Trial Function Method To Find Exact Solutions Of Nonlinear Partial Differential Equations

This dissertation constructed the Cole-Hopf transformation and a moregeneral trial function based on the homogeneous balance method and the ideaof trial function method, respectively. The new trial function is successfullyapplied to several kinds of nonlinear evolution equation, which makes asimple and effective method for solving nonlinear evolution equations. Bythis method one can get the hyperbolic function solutions, trigonometricfunction solutions and rational type solutions. Therefore the method not onlyextends the trial function method and enriches the solutions of some nonlinearevolution equations. The basic idea of this method is first to introduce theCole-Hopf transformation, then to chose appropriate trial function accordingto the given equations and tanking it into the original equation, finally todetermine the unknown coefficients by using the method of undeterminedcoefficients which leads the exact solutions for given nonlinear evolutionequations. The work in this dissertation includes the following:The first chapter briefly introduces the research area of nonlinearevolution equations and its developments, and then reviewed the concept ofsolitons and its development process. In the second chapter, the trial functionmethod is used to solve some nonlinear evolution equations with constantcoefficients. First we construct the multiple different forms of exact solutions of the combined KdV equation with the help of the Riccati equation. Second,the real function solutions of the BBM equation and KdV equation areconstructed by choice of the appropriate trial functions. Finally, we extend ourmethod to the (2+1)-dimensional equations and solved the(2+1)-dimensional Burgers equation and (2+1)-dimensional Boussinesqequation. The third chapter based on the results obtained in the second chapterto apply the trial function method to the variable coefficients nonlinearevolution equations and we get the more general trial function solutions of thevariable coefficients Burgers equation, KdV equation, KdV-Burgers and the(2+1)-dimensional KP equation, etc. The fourth chapter summarizes ourresults obtained in second and third chapter. We pointed our innovation andshortcomings which appears in the use of trial function method to solveconstant coefficients and variable coefficients nonlinear evolution equationsand to look ahead of solving nonlinear evolution equation in shortly.