Study On Analytic Solutions Of Nonlinear Differential Equations With Symbolic Computation

Many phenomena in natural science or engineering can be described with nonlinear differential equations. The study of analytical solutions to nonlinear differential equations plays a very important role in making clear the law of motion and analyzing the relation-ship under the nonlinear effect for all matters. However, to solve the nonlinear differential equations involves complicated differential and algebraic calculations inevitably, some of which is even beyond human’s abilities. Thus a new topic will be posed to researchers, that is how to complete the complicated differential and algebraic calculations by com-puter automatically. With the development of high performance computers and symbolic computation system in the last decades, symbolic computation research in the nonlinear science has been promoted largely and algorithms on both analytic solutions and analytic approximate solutions have emerged in the end. This dissertation based on the concept of mathematics mechanization makes some research on the algorithms of constructing exact solutions and analytical approximation ones. Our main work can be summarized as follows:Part I is devoted to the study of algorithm and method to construct analytic approx-imate solutions of nonlinear differential equations. The homotopy analysis is one of the important algorithms, proposed originally by a domestic scholar. For the universal exis-tence of nonlinear problems, this method will be dedicated to both theoretical and practical research of nonlinear science. To be more, as a modified method of the traditional one, the predictor homotopy analysis method (PHAM) can predict the multiplicity of solutions of nonlinear differential equations and calculate all branches of solutions only with the aid of one initial approximation guess, one auxiliary function and one auxiliary linear operator simultaneously. In this dissertation, PHAM will be applied to the study of nonlinear differ-ential equations with boundary conditions or initial conditions. We also provide a software package based on PHAM in computer algebraic system Maple, namely PHAM SOLVE, to analyze the multiplicity of solutions of nonlinear differential equations. It has user-friendly interface where the consequences are represented graphically.Part II is devoted to the study of algorithm and method to construct exact solutions of nonlinear evolution equations. The invariant subspace method is one of powerful ap-proaches for constructing exact solutions to nonlinear evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. Some nonlinear evolution equations are investigated with the help of the invariant subspaces method and exact solutions are constructed based on different invariant subspaces. The exact solutions include the polynomial type solutions, exponential function solutions, triangular function solutions and mixed type solutions.