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Inverse problems for singular differential equations
Abstract
In this dissertation we consider two inverse problems which arise in singular differential equations posed on unbounded domains. The first is the recovery of a Schroedinger potential $q(x)$ in the singular differential equation$${-}u\prime\prime(x)+(q(x)-\lambda)u(x)=f(x)\quad {-}\infty0\quad x>0$$subject to an initial condition and mixed boundary condition. This equation governs the transport of a chemical tracer through saturated porous media. Based on experimental evidence in the hydrogeology literature we assume D is bounded, continuous, and approaches a constant limit as $x\to\infty.$ The convection-diffusion equation can be written in its variational form, and it can be shown that the forward problem has a unique solution. The inverse problem was studied for the steady-state case, and a new Sinc-Galerkin method was developed to solve the steady-state forward problem. This problem was also formulated as a nonlinear least-squares problem, and the Tikhonov functional was minimized using the Levenberg-Marquardt method.
Subject Area
Mathematics
Recommended Citation
Mueller, Jennifer L, "Inverse problems for singular differential equations" (1997). ETD collection for University of Nebraska-Lincoln. AAI9805519.
https://digitalcommons.unl.edu/dissertations/AAI9805519