Some Mechanical Algorithms For Solving Differential Equations And Calculating The Size Of Solution Space

This dissertation considers the mechanical calculation of exact solutions to differential equations, especially for those nonlinear evolution equations(NLEEs) arising from the fields of fluid mechanics, aerodynamics, plasma physics, biophysics and chemical physics. Algorithms for our purposes as well as their implementations are presented.Chapter 1 is to introduce the related development of mathematical physics mechanization, emphasizing on the relation between differential equations and computer algebra.Chapter 2 concerns the construction of exact solutions of differential equation(s) under the uniform frame work of C — D pair theory introduced by Prof. H.Q.Zhang. The basic theory of C - D pair is presented. The methods for constructing the C - D pair are summarized at the same time. Based on the idea of AC = BD and the pseudo-differential division with remainders uniform and mechnical model of constructing exact solutions to nonlinear differential equation(s) is put forward. This model can cover not only the auxiliary equation methods (sine-cosine method, tanh method, Riccati equation method, projective Riccati method and so on) but also the well-known classical transformations such as backlund transformation, Daboux transformation.The mechanical algorithms— variable coefficient generalized projective Riccati method and new extend-Riccati equation method for constructing the exact solutions (including solitary solutions, soliton-like solutions, periodic solutions) of NLEEs is put forward in Chapter 3. The algorithm is implemented in symbolic computation software Maple. As applications of the algorithm and program, some higher dimensional nonlinear differential equation(s), such as the (2+1)-dimensional generalized shallow water wave equation, (2+l)-dimensional higher order Broer-Kaup system,(3+l)-dimensional KP equation, (2+l)-dimensional Broer-Kaup the high dimensional coupled Burgers equation, (2+1)-dimensional variable coefficient Broer-Kaup system are considered which prove the effectiveness of the algorithm. A further extension of the extend-Riccati equation method is also presented in this Chapter. Some new exact solutions of the KP equation are obtained by using this extended algorithm.Chapter 4 deals with the problem of how to diagonalize and solve a class of systems of differential equations. A new computation differential algbra geometrical method is presented, which extend the elimination method in systems of linear differential equations into the nonlinear case. This method can transform a class of systems of nonlinear PDEs into a single DE and canbe used to construct the general solutions. And with the help of the idea of inversion method, the sufficient and necessary condition for a class of systems of nonlinear PDEs can be diagonalized and linearizd has been presented.Chapter 5 is to introduce the theory of exterior system and its applications in partial differential equations. The algorithm to calculate Cartan characters of systems of PDEs and the involution algorithm are put forward. And the algorithm is implemented in symbolic computation software Maple.In chapter 6, based on combination of using Reid's initial condition algorithm and Cartan characters algorithm in theory of exterior system, the concept of dimension vector is brought forward and the computation algorithm is implemented in symbolic computation software Maple. As applications,the dimension vectors of some well-known systems of partial differential equa-tions,such as Maxwell equations, rotation equations are calculated out.The results are invariable under the equivalent transformations and consistent with physical laws.