A Method Of Fundamental Solutions For Anisotropic Heat Conduction Problem And Its Inverse Problems

Inverse problems of partial differential equations arise in many scientific and engineering applications. This thesis focuses on inverse problems of heat conduction in anisotropic materials which are of great importance in practice.In Chapter 2 and Chapter 3 , we will solve two kinds of inverse problems of anisotropic heat conduction, namely, IHCP (inverse heat conduction problem) and BHCP (backward heat conduction problem), respectively. For these two inverse problems, the method of fundamental solutions (MFS) is proposed and employed. The basic idea of the proposed method is : At first, to obtain the fundamental solution of the governing equation through variables transformation and make the anisotropy remain in the numerical scheme;Then to apply the MFS to solve the problems in both spatial and time domains directly. The interpolation matrixes arising from the MFS are highly ill-conditioned due to the highly ill-posed problems (both IHCP and BHCP). Thus, a regularization method should be employed. In this thesis, we choose truncated singular value decomposition to solve the resulting matrix equations, while the regularization parameter of TSVD is determined by the L-curve criterion. In the end, several numerical examples are presented to certify the efficiency and efficacy of the proposed method for the anisotropic inverse problems with both exact and noisy data. Meanwhile the convergency of the method and the stability with respect to the data noise, and the relationship between the numerical results and the parameter T are also analyzed.The geodesic distance based method of fundamental solutions for anisotropic heat conduction is presented in Chapter 4. This MFS is generally similar as the MFS mentioned above. Both of them are to embed the anisotropy in the numerical algorithm and then solve the problems directly. The difference between them is that any variables transformation is not required to get the fundamental solution in this MFS where the Euclidean distance is replaced by the geodesic distance. Similarly, numerical experiments are performed to demonstrate the efficiency and efficacy of the geodesic distance based MFS for piecewise smooth domain. In addition, the effect of the number of collocation points and the stability with respect to the anisotropy are discussed.In Appendix A. the radial basis function (RBF) methods are briefly introduced.These methods include the traditional RBF and the differential operator-geared RBF collocation methods, i.e. the MFS and the BKM (boundary knot method). For the sake of search convenience, some popular RBF, and fundamental solutions and nonsingular general solutions to some differential operators are also listed.