In this article we consider the approximation of an infinite-dimensional neutral-type stochastic differential equation by systems without delay. Continuity modulo Lemma is proved for neutral type stochastic delay evolution equation in a Hilbert space. An approximation system of stochastic differential equations without delay is proposed for neutral type stochastic delay evolution equation.
We describe all possible topological structures of Morse flows and co dimension one gradient flows generated by vector fields emerging in typical one-parameter bifurcations. Research is held on a 2-sphere with holes in the case where the number of singular points of flows is at most six. For that purpose, we construct topological invariants of these flows which are actually the graphs endowed with certain information. In this case, they are separatrix skeletons and distinguished graphs of flows. For example, the saddle-node singularity is specified by selecting a separatrix in the skeleton of the flow before the bifurcation, whereas the saddle connection is specified by a separatrix which connects two saddles. Apart from that, we construct special codes for these graphs so that we can regain a skeleton and a graph from the code and thus recover the whole flow on a certain surface.
In this paper, we consider the problem of seismic wave propagation based on the homogeneous Helmholtz equation with two cylindrical tunnel inclusions under perfectly welded interface conditions. The exact analytical solution of the model is obtained using a boundary method that involves series expansions of incident and reflected SH waves in terms of cylindrical wave functions, as well as coordinate transformations between different reference systems. We also examine the convergence of the approximate solution of the truncated system to the solution of the corresponding infinite system, and compute the relative error. This brief communication focuses on the role of introducing a normalizing coefficient in the truncation order of the resulting infinite block matrix equation in the study of seismic wave propagation boundary value problems for cylindrical inclusions in a half-space.
We study the problem of minimax estimation of functionals depending on the solutions of parabolic boundary value problems, where the observation model is based on time-averaging. Using the averaging method, we construct approximate minimax estimates and analyze the corresponding estimation errors. The obtained results show that the averaging approach provides effective tools for reducing the complexity of minimax estimation in parabolic systems with rapidly oscillating coefficients.
An explicit formula for solving the initial value problem for a general linear nonhomogeneous differential equation with constant coefficients is given. This formula is obtained using the Vandermonde matrix or the confluent Vandermonde matrix constructed from the roots of the characteristic equation associated with the differential equation.
