Studies On Numerical Methods For Two Ⅲ-Posed Problems Of Partial Differential Equations

Inverse problems has become one of the most important branches in the filed of Applied mathematics and enormously promoted the development of science and engineering filed in recent years. Inverse problems are usually ill-posed and small change in the given measurements may result in an enormous deviation of the solution. It is tricky to overcome the ill-poseness of inverse problems, which is also an important research subject. This thesis mainly investigates the solving numerical methods of two kinds of inverse problems, i.e. fractional-diffusion inverse heat conduction problems and inverse solid mechanics Cauchy problems.As well known, fractional-diffusion inverse heat conduction problems have been widely applied in various areas of engineering and sciences, such as in the fluid and continuum mechanics and superdiffusion, non-Gaussian diffusion and subdiffusion. In this paper, we investigate an fractional-diffusion inverse heat conduction problem with no conditions on some boundary. We propose a new scheme to deal with more gen-eral ill-posed problem by recovering the temperature and the heat flux functions at the active boundary from one measurement of the transient temperature function at some arbitrary interior space location of the semi-infinite domain. For the first spital deriva-tives, we use the multi-step FD scheme to improve the accuracy. We will also give a stable numerical solution based on radial basis functions (RBFs) to the time-fractional derivatives. To demonstrate the efficiency and stability of the proposed scheme, we provide some theoretical analysis and numerical results of the proposed method.Cantilever beam is one of the most widely used components in engineering systems. To solve displacements on the unknown boundary is an ill-posed Cauchy problem. We present a high-accuracy, stable and effective numerical method to solve inverse solid mechanics problems with Cauchy boundary data. A so-called general boundary control method (GBCM) is formulated to convert the inverse problem into a set of algebraic system equations similar to a forward problem, leading to a straight-forward procedure to deal with this type of ill-posed inverse problems. The present GBCM is based on the finite element method (FEM) that is widely used as an effective and routine method to formulate well-posed forward problem. To overcome the ill-posedness caused by the unavoidable random noises in the measure data, and obtain a stable and reliable solutions, we employ the Tikhonov regularization method with the L-curve method to choose the regularization parameter. A number of numerical examples are presented to demonstrate the effectiveness of the proposed inverse procedure. The examples show that the method can work well for the inverse solid mechanics problems with noisy Cauchy data and hence has a good potential to be applied to practical engineering problems.