Volume 8, Issue 1

- Necessary and/or sufficient conditions for the identifiability, or nonidentifiability, of coefficients associated with a general class of high-order, quasi-linear (partial) differential equations are established.

- The governing equations are Maxwell’s. The source of the current is either j = e y δ ( z − z 0 ) δ ( x ) f ( t ) or e ϕ δ ( z − z 0 ) δ ( r − r 0 ) f ( t ) (straight wire or loop of current, f ( t ) is a pulse). The conductivity σ ( z ) and permittivity ε ( z ) are unknown in the medium ( z > 0) and are known in the air ( z < 0). The source is in the air: z 0 < 0. The μ is a known constant. The data are the values of E ( x, y, z, t ) and H ( x, y, z, t ) for all t > 0, all x, y ∈ R 2 on the surface z = 0. For t < 0 the system is at rest: the electromagnetic fields vanish. Given the data, an exact inversion procedure is described to find σ ( z ) and ε ( z ).

- We consider an inverse heat conduction problem, the Sideways Heat Equation. This is a Cauchy problem for the heat equation in a quarter-plane, with data given along the line x = 1, where the solution is sought for 0 ≤ x < 1. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We discuss the stability and convergence properties of the wavelet- Galerkin method for solving the sideways heat equation. The wavelets are of Meyer type that have compact support in frequency space. Previous stability results for this method were suboptimal. We show that with additional assumptions concerning the smoothness of the solution, and concerning the definition of the wavelets, we can obtain almost optimal error estimates.

- We consider the problem of determining a connection on a vector bundle over a compact Riemannian manifold with boundary from the known parallel transport between boundary points along geodesics. The main result is the local uniqueness theorem: if two connections ∇' and ∇'' are C -close to a given connection ∇ whose curvature tensor is sufficiently small, then the coincidence of parallel transports with respect to ∇' and ∇'' implies the existence of an automorphism of the bundle which is identical on the boundary and transforms ∇' to ∇'' . A linearized version of the problem is also considered.

- We study the inverse spectral problems for second-order differential equations on a finite interval having a regular singularity inside the interval. For three classes of inverse problems the uniqueness theorems are proved. Necessary and sufficient conditions along with a procedure for the solution of the inverse problems are given.