Mechanical Algorithms With Implementations For Computation Of Symmetries And Exact Solutions To Differential Equations

This dissertation considers the mechanical calculation of (classical and nonclassical) Lie symmetries and exact solutions (including solitary solutions, soliton-like solutions, periodic solutions, period-form solutions, multi-solitary solutions and multi-soliton-like solutions) for 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 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 and at the same time, the existence of C -D pair for ordinary differential equations (ODEs) is proved by using the Ore polynomial theory. An algorithm for constructing the C ?D pair of ODEs is performed. The algorithm is implemented in Mathematica.A mechanical algorithm ?variable coefficient generalized Tanh method for constructing the exact solutions (including solitary solutions, soliton-like solutions, periodic solutions, period-form solutions, multi-solitary solutions and multi-soliton-like solutions) of NLEEs is put forward in Chapter 3. The algorithm is implemented in symbolic computation software Maple. As application of the algorithm and program some high dimensional NLEEs, such as the (3+l)-dimensional Jumbo-Miwa equation, the high dimensional coupled Burgers equation, the Boiti-Leon-Pempinelii equation and the (2+l)-dimensional Broer-Kaupare are considered which prove the effectiveness of the algorithm and program. A further extension of the variable coefficient generalized Tanh method is also presented in this Chapter. Some new exact solutions of the variable coefficient generalized KP equation are obtained by using the extended algorithm.Chapter 4 mainly deals with the mechanical calculation of determine equations associated with classical and nonclassical Lie symmetries for differential equation(s). Firstly, an ordinal feedback information algorithm is performed, which largely overcomes the difficulty of "intermediate expression's explosion" and hence improve the efficiency of calculation. Secondly, based on our algorithm and the algorithm for calculating nonclassical symmetries of differential equations in reference [122], a Maple package LIESYM is presented. LIESYM is easy to operate, and has the merit of versatility and efficiency. The package also avoids infinteloops that commonly encountered in the calculation of determine equations associated with nonclassical symmetries. The validity and effectiveness of the algorithm and programe are illustrated. Finally, group classification of a generalized KP equation is undertaken. Group invariant solutions of the equation are obtained by combining the symmetry reduction method with the variable coefficient generalized Tanh method.