Study On Analytic And Analytic Approximating Solution Of Differential Equation With Symbolic Computation

Many phenomena in nature can be described with nonlinear differential equations. The study of analytical solutions to nonlinear differential equations plays a very important role in penetrating the inner structure, in analyzing the relationship of things as Well as in interpreting various physical phenomena. The naissance of high performance computer greatly promotes the study on symbolic computation of nonlinear differential equations and then coming forth abundant algorithms and methods to construct the analytical solutions of nonlinear differential equations. Armed with the computer algebraic system Maple, this dissertation concentrates on the nonlinear differential equations and do much research on various algorithms and approaches to construct exact solutions and analytical approximation ones. Our main works are summarized bellow.Part I is devoted to study the algorithm and method to construct exact solutions of nonlinear differential equations. We do research from the following two aspects.We improve and integrate several algebraic methods of constructing the exact solitary wave solutions to nonlinear evolution equations such as the Riccati equation method, the coupled Riccati equation method, the deformed mapping method, and the ansats making method and then propose an approach named "elliptic equation method." Based on Wu elimination method, we provide an software package RAEEM in computer algebraic system Maple for seeking for exact traveling wave solutions to nonlinear differential equations, which can output automatically a series possible exact traveling wave solutions for inputted equation including the polynomial type solutions, rational function solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, Jacobi elliptic function solutions and the Weierstrass elliptic function solutions etc..The study of Backlund transformation method is very important in the sense of integrability and the solving of exact solutions to nonlinear differential equations. Especially, once the nonlinear superposition formula of solutions is obtained starting from the Backlund transformation, one can construct new solutions of differential equations only by algebraic computation. Benefiting from the existed constructing Backlund transformation method, we propose an mechanization algorithm to build a kind of self-Backlund transformations for 1+1 nonlinear evolution equations. Similarly, utilizing the idea of Wu Wentsun mechanization, we give a corresponding implementation software package AutoBT in Maple, which can not only output the self-Backlund transformations of all possible specified types and the corresponding parameters constraints, but also can output the nonlinear superposition formula of solutions. Part II is devoted to study the algorithm and method to construct the analytic and approximating solutions of nonlinear differential systems. The homotopy analysis method is effective in constructing the analytic and approximating solutions of nonlinear differential systems, which is developed in recent several years. Differing from the perturbed method, the effectiveness of the homotopy analysis method is independent of whether or not the considering nonlinear problem having small parameter. Moreover, It is differ from all other traditional perturbed and unperturbed methods, such as the manual small parameter method, the 5 expansion method and the Adomian decomposition method etc. The homotopy analysis method itself provides a kind of convenient tool in controlling and adjusting the convergence speed and region of the solution series. This method has been widely used in solving many problems of applied mathematics and mechanics.The experimental and theoretical researches concerning composite media have received much attention because of their potentially wide applications in engineering and physics. The perturbation method is a powerful tool for dealing with weakly nonlinear problems of composite media. However, it is still very difficult to solve strongly nonlinear problems of composite media. The new homotopy analysis method is a powerful tool for solving strongly nonlinear problems. The authors in [85] and [86] have constructed the analytic approximating solutions of strongly nonlinear composite media by the homotopy analysis method. However, for the simplicity of computation, they first predigest the original system into an ordinary system with the help of the mode expansion method and just keep the first mode, this may evoke great errors between the original system with the reduced ordinary differential system. To get the higher precision, in this paper we choose linear partial differential equations as linear operators, and directly construct the homotopy analysis solutions for the original system. Our obtained solutions are obviously superior to the known ones. Another innovation is that we extend the homotopy analysis method to solving fractional differential systems.