Neumann Solution Research Articles

Thermal stress analysis of a semi-infinite solid cylinder under consideration of phase transformation. Approximation by means of stephan problem.

As one of analytical problems of thermal stresses with a moving boundary, we analyzed thermal deformations and thermal stresses caused by a phase transformation. We assumed that a cooled end surface of a semi-infinite solid cylinder, which is kept at a constant temperature, yields the phase transformation with respect to the axial direction; then as a result, the solid cylinder could be seprated into two dissimilar solid phases with thermally and thermoelastically different material properties by a moving interface. In the analytical developments, we adopted Neumann's solution for one-dimensional temperature distribution, and then we analyzed the axisymmetric thermoelastic field with aid of the thermoelastic displacement potential function and Love's displacement function. As an illustration, numerical calculations are carried out for Martensite Transformation, and the influence of the latent heat on the temperature and thermoelastic fields were examined precisely.

Truncation and accumulated errors in wave propagation

Abstract The approximation of the truncation and accumulated errors in the numerical solution of a linear initial-valued partial differential equation problem can be established by using a semidiscretized scheme. This error approximation is observed as a lower bound to the errors of a finite difference scheme. By introducing a modified von Neumann solution, this error approximation is applicable to problems with variable coefficients. To seek an in-depth understanding of this newly established error approximation, numerical experiments were performed to solve the hyperbolic equation ∂U ∂t = −C 1 (x)C 2 (t) ∂U ∂x , with both continuous and discontinuous initial conditions. We studied three cases: (1) C 1 ( x )= C 0 and C 2 ( t )=1; (2) C 1 ( x )= C 0 and C 2 ( t = t ; and (3) C 1 (x)=1+( solx a ) 2 and C 2 ( t )= C 0 . Our results show that the errors are problem dependent and are functions of the propagating wave speed. This suggests a need to derive problem-oriented schemes rather than the equation-oriented schemes as is commonly done. Furthermore, in a wave-propagation problem, measurement of the error by the maximum norm is not particularly informative when the wave speed is incorrect.

Freezing and thawing of soils and permafrost containing unfrozen water or brine

An analytical solution is presented for freezing and thawing of soils and permafrost containing unfrozen water or brine and with temperature dependent thermal properties. Latent heat effects are incorporated into an apparent heat capacity. The partially frozen soil is divided into layers, each with constant thermal properties and with fixed temperatures at the layer boundaries which move with time in a multiple moving boundary problem. Solutions are obtained for the positions of the layer boundaries and for the temperature distribution within each layer. The theory is used to predict the maximum depth of ice penetration and the temperature profile in a large artificial island. Maximum ice penetration in the island is greater than that determined from the two‐layer Neumann solution. Predicted temperature profiles are relatively smooth and do not exhibit a sharp break at the phase boundary. The solution procedure is also applicable to other heat conduction problems in permafrost containing unfrozen water or brine.

The Neumann solution of the multiple scattering problem in a plane-parallel medium—IIb. The resolvent function in a semi-infinite space

We solve the problem of multiple scattering of radiation in a plane-parallel, semi-infinite medium by means of its Neumann solution. The n th iterated function is given for any n ⩾/ 1, which leads to a rigorous description of the n -times scattered radiation field in a half-space. The resolvent function is then calculated as the sum of its Neumann series.

The Neumann solution of the multiple scattering problem in a plane-parallel medium—IIc. The specific intensity in a semi-infinite space

Abstract Using a new formulation of the resolvent function [Rutily and Bergeat JQSRT 38, 61 (1987)], we solve the Milne integral equation for the mean intensity in a semi-infinite plane-parallel atmosphere. The specific intensity is then derived at any optical depth. By expanding the intensity in power series in the albedo, we derive the corresponding intensities of a n-times scattered radiation field propagating in a half-space (n ⩾/ 1).

