Boundary Knot Method For The Cauchy Problem Associated With The Non-homogeneous Biharmonic Equation

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.