Some boundary properties of a solution of the Poisson equation

Certain boundary properties of a solution u of the boundary-value problem for the Poisson equation Δ u = f in a disk are studied. In particular, various estimates for integral norms of the solution through the Green capacity of the condenser composed of the support of the function f and the boundary of the disk and also through the growth rate of the function f are given. The proofs are based on the theorem on coverings of supports of Borel measures outside of which the Green potentials of these measures are bounded by unity. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

On finite element approximation of the gradient for solution of Poisson equation

Summary A nonconforming mixed finite element method is presented for approximation of w with w=f,w| _r =0. Convergence of the order is proved, when linear finite elements are used. Only the standard regularity assumption on triangulations is needed.     

Fast parallel solution of the Poisson equation on irregular domains

We present a fast parallel solution method for the Poisson equation on irregular domains. Due to a simple embedding method using harmonic polynomial approximation, a dominant part of the computation becomes solving one Poisson problem on a disk.     

Bounds for the solution of the Poisson—Boltzmann equation about a cylindrical particle

Pointwise upper and lower bounds for the solution of a Dirichlet problem involving the Poisson-Boltzmann equation in cylindrical coordinates are derived from the theory of maximum principles in differential equations. Simple analytical bounding curves are obtained for various illustrative examples.     

Finite element solution of vector Poisson equation with a coupling boundary condition

The vector Poisson equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown. A rigorous analysis of such a vector Poisson problem and uncoupled solution methods have been presented for domains of C^1,1 and Lipschitz regularity in [1] and [2], respectively. In this work, the finite element approximation of the two uncoupled solution methods is studied, and a convergence analysis of the numerical schemes is provided together with some numerical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 71–83, 2000     

On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.     

On third Poisson structure of KdV equation

The third Poisson structure of the KdV equation in terms of canonical free fields and the reduced WZNW model is discussed. We prove that it is diagonalized in the Lagrange variables which were used before in the formulation of 2d gravity. We propose a quantum path integral for the KdV equation based on this representation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 461–466, June, 1995.     

An asymptotic solution of the Dirichlet problem for the Poisson equation on a nonperiodic frame

In this paper, the Dirichlet problem for the Poisson equation is considered in a nonperiodic framelike domain that consists of thin short strips or cylinders. We construct a complete asymptotic expansion for the solution. We obtain an estimate for the difference between the exact solution and the asymptotic one. Bibliography: 9 titles. Dedicated to Olga Arsenievna Oleinik Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

On a separating solution of a recurrent equation

We consider the recurrent equation

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

Numerical solution of the Poisson equation over hypercubes using reduced Chebyshev polynomial bases

We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations. A special attention is paid on conditioning problems.     

Approximate solution to a time optimal boundary control problem for the wave equation

The paper provides some examples of mutually dual unconstrained optimization problems originating from regularization problems for systems of linear equations and/or inequalities. The solution of each of these mutually dual problems can be found from the solution of the other problem by means of simple formulas. Since mutually dual problems have different dimensions, it is natural to solve the unconstrained optimization problem of the smaller dimension.     

On the decay of a solution of a nonuniformly parabolic equation

For a uniformly parabolic second-order equation with lower-order terms in an unbounded domain, we obtain an upper bound for the decay rate of the solution of the mixed problem with alternating boundary conditions of the first and third types. We prove that the bound is sharp in the case of an equation without lower-order terms in a wide class of domains of revolution. In addition, we show that a solution of a nonuniformly parabolic equation can decay much more rapidly than a solution of a uniformly parabolic equation.     

On the riccati difference equation of optimal control

Necessary and sufficient conditions for the existence of a stabilizing solution to the Riccati difference equation of quadratic optimal control are derived. The results are based on a recent spectral characterization of stabilizability which allow for the time-invariant derivation to go through mutatis mutandis. It is also shown that if the system's dynamics are antistable or observable, then the solution is positive-definite.     

On positiveness of the fundamental solution of a difference equation

We obtain sharp and effective conditions of positiveness of the fundamental solution for linear scalar difference equations. Two approaches are realized, namely, an analytic one (in terms of properties of the characteristic equation) and a geometric one (in terms of a domain in the parameter space of the problem).     

Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.     

On non-explosion for the solution of a stochastic differential equation

In the present work we give general sufficient conditions in terms of Lyapunov functions which ensure existence of a weak solution for a stochastic differential equation for which conditions of existence hold in every bounded region.