The Neumann solution of the multiple scattering problem in a plane-parallel medium—IIa. Semi-infinite spaces and H-function

Abstract The aim of this paper is first to expand the H -function in power series in the albedo a and then to derive the general equations involving the n th order terms in this expansion. When multiplied by a n and summed over n from n = 1 to + ∞, they yield the classical equations of the existing theory including the H -equation. This approach is thus a new way to tackle the theory of multiple scattering in a half-space.

Absorbing set forn-person games

This study tries to modify von Neumann and Morgenstern (vN-M) solution. As with John Harsanyi in 1974, vN-M solution is viewed in a dynamic sense. The final outcome of a game not only depends on the ability of players and standards of behavior of the particular society but also upon which imputation is proposed first. An absorbing property is obtained as a result of modifying the bargaining process of Harsanyi. This modified solution concept maintains the internal stability condition of vN-M solution, and replaces their external stability condition by this absorbing property.

An nth order solution of the Neumann series in scattering from rough interfaces

Scattering from rough interfaces has been treated by the integral representation of the Helmholtz equation. This equation involves surface integrals and is well suited for rough interfaces which are piecewise continuous. The well‐known Kirchoff approximation involves replacing the unknown surface pressure field and its gradient by the incident field as a first‐order approximation to the integral equation. This leads to an expression which involves direct integration and will be noted to represent the first term in the Neumann solution of an integral equation of the second kind. Physically, it corresponds to one encounter of the field with the interface. It can be shown that the nth solution corresponds to n encounters with the interface. We derive expressions for the nth solutions of the Neumann series. The expressions are exact for each order for the three‐dimensional case and include an out‐of‐plane scatter.

A Characteristic Dimensionless Time in Phase Change Problems

In this paper a comparison is made between an approximate analytical solution and the numerical finite difference solution for the one dimensional solidification of a phase change material of finite size. The analytical model is not only capable of handling materials with a fixed melting temperature but is also extended to cope with materials with a transition range. In the approximate analytical model, use is made of the well known Neumann solution for the solidification in a semi-infinite region. A characteristic dimensionless time has been derived that can be used in a simplified description of the solidification of a phase-change material. With this description the testing of latent heat storage devices can be simplified and the results can also be used in simulation programs of solar energy installations with a latent heat storage.

An asymptotic, large time solution of the convection Stefan problem with surface radiation

