). Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the ITEM and IGEM for hyperbolic function, trigonometric function and rational function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.