Study On Symbolic Computation Of Analytic Approximating Solutions For Nonlinear Differential Equations

In various fields of science, many phenomena can be modeled by nonlinear equa-tions, therefore the research on analytic and analytic approximate solutions of nonlinear differential equations is very important for many scientific workers. In this paper, based on four kinds of Adomian polynomial algorithms proposed by Rach and the idea of Wu Wentsun's mathematical mechanization, we present a new algorithm to construct analytic approximate solutions for nonlinear differential equations, and also develop a software package armed with the computer algebraic system Maple. Our main work includes the following three parts.Partâ… is devoted to introduce the research background related to the dissertation. Firstly, the development of the methods to construct analytic approximate solution for nonlinear differential equations is outlined briefly. Then, various achievements and de-velopment of analytic approximate methods are reviewed.Partâ…¡is devoted to study the new Adomian-Pade algorithm to construct analytic ap-proximate solutions of nonlinear differential equations. In this part our research includes two aspects, firstly, based on the basic idea of Adomian decomposition method, together the new algorithms of Adomian polynomials proposed by Rach, we improved some key steps of the Adomian decomposition method. Further to improve the efficiency of this algorithm by applying some other techniques, these skills also effectively controlled the fasting expansion of the middle expressions. Another thing, as the Adomian decomposi-tion method is usually convergent in a very limited area, and outside the convergence area the errors increased very quickly. we embedded the Pade technique into our improved Adomian decomposition method to enlarge the convergence area.In partâ…¢, based on the improved algorithm, a Maple package named NAPA is developed. This package is user-friendly and efficiently, as long as one input an equation as well as the interface parameters, our package will output analytic approximate solutions within a few seconds. Besides, several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. Meanwhile, in this part the interface and procedures are introduced in details. Finally, several different types of examples are given to illustrate the validity and promising flexibility of the package.