An asymptotic, large time solution has been obtained for the convection Stefan problem with surface radiation. The moving boundary problem has been reformulated as a fixed boundary problem where Lagrange-Burmann expansions are used to complete the variable transformation. An asymptotic solution of the problem is obtained by requiring that the asymptotic expansions assumed for the interface position X ( t ) and wall temperature u w ( t ) for large times are consistent with the resulting interfacial Lagrange-B√ľrmann expansions. It is found that the asymptotic expansions admit Neumann's solution as the leading terms and that logarithmic terms start intervening at the third-order terms of the expansions for nonzero Stefan number. Une solution asymptotique du probl√®me de convection de Stefan pour un temps grand est obtenue dans le cas d'un rayonnement de surface. Le probl√®me de fronti√®re mobile est reformul√© en un probl√®me de fronti√®re fixe o√Ļ les d√©veloppements de Lagrange-B√ľrmann sont utilis√©s pour compl√©ter la transformation. Une solution asymptotique du probl√®me est obtenue sous r√©serve que les d√©veloppements asymptotiques admis pour la position X ( t ) de l'interface et la temp√©rature de paroi u w ( t ) pour des temps grands sont compatibles avec les d√©veloppements de Lagrange-B√ľrmann √† l'interface. On trouve que les d√©veloppements asymptotiques admettent la solution de Neumann comme terme principal et que les termes logarithmiques commencent √† intervenir dans les termes de troisi√®me ordre des d√©veloppements pour un nombre de Stefan nul. Eine asymptotische Langzeitl√∂sung wurde f√ľr das konvektive Stefanproblem mit Oberfl√§chenstrahlung ermittelt. Das Problem beweglicher Begrenzungen wurde in ein Problem mit festen Grenzen umformuliert. Bei der Variablen-Transformation wurden Lagrange-B√ľrmann-Entwicklungen angewandt. Eine asymptotische L√∂sung des Problems erh√§lt man mit der Annahme, daő≤ die asymptotischen Entwicklungen f√ľr die Grenzfl√§chenposition X ( t ) und die Wandtemperatur u w ( t ) f√ľr lange Zeiten mit den nach Lagrange-B√ľrmann ermittelten √ľbereinstimmen. Die asymptotischen Erweiterungen beeinhalten als Hauptterme die Neumann'sche L√∂sung. Logarithmische Ausdr√ľcke beginnen bei Termen dritter Ordnung der Entwicklungen einzugreifen, wenn die Stefanzahl von Null verschieden ist. –Ē–Ľ—Ź –ļo–Ĺ–≤e–ļ—ā–ł–≤–Ĺo–Ļ –∑a–īa—á–ł C—āe—Ąa–Ĺa, –≤–ļ–Ľ—é—áa—é—Če–Ļ –ł–∑–Ľy—áe–Ĺ–łe –Ņo–≤ep—á–Ĺoc—ā–ł, –Ņo–Ľy—áe–Ĺo ac–ł–ľ–Ņ—āo—ā–ł—áec–ļoe pe—ąe–Ĺ–łe, c–Ņpa–≤e–ī–Ľ–ł–≤oe –Ĺa –Īo–Ľ—Ć—ą–ł—á –≤pe–ľe–Ĺa—á. C –Ņo–ľo—Č—Ć—é –Ņpeo–Īpa–∑o–≤a–Ĺ–ł—Ź –õa–≥pa–Ĺ–∂a‚ÄĒ–Ď—ép–ľa–Ĺa co–≤ep—ąe–Ĺ –Ņepe—áo–ī –ļ –∑a–īa—áe c –Ĺe–Ņo–ī–≤–ł–∂–Ĺ—č–ľ–ł –≥pa–Ĺ–ł—Üa–ľ–ł. –ėc—áo–ī—Ź –ł–∑ yc–Ľo–≤–ł—Ź, —á—āo –Ņpe–ī–Ņo–Ľa–≥ae–ľ—če ac–ł–ľ–Ņ—āo—ā–ł—áec–ļ–łe pa–∑–Ľo–∂e–Ĺ–ł—Ź –ī–Ľ—Ź –Ņo–Ľo–∂e–ł–Ĺ—Ź–ľ –Ľa–Ĺ–ł—Ü—č pa–∑–īe–Ľa X ( t ) –ł —āe–ľ–Ņepa—āyp—č c—āe–Ĺ–ļ–ł –Ņp–ł –Īo–Ľ—Ć—ą–ł—á –≤pe–ľe–Ĺa—á co–≥–Ľacy—é—āc—Ź c pa–∑–Ľo–∂e–Ĺ–ł—Ź–ľ–ł –õa–≥pa–Ĺ–∂a‚ÄĒ–Ď—ép–ľa–Ĺa –ī–Ľ—Ź –≥pa–Ĺ–ł—Ü—č pa–∑–īe–Ľa –Ņo–Ľy—áe–Ĺo ac–ł–ľ–Ņ—āo—ā–ł—áec–ļoe pe—ąe–Ĺ–łe. Ha–Ļ–īe–Ĺo, —á—āo pe—ąe–Ĺ–łe He–Ļ–ľa–Ĺa –Ņpe–īc—āa–≤–Ľ—Źe—ā co–Īo–Ļ –≥–Ľa–≤–Ĺ—č–Ļ —á–Ľe–Ĺ ac–ł–ľ–Ņ—āo—ā–ł—áec–ļo–≥o pa–∑–Ľo–∂e–Ĺ–ł—Ź –ł —á—āo –≤ —á–Ľe–Ĺa—á —āpe—ā—Će–≥o –Ņop—Ź–ī–ļa –ľa–Ľoc—ā–ł –Ņo—Ź–≤–Ľ—Ź—é—āc—Ź –Ľo–≥ap–ł—Ą–ľ–ł—áec–ļ–łe –≤—čpa–∂e–Ĺ–ł—Ź.

