Curvilinear Co-ordinate Systems in which the Helmholtz Equation Separates

Part 1 of this paper describes a number of the more complicatedco-ordinate systems in which the Helmholtz equation is separable,and discusses the geometrical relationships between them andthe ellipsoidal system, which is the most general of such systems.Part II describes the ellipsoidal system in detail, using theJacobian from in which nearly all recent work has been done,with particular emphasis on the difficulties which occur inapplications. Part III examines the analytic processes of degeneracyfrom the ellipsoidal system to others, including the degeneracyof the separated ordinary differential equations. The Appendixgives a summary of the properties of Jacobian elliptic functionsused in Parts II and III.     

Variations of Constrained Domain Functionals Associated with Boundary-Value Problems

We study variational formulas for maximizers for domain functionalsF(x0, u(x0)), x0, and F(x,u(x))dxover all Lipschitz domains satisfying the constraintg(x) dx=1. Here, u is the solution ofa diffusion equation in . Functional variations arecomputed using domain variations which preserve the constraint exactly. Weshow that any maximizer solves a moving boundary problem for the diffusionequation. Further, we show that, for problems with symmetry, the optimaldomains are balls.     

Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.     

Nonlocal Boundary-Value Problems for Elliptic Systems in Angles

In this paper we consider elliptic systems in plane angle with nonlocal boundary conditions. We prove the existence and uniqueness of the solution in weighted space. For a proof, we study a system of ordinary differential equations with a parameter and nonlocal boundary conditions.     

Boundary-Value Problems for a Class of Operator Differential Equations of High Order with Variable Coefficients

In this paper, we obtain sufficient conditions for the existence of a unique regular solution of the boundary-value problems for operator differential equations of order 2k with variable coefficients. These conditions are expressed solely in terms of operator coefficients of the equations under study.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

Boundary-Value Problems for Second-Order Equations and for Systems of First-Order Equations

The oblique derivative problem for harmonic functions under violation of the Shapiro–Lopatinsky condition is considered as well as some multi-dimensional analogues of the Cauchy–Riemann system. These problems are reduced to nonelliptic pseudo-differential equations. A method generalizing the regularization of singular integral equations is also presented.     

Averaging in Boundary-Value Problems for Systems of Differential and Integrodifferential Equations

Ukrainian Mathematical Journal - The averaging method is applied to the investigation of the problem of existence of solutions of boundary-value problems for systems of differential and...     

Boundary-Value Problems in a Cut Plane

We present a survey of the theory of boundary-value problems in the plane with curvilinear cuts. The problem of linear conjugation, the Riemann-Hilbert problem, and the Riemann-Hilbert-Poincare problem are considered in detail both in the classical setting and for a cut plane, with an emphasis on problems with shifts. The main focus is on the solvability conditions and index formulas in various function classes. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities

We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 102–110, January, 2005.     

Sharp Regularity for Elliptic Systems Associated with Transmission Problems

The paper concerns regularity theory for linear elliptic systems with divergence form arising from transmission problems. Estimates in BMO, Dini and Hölder spaces are derived simultaneously and the gaps among of them are filled by using Campanato–John–Nirenberg spaces. Results obtained in the paper are parallel to the classical regularity theory for elliptic systems.     

Limit Theorems for the Solutions of Linear Boundary-Value Problems for Systems of Differential Equations

Ukrainian Mathematical Journal - We establish sufficient conditions for the convergence of a sequence of solutions of general boundary value problems for systems of linear ordinary differential...     

On a Numerical Algorithm for Solving Boundary-Value Problems for Systems of Hyperbolic Equations of the First Order

In this article, we develop and substantiate a two-step, explicit–implicit numerical algorithm for solving boundary-value problems for systems of partial hyperbolic differential equations of the first order. The algorithm is based on finite-difference schemes with the differences taken against the flow. The approximation error of the solution has order two in the time variable and order one in the spatial variables.