The Study Of Analytic Solutions For Nonlinear Differential Systems With Symbolic Computation

In the70’s of last century, the Chinese mathematician Wentsun Wu advocated a new field of research-the mathematical mechanization. International scholars actively promot-ed the symbol computation in the1980s, and they also developed the theory and practice of computer-aided analysis, calculations and reasoning. So some excellent computer algebra systems are developed, such as Reduce, MACSYMA, Mathematica and Maple. Especial-ly, Mathematica and Maple have been widely used in mathematics and engineering. It is late that the Chinese scholars start to study on the symbol computation in the field of application research, particularly for the computing software. Furthermore, the applica-tion research of symbol computation is far behind that of the western developed countries. Therefore, computing software is highlighted to be a promising research direction in the support of the National Twelfth Five-Year Development Plan.A lot of problems ultimately are attributed to nonlinear differential equations in science and engineering. Therefore, the studies of nonlinear differential equations are always impor-tant scientific topic. With the mathematical mechanization, this dissertation concentrates on the nonlinear differential equations and do much research on various mechanization al-gorithms to construct analytical solutions of nonlinear differential equations (especially the initial value or the boundary value problems in nonlinear differential equations). Further-more, a software package is developed to automatically construct some special types of analytical solutions. The innovation of this dissertation is that the double decomposition method and the two-step decomposition method are embedded into the classic Adomian decomposition mehtod for the first time, then we propose a new algorithm to construct an-alytical approximate solutions of nonlinear differential equations with initial or boundary conditions. Besides, we develop the corresponding software package. Our main works are summarized bellow.1. Analytical approximate solutions:The Adomian decomposition method is one of the most effective methods to construct analytical approximate solutions of nonlinear dif-ferential equations. The Adomian decomposition method is widely applied to solve various nonlinear differential equations, because of the simplicity of calculations and the ability of solving strongly nonlinear problems. Based on the classic Adomian decomposition method, Adomian and his colleagues delivered the modified Adomian decomposition method, the two-step decomposition method, and even the double decomposition method which can solve boundary value problems in nonlinear differential equations with less computation. An essence is that the two-step decomposition method attempts at constructing exact so-lutions of differential equations on the basis of the classic Adomian decomposition method. In2008, Rach developed the novel notion to derive the unifying formula for the family of classes of the Adomian polynomials, but also the new definition of Adomian polynomials is more efficient. On the basis of the Rach’s new definitions of the Adomian polynomi-als, two-step decomposition method and Pade technique, a new algorithm is proposed to construct analytical approximate solutions of nonlinear differential equations with initial conditions. Then combined with the double decomposition method, a new algorithm is presented to construct analytical approximate solutions of nonlinear differential equations with boundary conditions. Furthermore, we extend these algorithms to construct analyti-cal approximate solutions of fractional differential equations. Finally, we develop a Maple software package ADMP, which automatically construct analytical approximate solutions of nonlinear differential equations (including fractional differential equations) with initial or boundary conditions. ADMP also can solve nonlinear differential equations with Robin boundary conditions or fractional initial conditions.2. Exact solutions:The invariant subspaces method is one of the most effective methods to construct exact solutions for nonlinear differential equations. We apply the invariant subspace method to construct exact solutions of the one-dimensional reaction-diffusion equation and analyze its inner behavior. Furthermore, based on the invariant subspaces of the one-dimensional reaction-diffusion equation, we construct exact solutions of two-dimensional reaction-diffusion equation. Finally, natural phenomena are successfully interpreted by pattern and temporal or spatial sequence diagrams. The theory of the invari-ant subspace method is simple, but its calculation is rather complicated. By the invariant subspace method, we develop a Maple software package ISM, which can output automat-ically a series possible exact solutions for system input and the corresponding parameters constraints. Applying it, we have succeeded in solving many nonlinear differential equation-s, especially for constructing exact solutions, including polynomial solutions, trigonometric solutions, exponential-trigonometric solutions, etc. We note that Eq.(4.72) is complexi-tions consisting of trigonometric functions and exponential functions. While Eq.(4.25) is positions consisting of different trigonometric functions.