On a class of moving boundary problems in non-linear heat conduction: Application of a Bäcklund transformation

A reciprocal Backlund transformation is employed to reduce a Stefan problem for a non-linear heat conduction equation to a form which admits a class of exact solutions analogous to the classical Neumann solution.

An approximate solution for planar solidification with Newton cooling at the surface

The classical moving boundary problem for the planar freezing of a semi-infinite saturated liquid with Newton cooling at the wall is well known not to admit an exact solution. Existing perturbation solutions are valid only when the Stefan number is large. Further, since the implementation of the Newton cooling condition involves the step size, numerical solutions are only accurate if extremely small sizes are taken which involves large computing times. Here two new approximate analytic solutions are obtained, the first is an initial or starting solution while the second is valid for subsequent times. In the limit of large Stefan numberα the pseudo-steady state and first order corrected motions arise from both approximations. Further, in the limit of no Newton cooling at the wall the large time solution gives rise to precisely the well known Stefan or Neumann solution. The validity of the approximations is illustrated numerically with reference to previous work and known upper and lower bounds.

The Neumann solution of the multiple scattering problem in a plane-parallel medium—I. The infinite medium

Abstract This paper deals with the study of the radiation field arising from a plane distribution of isotropic sources in a plane-parallel, infinite medium. The classical Neumann method is used together with the theory of singular integrals of the Cauchy type. The main advantage of this technique is that is available for a general treatment of radiative transfer problems in semi-infinite and finite spaces.

Improved lower bounds for the motion of moving boundaries

AbstractIn a recent paper the authors give upper and lower bounds for the motion of the moving boundary for the classical Stefan problem for plane, cylindrical and spherical geometries. On comparison with the exact Neumann solution for the plane geometry and no surface radiation the bounds obtained are seen to be quite adequate for practical purposes except for the lower bound at small Stefan numbers. Here improved lower bounds are obtained which in some measure remove this inadequacy. Time dependent surface conditions are also examined and the new lower bounds obtained for the classical problem are illustrated numerically.

A large time solution of a crystal growth stefan problem

Abstract An asymptotic large time solution has been obtained for a Stefan problem of crystal growth. The analysis presented is valid at large times when the rate of crystal growth is limited by the diffusion process in the environmental melt system so that a thermodynamic equilibrium state prevails at the interface. The first three terms of the asymptotic expansions obtained show that the leading term reduces to a similarity solution of Ghez, a counterpart of Neumann's solution in the thermal Stefan problem and that logarithmic terms start intervening at the third power of the expressions.

Freezing of a Semi-Infinite Medium With Initial Temperature Gradient

Exact solutions to problems of conductive heat transfer with solidification are rare due to the nonlinearity of the equations. The heat balance integral technique is used to obtain an approximate solution to the freezing of a semi-infinite region with a linear, initial temperature distribution. The results indicate that the constant temperature Neumann solution is acceptable for soil systems with a geothermal gradient unless extremely long freezing times are considered. The heat balance integral will yield good solutions, with simple numerical work, even for non-constant initial temperatures.

