Analytical and approximate solutions of hirota satsuma coupled KdV equation

This study has been organized as nine sections. In Section 1, works that are related to nonlinear partial differential equations and systems and an introduction of our work have been given. In Section 2, some basic definitions and theorems, which will be used in the next sections, have been given. In Section 3, information related to generalized Hirota-Satsuma coupled KdV equation, which will be used in our work, has been given. In Section 4, Adomian decomposition method (ADM) and the resulting non-linear terms, how to create based on Adomian polynomials, have been shown. Extended Adomian decomposition method (GADM) are provided in this secion. In Section 5, variational iteration method (VIM) that we will use in our work and extented variational iteration method (GVIM) have been defined and some works related to these methods have been given. In Section 6, series solutions of HS coupled KdV equation have been obtained by using GADM and GVIM that have been analyzed in the previous sections. In Section 7, numerical results obatined by GADM and GVIM for analytical solutions u(x, t), v(x, t) and w(x, t) of HS coupled KdV equation have been compared with the results that obatined by DTM and HPM and these results have been given by tables and figures.In Section 8, basic structures of (G?/G)? expansion method, extended (G?/G)? expansion method and improved (G?/G)? expansion method have been given. Then improved (G?/G)? expansion method have been applied to Hirota-Satsuma coupled KdV equation. In Section 9, results that obatined by the mentioned methods have been ealuated. Key Words: Generalized Hirota-Satsuma Coupled KdV equation, (G?/G)? expansion method, Modified Adomiam method, Modified Variational iteration method, Adomian Polynomials, Soliton.