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

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.

This paper investigates the numerical computation of a Cauchy problem for Laplace's equation which is a typical ill-posed problem. By using Green's formula, the problem is transformed to a moment problem. For numerical computation of the moment problem, an error estimation and several numerical examples for verification are presented. Necessary and sufficient conditions for the existence of the solution of the Cauchy problems for Laplace's equation in two-dimension are also given.

Generalized HPC method for the Poisson equation

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.

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

Extinction and positivity for the evolution [formula omitted]-Laplacian equation in [formula omitted

In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.

Using examples of boundary value problems for the Helmholtz and Laplace equations, the method is discussed, which allows significantly expanding the class of mathematical physics problems that allow a complete analytical description of fields in non-canonical domains. The basic concept that defines the procedure for constructing analytical solutions is the concept of a general solution of a boundary value problem. Examples are considered that reveal the essence of the method and illustrate the construction of such solutions using partial solutions in different coordinate systems. It is shown that the theorem on the uniqueness of the solution of a boundary value problem does not exclude the possibility of different analytical forms of its representation. Specific problems that arise due to the ambiguity of the form of general solutions and the possibility of different formulations of functional equations that determine the boundary conditions and conjugation conditions of conjugation on the boundaries of partial domains are indicated. The procedures for finding the values ​​of coefficients in general solutions are discussed.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

A fundamentally new—nonsaturable—method is constructed for the numerical solution of elliptic boundary value problems for the Laplace equation in $${{C}^{\infty }}$$-smooth axisymmetric domains of fairly arbitrary shape. A distinctive feature of the method is that it has a zero leading error term. As a result, the method is automatically adjusted to any redundant (extraordinary) smoothness of the solutions to be found. The method enriches practice with a new computational tool capable of inheriting, in discretized form, both differential and spectral characteristics of the operator of the problem under study. This underlies the construction of a numerical solution of guaranteed quality (accuracy) if the elliptic problem under study has a sufficiently smooth (e.g., $${{C}^{\infty }}$$-smooth) solution. The result obtained is of fundamental importance, since, in the case of $${{C}^{\infty }}$$-smooth solutions, the solution is constructed with an absolutely sharp exponential error estimate. The sharpness of the estimate is caused by the fact that the Aleksandrov $$m$$-width of the compact set of $${{C}^{\infty }}$$-smooth functions, which contains the exact solution of the problem, is asymptotically represented in the form of an exponential function decaying to zero (with growing integer parameter $$m).$$

Flow patterns of steady rainfall seeping through bedded land or a hillside with a barrier at great depth

The parametric limit process for Laplace's tidal equations (LTE) is considered, starting from the full equations of motion for a rotating, gravitationally stratified, compressible fluid. The boundary-value problem for free oscillations of angular frequency σ is not well posed if σ2<N2+ 4ω2, whereNis the Väisälä frequency and ω is the rotational speed of the Earth, and the governing partial differential equation is elliptic/hyperbolic on the polar/equatorial sides of the inertial latitudes given by ± σ =f(vertical component of 2ω) if σ < 2ω [Lt ]N. The solution of this ill-posed problem is considered for a global ocean of uniform depth, with the effects of ellipticity, the ‘traditional’ approximation and stratification measured by the small parametersm= ω2a/g, δ =h/aands=hN2/g(g= acceleration due to gravity,h= depth of ocean,a= radius of Earth). LTE represent the joint limitm, δ,s↓ 0 and yield bounded solutions for all latitudes. It is argued that the parametric expansion inmis regular. The joint expansion in δ andswith LTE as the basic approximation is singular at the inertial latitudes if σ < 2ω, which difficulty is traced to the failure of LTE to provide an adequate description of the characteristics in the hyperbolic domain. It is shown that an alternative formulation, in which the buoyancy force is retained in the basic equations in the joint limits↓0, δ↓0 withN[Gt ] 2ω, yields solutions that are uniformly valid in the neighbourhoods of the inertial latitudes. The resulting representation comprises a barotropic mode, which satisfies LTE, and an infinite discrete set of baroclinic modes, each of which has Airy turning points at the inertial latitudes and is trapped between them. The barotropic and baroclinic modes are coupled by the Coriolis acceleration associated with the horizontal component of the Earth's rotation. The relative effects of this coupling are uniformlyO(δ) if σ > 2ω, but it induces currentsO(δ/s1/4) and vertical displacementsO(δ/s3/4) between the inertial latitudes if σ < 2ω [Lt ]N. It appears that resonant amplification of the baroclinic modes forced by the barotropic modes could imply internal displacements that dominate those of the basic motion.