【作者】 张晶晶；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 应用数学， 2009， 硕士

【摘要】 本文研究微分方程(组),特别是在力学、空气动力学、等离子体物理、生物物理和化学物理等现代科学技术中引出的非线性偏微分方程(组)的守恒律的机械化算法,包括偏微分方程(组)的守恒律的两种新算法,并予以程序实现。并介绍了守恒律分类的一种方法,运用这一方法对方程u_t=(A(u)u_x)_x+B(u)u_x+C(u)的守恒律进行了分类,得到了它的守恒律分类的结果和势系统。第一章介绍了数学机械化思想与符号计算以及一些有关这些学科的国内外学者所取得的成果,回顾了李群理论的起源,简单介绍了Lie对称和守恒律的历史和发展。同时,叙述了群分类及守恒律分类的近期研究情况。第二章介绍了求解微分方程的“AC=BD”理论及其应用,通过实例说明了这一理论的使用方式和适用范围,并通过具体的变换给出了C-D对的构造方法。第三章首先介绍了守恒律的相关理论,并提出计算微分方程守恒律两个机械化算法——原方程不求导方法和原方程求导方法。第一种算法以破裂孤立子方程为例,介绍了算法求守恒律的过程;第二种算法以形变Boussinesq方程为例,介绍了如何用此种算法来求守恒律。通过这些例子说明了算法的有效性。第四章介绍了守恒律分类的一种方法,运用这一方法对方程u_t=(A(u)u_x)_x+B(u)u_x+C(u)的守恒律进行了分类,得到的守恒律分类的结果中包含了Popovych,R.O.和Ivanova,N.M.对微分方程u_t=(A(u)u_x)_x+B(u)u_x的守恒律进行分类得到的结果,还得到了微分方程的势系统。更多还原

【Abstract】 In this dissert,the mechanical algorithm of conservation laws to differential equations is considered,especially for those nonlinear evolution equations arising from the fields of fluid mechanics,aerodynamics,plasma physics,biophysics and chemical physics.A method of the classification of the conservation laws is introduced,and it is applied to classify the conservation laws of the differential equations u_t=(A(u)u_x)_x+ B(u)Ux + C(u).The results of the classification of the conservation laws of the differential equations and the potential systems of the differential equations are obtained.Chapter 1 is devoted to introducing the related development of mathematical physics mechanization at home and abroad,the development of Lie Group,symmetry and conservation law.In addition,the development of group classification and classification of conservation laws is also presented.The constru tion of exact solutions to nonlinear partial differential equations under the instruction of the AC = BD theory is given in Chapter 2.Some examples are chosen to illustrate the principle and the application range.And some methods of constructing the operators of C and D are introduced.The mechanical algorithm for constructing the conservation laws of differential computation software Maple in Chapter 3.As applications of the algorithm and program, some higher dimensional nonlinear differential equation(s),such as Modified equation and Breaking soliton equation are considered,which is devoted to prove the effectiveness of the algorithm.In Chapter 4,a method of the classification of the conservation laws is introduced, which is applied to classify the conservation laws of the differential equations u_t=(A(u)u_x)_x+B(u)u_x+C(u).The results cover all results of the classification of the conservation laws of the differential equations u_t=(A(u)u_x)_x+B(u)u_x,which are got by Popovych,R.O.and Ivanova,N.M.And the potential systems of the differential equations are also obtained in this part.更多还原