The Solving Methods Of The Cauchy Problem For Elliptic Equation

The Cauchy problem of elliptic equations is widely used in many fields,such as geology,biological electric field,plasma physics and many other fields.The Cauchy problem of the Laplace equation and the Cauchy problem of the Helmholtz equation are two special cases of the Cauchy problem of elliptic equations and they all have a wide application background.For example,the Cauchy problem of Laplace equation has an important application in the wave equation for the initial value problem of the non empty initial epidemic and the data interpretation and data processing of geophysical exploration.The Cauchy problem of Helmholtz equation is often used to describe the vibration of a structure,the acoustic cavity problem,the radiation wave,the scattering of a wave.However,it is seriously ill-posed,which destroys the continuous dependence of solution on data,that is,its solution will cause huge errors due to the tiny perturbation of Cauchy data.Its ill-poseness makes it difficult for us to solve it by classical method,which brings great hindrance to the research work.Therefore,we need some effective ways to solve this ill-posed problem.In this thesis,we study three kinds of the Cauchy problem of elliptic equation: the Cauchy problem of Laplace equation,the Cauchy problem of Helmholtz equation and the Cauchy problem of variable coefficient elliptic equation.We analyse why they are ill-posed,give the corresponding regularization solution method and obtain the convergence estimate between the exact solution of the problem and the regular approximate solutions.We also verify the effectiveness of the regularization method with numerical examples.(1)For the Cauchy problem of the Laplace equation,We give the regularized approximate solution of the exponential variational form and the logarithmic variational form for the problem.We also get the convergence estimate under regularization parameter posterior selection rules.Numerical experiments show the effectiveness of the method.(2)For the Cauchy problem of the Helmholtz equation,we give the regular approximate solution by the smoothing the data.We get the convergence estimates under regularization parameter priors selection rules and regularization parameter posterior selection rules.Numerical examples verify the effectiveness of the method.(3)For the Cauchy problem for the elliptic equation with variable coefficients,we get the regular approximate solutions of the problem through the smoothing regularization method based on the Gaussian kernel.We also get the convergence estimate under regularization parameter posterior selection rules.Numerical example verify the effectiveness of the method.