Research Of Well-Posedness And Numerical Solution For The Inverse Parabolic Equations With Convection Terms

The inverse problem has a wide range of application in social science,engineering technology and natural science. It has become one of the fastest growingareas in applied mathematics. A lot of researchers have devoted to this area．By now,inverse parabolic equations without convection terms have been studied well. It’sknown that convection terms may make the problem more complex. Based on it, thispaper focuses on the problem with convection terms.In the first part of this thesis, the author investigates the inverse parabolicproblem subject to the integral over-specification. First, the well-posedness of thisproblem is proved: the iterative method is employed to confirm the existence anduniqueness of a smooth solution pair in Holder classes, and the regularity theory ofparabolic partial differential equations is used to obtain continuous dependence uponthe data. Second, the numerical solution of this problem is studied, and four differentkind of finite difference schemes of the problem are given, i.e. FTCS、BTCS、CN andSaulyev-1.In the second part of this thesis, the author considers the inverse parabolicproblem subject to the temperature over-specification. First, the well-posedness of thisproblem is discussed. Second, the numerical methods of this problem is analyzed, andfour different kind of finite difference schemes of the problem are given, i.e. FTCS、BTCS、CN and Saulyev-1.In the third part of the thesis, taking forward difference scheme for example, theauthor analyzes the stability and convergence of numerical methods. The analysisconfirms the meaningful of the numerical experiments, and the results of numericalexperiment shows the effectiveness and fast convergence of the difference schemes.Finally, the thesis is summarized and the future of the problem is described.