硕士论文代写-《求解线性与非线性二阶初边值问题的逼近解析解》
【摘要】 二阶常微分方程初边值问题，包括线性与非线性情形，摄动情形以及方程组的情形，在许多领域都有非常广泛的应用但几乎不可能给出解析解，除非是对非常非常简单的线性情形。幸运的是，近十多年来，学者们在寻找逼近解析解方面却有了很大的突破和进展，提出了一些好的方法，尤其是中国学者廖世俊创立的同伦分析法以及何吉欢创立的同伦摄动法利用同伦摄动法给出微分方程的逼近解析解，最主要的关键是同伦的构造但同伦的构造不是唯一的，而是有很多选择有些同伦的构造理论上虽然可行，但在计算时根本无法进行，其计算难度与求解原问题没有本质区别，甚至更难：有的虽计算上可行，但计算量很大因此对各种可能的同伦构造进行比较分析，然后找出计算可行，尤其是逻辑结构简单，易于编程，计算量小的同伦构造，或者另辟蹊径，构造新的同伦，进而得到精确的逼近解析解就具有很大的意义本学位论文针对带Dmchlet边界条件的两点摄动边值问题、带Neumann边界条件的两点摄动边值问题、非线性两点边值问题以及带初值条件的二阶非线性方程组，构造了各种有效可行的同伦，有的是第一次构造，有的是克服了已有同伦方法的缺点，进而给出了高精度的逼近解析解大量数值算例验证了我们方法的有效性。
【Abstract】 Second-order initial and boundary value problems occur in a wide variety of problems inscienti?c and engineering ?elds. It is well known that analytical solutions to these problemscannot be obtained generally except for very simple cases. // Fortunately several good methodshave been proposed to give approximate analytical solutions to these problems in the lastdecade such as the homotopy analysis method by Liao and homotopy perturbation methodby He.When we use homotopy perturbation method to solve these problems analytically, thekey step is the homotopy construction. However, it is worth to note that, even for a sameproblem, the homotopy can be freely constructed in many ways and the initial approximatesolution can also be freely selected. As we will see, homotopies constructed in some way maybe impossible to conduct real computation or the computational cost is in fact quite expensiveeven if they are theoretically ?exible. Therefore it is signi?cant to determine or create anoptimal homotopy in the sense that whose logical structure is simple and computational costis cheap by comparing and analyzing various possible construction ways.In this thesis, for the two-point perturbed boundary value problem with Dirichlet andNeumann boundary conditions, nonlinear two-point boundary value problems and systemsof second-order nonlinear initial value problems, various possible homotopy constructionways are discussed and compared, then some e?cient homotopies for these problems areconstructed and so that accurate analytical solutions are given. Numerical examples aretested to illustrate the e?ciency of our construction methods of homotopies.
代写硕士论文-【关键词】 同伦摄动方法； 同伦构造； 两点摄动边值问题； 非线性两点边值问题； 二阶初值问题的非线性方程组； 逼近解析解；
【Key words】 Homotopy perturbation method； Two-point perturbed boundary valueproblem； Nonlinear two-point boundary value problems； Non-linear system of second-orderinitial value problems；