《抛物型方程反问题的数值解法研究》-毕业论文.doc

精品 I 摘要 在自然科学与工程技术领域中有许多问题都可以用偏微分方程来描述，研究 偏微分方程的数值解是解决上述问题的有力工具。而偏微分方程的数值解的研究 己成为一门专门的学科，国内外有很多学者在这个领域进行研究，并利用各种数 值方法和最新的研究结果来解决工程中的实际问题。 当偏微分方程中的算子、右端项、边界条件、初始条件从过去的已知变成未 知，而原方程的解仍然未知时，就构成了偏微分方程的反问题。由于反问题的不 适定性与非线性性，使得它的理论研究与数值求解都比正问题困难的多，而且涉 及面广。所以如何解决这些问题，成为广大数学工作者、自然科学工作者及工程 技术人员努力开拓的一个崭新的学科领域。 本文应用有限差分方法，研究了第二边值条件下抛物型偏微分方程反问题的 数值解法，在第二边值条件的基础上，假设其中一个边界条件也是未知的，然后 利用附加条件同时确定抛物型偏微分方程中的多个未知参数的数值解法做了进 一步的研究，并进行了相应的数值实验。数值结果表明，在解决形式复杂的抛物 型反问题中，本文的处理方法具有精度高、稳定性好等优点。另外本文还讨论了 一类抛物型偏微分方程反问题的一种差分解法—向前差分格式，利用极值原理证 明了该差分格式的稳定性和收敛性，并给出了在离散L ∞ 模意义下收敛阶数为 2 O (τ+h)，数值例子验证了理论分析结果。 关键词：抛物型偏微分方程；反问题；差分格式；稳定性；收敛性；未知参数； 最小二乘法II ABSTRACT There are many problems can be described by the partial differential equation in the natural science and engineering technology field,studying the numerical solution of these partial differential equations is a strong tool for solving these problems. How to get the numerical solution of these partial differential equations has been become a special subject and many researchers at home and abroad study in this field. All kinds of numerical methods and the recent research results are used to solve this kind of problems. But in fact,if the operator,the right term,the boundary condition or the initial condition is partially unknown and the solution of the equation is unknown either, the partial differential equation becomes an inverse problem.The theory and the solving solution of the inverse problem are more difficult than those of the direct problem and be related with many aspects because the inverse problems is nonlinear and ill-posed,and how to solve these problems becomes a new field that natural science researchers and engineering technicians try to study. In this paper,we used the finite difference method to study the numerical solution of an inverse parabolic problem with Neumann boundary conditions.If one of boundary conditions is considered as unknown function,it is desirable to be able to determine more than one parameter from the given dat