Undetermined Coefficient Problems for Quasi-Linear Parabolic Equations

Our research concerns undetermined coefficient problems in partial differential equations, in particular those problems where the unknown coefficients depend only on the dependent variables. The problems modeled by these equations are related to the determination of unknown physical laws or relationships. The nonlinear terms which we seek to recover in our model problems correspond to material properties that have physical significance. These include temperature dependent specific heats, conductivities, and reaction terms. The main analytical tool is the Fixed Point Projection method, which the investigators have developed for use in elliptic and parabolic inverse problems. This method involves projecting the underlying differential operator onto that subset of the domain where the overposed data is given, and reformulating the inverse problem as an equivalent fixed point problem, which is then solved by iteration. Analytical as well as numerical results have been obtained by the investigators, and the FPP method is currently being extended to hyperbolic inverse problems. kr