Study On Approximate Analytic Solutions Of Differential Equations With Symbolic Computation

In the dissertation, under the guidance of mathematical mechanization, two kinds of analytical approximation methods, which are Adomian decomposition method (ADM) and homotopy analysis method (HAM) for strong nonlinear problems around the non-linear equations, are investigated by means of symbolic computation. The application and mechanization of them are discussed, respectively.Chapter 1 is the research background related to the dissertation. The development of computer algebra and the theory of solitons are briefly outlined. Subsequently, the recent development and achievement of analytical approximation methods are summarized at home and abroad.In Chapter 2, the two-soliton solutions of modified Korteweg-de Vries (mKdV) equation and Kadomtsev-Petviashvili (KP) equation can be obtained by the modified ADM, respectively. By means of the transformation of the independent variables and trav-eling wave transformation, the short-wave model for Degasperis-Procesi (DP) equation is reduced to an ordinary differential equation the solution of which in closed form can be obtained by ADM. Then by means of the transformations back to the original variables, the loop-soliton solution of the short-wave model for DP equation can be derived. The results indicate the validity of ADM for constructing the special type of soliton solution of nonlinear differential equations. The discrete variable in nonlinear differential-difference equation is successfully overcome and ADM is extended to solving some classical sys-tems of differential-difference equations. The soliton solutions of them can be obtained with high accuracy by combining ADM and Pade approximants. Meanwhile, the pos-sibility of spurious poles of rational approximation is discussed and a criterion for the choice of the order of Pade approximants is given. The obtained results degree well with the exact solutions. This demonstrates the validity of ADM in strong nonlinear problems.In Chapter 3, By means of the transformation of the independent variables and trav-eling wave transformation, the partial differential equation is reduced to an ordinary dif-ferential equation, which can be solved by HAM. Then by means of the transformations back to the original variables, the solution of the original equation is obtained. The one-loop soliton solution of the short-wave model for DP equation and one-cusp soliton for Camassa-Holm (CH) equation can be obtained. This indicates the validity of HAM for constructing the special type of soliton solution of nonlinear differential equations. The discrete variable in nonlinear differential-difference equation is successfully overcome and HAM is extended to solving the discrete mKdV equation. The bright soliton solution can be obtained. A technique for choosing the initial guess is also shown. The obtained results degree well with the exact solution. This demonstrates the validity of HAM in strong nonlinear problems.In Chapter 4, Based on the existed algorithm for the calculation of ADM polynomials proposed by Biazar, an software package is developed to construct approximate analytic solutions of differential equations and integral equations automatically in computer alge-braic system Maple. Avoiding the huge size of the calculation of ADM polynomials, the algorithm needs less time without any need to formulas other than elementary operations than that based on ADM. Many examples are presented to illustrate the implementation of the package.