Band 17, Heft 3

A numerical solution for the inverse kinematic problem of seismics with inner sources is suggested. We assume the coordinates of centers and the times of beginning of earthquakes to be known in some interior area (a focal zone) of the Earth. The purpose is to determine a velocity structure of the Earth in the focal zone. For the solution of the problem, two algorithms are suggested. The first algorithm is based on the idea of inversion of the wave front and its local approximation by a plane or a sphere. Main tools of the second algorithm are the procedures of interpolation, approximation and smoothing of functions based on the modern algorithms of multidimensional splines. Properties of the algorithms by means of test media with known velocity structure are investigated.

We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

We show a strategy recently developed to prove global in time existence and uniqueness results for integrodifferential inverse problems. The models we discuss in this paper are: which are: the strongly damped wave equation with memory, the heat equation with memory and a model in the theory of combustion with memory, respectively. Here f is a given nonlinear function and Ω is a bounded domain in ℝ 3 . We determine u and the convolution memory kernel h under suitable initial–boundary conditions and assuming to know an additional restriction on the state variable u , for example of type where φ and g are given functions representing the type of device used to measure u .

In this paper, we consider the stability of an inverse spectral problem for a nonsymmetric ordinary differential operator. We give an estimate for deviation in the coefficients of this operator when the spectral data perturb. Our result shows that if two spectral data are sufficiently close to one another, then the corresponding two differential operators must be close each other in the sense of C ϑ -norm ( ϑ = 0, 1).