Research On Direct Problem And Inverse Problem Of Some Distributed Parameter Systems

Distributed parameter systems are often described by partial differential equations(PDEs). This dissertation studies mainly the qualitative theory, parameter identification and numerical method problems in PDEs. Therein we place emphasis on the existence and uniqueness of solution, the blow-up in the finite times of solution, the new exact traveling solutions of equations, the identifiability of the identification problems, the new numerical method of equations. The main results, obtained in this dissertation, may be summarized as follows:1. The initial boundary value problems of the nonlinear Sobolev-Galpern equations are studied in Chapter one. Firstly, the original problem is transformed into an equivalent operator equation by a Green's function. Secondly, the existence and uniqueness of the generalized local solution is obtained using the contraction mapping principle. Finally, the blow-up problems of solutions in finite time under different boundary conditions are proved with the aid of Jensen's inequality.2. In Chapter two, using the traveling function transformation, the Jacobi elliptic function expansion method is improved. Some new periodic solutions of a class nonlinear wave equation are obtained by this method. Several important nonlinear equations of mathematical physics such as sine-Gordon equation, φ~4 equation, Klein-Gordon equation, Landau-Ginzburg-Higgs equation and nonlinear telegraph equation are the special cases of the nonlinear wave equation presented in this paper.3. The identification problem described by coupled distributed system is investigated in Chapter three. The theories of optimal control of distributed parameter system are introduced to investigate the parameters identification problem involving the three dimensional population system. The existence, the uniqueness and the boundness of the solution of the system of the partial differential equations are proved by using the monotone method. The existence of the parameters identification problem is established, and the continuous dependence of the solutions of the state equations on the identified parameters is presented.4. A posteriori error estimate of finite element method for Sobolev equation with veriable coefficient and convection-dominated term is presented in Chapter four, which can be used to adjust space mesh. Finally, an adaptive method to adjust local space mesh based on the posterioriestimate is given.