Integral representations on Hermitian manifolds: the ∂̄-Neumann solution of the Cauchy-Riemann equations

Its significance for complex analysis lies in the fact that if N ƒ G dom D solves O(Nf) = ƒ and df = 0, then u — d Nf is the unique solution of du = ƒ which is orthogonal to kerd. J. J. Kohn has established existence and regularity properties of the solution operator iV, giving the solution of minimal norm—the so called <9-Neumann operator—in case D is strictly pseudoconvex [5], and in more general cases as well [6]. The proofs are based on a priori estimates in L-Sobolev spaces, and therefore they do not give any explicit information about the kernels of N or d N. In recent years there has been much interest in finding more explicit and concrete representations of the abstractly defined operators N and d N (see [2, 3, 9, 10, 12]). In [7] we began to study d N by using the calculus of Cauchy-Fantappie kernels in C n , in analogy to the work of Kerzman and Stein [4] and Ligocka [8] for the Szego, respectively, Bergman kernel; in contrast to the scalar case, the incompatibility of the Euclidean metric with the complex geometry of the boundary of D turned out to be a major obstruction in the general case. In the present paper we overcome this obstruction by generalizing the results in [7] to arbitrary Hermitian manifolds; this enables us to then introduce a special Levi metric—similar to the one in [2]—and to establish the required symmetry properties of the kernels. Our main result gives a new and completely explicit integral representation of the principal part of d N on

The ∂̄-Neumann solution to the inhomogeneous Cauchy-Riemann equation in the ball in 𝐶ⁿ

Let ϑ \vartheta denote the formal adjoint of the Cauchy-Riemann operator ∂ ¯ \overline \partial on C n {{\mathbf {C}}^n} , and let N N denote the Kohn-Neumann operator on the unit ball in C n {{\mathbf {C}}^n} . The operator ϑ ∘ N \vartheta \; \circ \;N provides a natural fundamental solution for ∂ ¯ f = g \overline \partial f = g on the ball. It is our purpose to present the kernel P P for this operator ϑ ∘ N \vartheta \; \circ \;N explicitly—the coefficients are exhibited as rational functions.

An inequality for the coefficient 𝜎 of the free boundary 𝑠(𝑡)=2𝜎√𝑡 of the Neumann solution for the two-phase Stefan problem

We consider a semi-infinite body (e.g. ice), represented by ( 0 , + ∞ ) \left ( {0, + \infty } \right ) , with an initial temperature − c &gt; 0 - c &gt; 0 having a heat flux h ( t ) = − h 0 / t ( h 0 &gt; 0 ) h\left ( t \right ) = - {h_0}/\sqrt t \left ( {{h_0} &gt; 0} \right ) in the fixed face x = 0 x = 0 . If h 0 &gt; c k 1 / π a 1 {h_0} &gt; c{k_1}/\sqrt {\pi {a_1}} there exists a solution, of Neumann type, for the resulting two-phase Stefan problem. If we connect it with the Neumann problem (on x = 0 x = 0 the body has a temperature b &gt; 0 b &gt; 0 we obtain the inequality erf ( σ / a 2 ) &gt; ( k 2 b a 1 / k 1 c a 2 ) \left ( {\sigma /{a_2}} \right ) &gt; \left ( {{k_2}b{a_1}/{k_1}c{a_2}} \right ) for the coefficient σ \sigma of the free boundary s ( t ) = 2 σ t s\left ( t \right ) = 2\sigma \sqrt t , where k i {k_i} , and a i 2 a_i^2 are respectively the thermal conductivity and thermal diffusivity coefficients of the corresponding i i phase i = 1 : i = 1: solid phase, i = 2 : i = 2: liquid phase). If h 0 &gt; c k 1 / π a 1 {h_0} &gt; c{k_1}/\sqrt {\pi {a_1}} there is no solution of the initial problem and if h 0 = c k 1 / π a 1 {h_0} = c{k_1}/\sqrt {\pi {a_1}} the problem has no physical meaning and corresponds to the case where the latent heat of fusion L L tends to infinity.

Global Stability of the Neumann Solution of the Two-phase Stefan Problem

The global stability of the Neumann solution of the two-phase Stefan problem is proved in the framework of the linear theory of perturbations. As a corollary of this result the spatial stability of the free boundary in a one-dimensional two-phase Cauchy-Stefan problem is proved, assuming that the initial temperature has finite limits at infinity, and that its initial perturbation is located in a strip of a finite width.