Numerical Computation of a Cauchy Problem for Laplace's Equation

Geophysical Inverse Theory and Regularization Problems

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC 1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green's formulation, the problem can be transformed into a moment problem. Then we propose a numerical algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimate and convergence analysis have also been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.

SynopsisIn this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with [formula omitted]-Laplacian

In the present paper, we consider a Cauchy problem for the Laplace equation in a rectangle domain. A new filtering method is presented for approximating the solution of this problem, and the Hölder-type error estimates are obtained by the different parameter choice rules. Numerical illustration shows that the proposed method works effectively.

An adaptive method of fundamental solutions for solving the Laplace equation

Generalized HPC method for the Poisson equation

Nevanlinna matrices for the strong Stieltjes moment problem

In this paper we weaken the sufficient conditions of harmonicity for functions of two variables. I.I. Privalov had shown that a continuous function that satisfies the Laplace equation in each point of the domain is harmonic. For function of two variables the Privalov’s condition on continuity can be weakened. G.P. Tolstov replaced the continuity condition by the boundness condition, later the author had shown that summability is sufficient. At the same time summability condition can not be weakened substantially.In this paper, while we keep the summability condition, we provide the sufficient condition for harmonicity of the functions, that satisfy less restricted condition than the Laplace equation in all points of the domain. We assume that arbitrary close to any point ζ there exists a collection of four nodes for which a difference relation of Schwartz type for the Laplace equation can be made arbitrary small by the absolute value. Nodes are the ends of two mutually perpendicular segments, that intersect at point ζ. We need to impose a certain weakened continuous assumption on function itself, in case of function that satisfy the traditional Laplace condition this continuity condition follows from the existence of the partial derivatives.

This chapter contains sections titled: Poisson's and Laplace's Equations Solutions to Poisson's and Laplace's Equations Green's Functions Uniqueness of the Electrostatic Solution Method of Images

An asymptotic local solution for axi-symmetric heat conduction problems with temperature discontinuity on the boundary

Often in the analysis of electromagnetic field problems reducible to Laplace's equation in two dimensions, solution is required for only a few points. A numerical method which is applicable to the solution of such problems, provided that the boundary is circular, is given. Excellent results have been obtained also for some cases with piecewise-linear boundaries.

Starting from the triple periodic analogues of simple and double layer potentials, existence and uniqueness theorems for the triple periodic solutions of the fundamental boundary-value problems for the Laplace equation are proved.

Lower and upper solutions for the periodic problem associated with a ϕ – Laplacian equation