![]() | Only 14 pages are availabe for public view |
Abstract The purpose of this thesis is studying the Painlevé analysis and the symmetry method to various nonlinear partial di¤erential equations to get exact solutions for them: This thesis is divided into four chapters. In Chapter (I): We con.ned our attention to the survey and development of the di¤erent techniques that have been utilized in chapters II-VI to obtain exact solutions of nonlinear partial di¤erential equations of physical and engineering interests: In Chapter (II): The cubic nonlinear Klein-Gordon equation and the general- ized Kuramoto-Sivashinsky equation have been analyzed via Painlevé analysis: Firstly, we have carried out Painlevé analysis to cubic nonlinear Klein-Gordon equation then, we have shown that the integrability condition is not satis.ed for arbitrary func- tion _(x; t): Therefore, cubic nonlinear Klein-Gordon equation fails the Painlevé test: Using Bäcklund and Kruskal.s transformations, we have obtained exact solutions: Furthermore, the application of the Painlevé analysis to the generalized Kuramoto- Sivashinsky equation has led to non-integer roots for the resonance involved in the method: Therefore, the generalized Kuramoto-Sivashinsky equation fails the Painlevé test: Using Bäcklund and Kruskal.s transformations, we have obtained new exact solutions: In Chapter (III): We con.ned our attention to solve the variable coe¢ cients generalized Klein-Gordon equation and we have obtained new exact solutions when we applied Painlevé analysis since we have found that this equation fails the Painlevé test: Using Painlevé expansion and solving the integrability condition, we have ob- tained new exact solutions: ii In Chapter (IV): The Quasi-linear wave equation has been analyzed via the symmetry method: We have carried out the symmetry method for some cases for the Quasi-linear wave equation and, we have got di¤erent group theoretic reductions for these cases: Some of these reductions have led to exact solutions: iii |