Nonlinear Parabolic Equations Of Certain Types Of Inverse Problem

Inverse problems, which are of interdisciplinary and frontier characteristics, have great significance both in theoretical researches and in applied field. They arise from a variety of practical backgrounds. The difficulty of studying inverse problems is that most of the inverse problems are ill-posed in Hadamard's sense and nonlinear, while the corresponding forward problems are well-posed and linear. Therefore solving the inverse problems is more difficult than solving the direct ones.There are many problems which can be formulated into inverse problems of partial differential equations in the natural sciences and engineering technology. Due to the importance of inverse problems, we study the well-posed analysis and numerical simulation for a few kinds of nonlinear parabolic inverse problems.In this paper, we mainly investigate the well-posedness and numerical algorithms about some nonlinear parabolic inverse problems.The second chapter discusses one-dimensional semilinear parabolic inverse problems using semi-discrete central scheme to construct the numerical method and do numerical simulation. The numerical results coincide with the exact solutions, and confirm the efficiency of the numerical method.In the third and fourth chapters we deal with the blow-up property of nonlinear parabolic and verify the importance of the time choices in the inverse problems. Because of the ill-posedness of the problems, we propose a regularization method for a class of high-dimensional semilinear and quasilinear parabolic inverse problems respectively. Combining the method of the Fourier transformation, the inverse problems are converted into integral equations. Then by adding a small disturbance in the resultant integral equations, the regularized solutions are constructed. Under some reasonable hypotheses, we prove the conditional well-posedness and numerical stability, and give the error estimation of the approximate solutions.In the fifth chapter, we study the nonlinear parabolic partial differential equations about heat and moisture transfer in the textile material. Finally, we also propose related inverse problems.The novelty of this thesis consists in overcoming the difficulty of the nonlinearity, high-dimension and ill-posedness, and we derive the theoretical results and numerical algorithms, and presentation of inverse problems for the textile material design for the first time.