Physical Sciences and Mathematics Commons^™

Determination Of A Potential From Cauchy Data: Uniqueness And Distinguishability, Lester Caudill

Department of Math & Statistics Faculty Publications

The problem of recovering a potential q(y) in the differential equation:

−∆u + q(y)u = 0 (x,y) &∈ (0, 1) × (0,1)
u(0, y) = u(1, y) = u(x, 0) = 0
u(x, 1) = f(x), u_y(x, 1) = g(x)

is investigated. The method of separation of variables reduces the recovery of q(y) to a non-standard inverse Sturm-Liouville problem. Employing asymptotic techniques and integral operators of Gel'fand-Levitan type, it is shown that, under appropriate conditions on the Cauchy pair (f, g ), q(y) is uniquely determined, in a local sense, up to its mean. We characterize …

A Direct Method For The Inversion Of Physical Systems, Lester Caudill, Herschel Rabitz, Attila Askar

Department of Math & Statistics Faculty Publications

A general algorithm for the direct inversion of data to yield unknown functions entering physical systems is presented. Of particular interest are linear and non-linear dynamical systems. The potential broad applicability of this method is examined in the context of a number of coefficient-recovery problems for partial differential equations. Stability issues are addressed and a stabilization approach, based on inverse asymptotic tracking, is proposed. Numerical examples for a simple illustration are presented, demonstrating the effectiveness of the algorithm.