【作者】 刘晓宇；

【导师】 祝家麟；

【作者基本信息】 重庆大学 ， 计算数学， 2010， 硕士

【摘要】 在科学和工程技术中,许多实际问题归结为求解偏微分方程的反问题。本文考虑对于椭圆型偏微分方程的定解问题而言,给出的边界条件不足的Cauchy型反问题。用边界元法来求解椭圆方程型方程的边值问题,需要已知足够的边界条件,否则问题就是不适定的,会产生很大误差,若采用边界结点法则可以克服这种不适定性[46]。这里的边界结点法是指除了在所研究的区域的边界上分布结点外,还要通过在区域之外分布若干虚拟源点,在所研究的区域之内选取若干内点,以此来求解未给出边界条件的那部分边界上结点的未知函数值。这种方法要通过选取合适的径向基函数的线性组合来表示特解,再利用微分算子(Laplace算子、重调和算子等)的基本解形成满足已知边界条件的线性组合来表示问题的通解,这样的解适合整个边界以及区域内部。本文主要针对二维的非齐次双调和方程的Cauchy反问题用边界结点法求解,利用部分已知的边界条件来推导解的线性组合的待定系数,从而得出适用于全部求解域的解的表达式。求待定系数时,由于所选取的虚拟源点和边界结点数目不匹配,因此我们将该问题转化为一个最小二乘问题;对于Cauchy问题的不适定性,本文使用常用的正则化方法即奇异值分解法,来求解该最小二乘问题所对应的病态线性方程组。本文的数值试验考察了边界光滑和分片光滑的不同区域的情况,并对给出的准确数据以及有噪声的数据所得到得结果进行了分析,分析了几个影响结果准确度的参数。数值试验的结果表明了使用边界结点法求解非齐次双调和方程Cauchy问题的有效性,计算效率高、结果精确度高,还观察到计算结果的误差随着数据中的噪声的减小而收敛。更多还原

【Abstract】 Many practical problems in engineering and science can be classified as inverse problems of partial differential equations. In this thesis, we consider the inverse problem of the Cauchy kind that to the problem for determining solution of elliptic partial differential equations, the boundary conditions are not sufficient.When using boundary element method to solve the boundary value problem of elliptic equations, we need to know sufficient boundary conditions, otherwise, the problem is ill-posed, and the solution is not proper. This ill-posed problem can be avoided by using the Boundary Knot Method[46]. By the Boundary knot method, besides the boundary knots, we place some virtual source points outside the domain, and select some inter points inside the domain in order to help us get the unknown values on the boundary; we should choose the linear combination of some proper radial basis functions to express the particular solution, and using the fundamental solution of partial differential operators such as Laplace operator and biharmonic operator etc. to produce a linear combination to represent the general solution, which satisfies the known boundary condition.In this thesis, we use the Boundary Knot Method to solve the 2-D Cauchy inverse problem of non-homogeneous biharmonic equation. We obtain the coefficients in the combination expression depending on the known boundary data to form the representation of the solution. If the number of the virtual source points cannot match the number of boundary knots, we need to use the least square method. Meanwhile, since the Cauchy problem is ill-posed, we choose the singular value decomposition method to solve the linear equations corresponding to the least square problem.Numerical examples were performed to demonstrate the efficiency and efficacy of the boundary knot method. The examples with both smooth and piecewise smooth boundary and with both exact and noisy known data are tested. Several parameters were checked to find out the influence to the numerical results. Numerical tests show that the method is computationally efficient, accurate, stable and convergent with respect to the decrease of the noise in the data.更多还原