Spatial decay estimates for the heat equation via the maximum principle

Horgan, Cornelius O.; Wheeler, Lewis

doi:10.1007/bf01590509

Cited by 23 publications

(10 citation statements)

References 8 publications

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“…This was established by Knowles [21] using arguments based on differential inequalities for quadratic functionals. A similar result was obtained by Horgan and Wheeler [18] using the maximum principle. A stronger result, analogous to that of Boley [3], Boley and Weiner [4], was established by Horgan et al [16], who showed that the spatial decay of end effects at each fixed time t in the transient problem is faster than that for the steady-state case.…”

Section: Introductionsupporting

confidence: 72%

Spatial decay of transient end effects in functionally graded heat conducting materials

Horgan¹,

Quintanilla²

2001

Quart. Appl. Math.

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Abstract.The purpose of this research is to investigate the influence of material inhomogeneity on the spatial decay of end effects in transient heat conduction for isotropic inhomogeneous heat conducting solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications.The spatial decay of solutions to an initial-boundary value problem with variable coefficients on a semi-infinite strip is investigated.It is shown that the spatial decay of end effects in the transient problem is faster that that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. Explicit decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation with constant coefficients. The results may be interpreted in terms of a Saint-Venant principle for transient heat conduction in inhomogeneous solids.

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Section: Introductionsupporting

confidence: 72%

Spatial decay of transient end effects in functionally graded heat conducting materials

Horgan¹,

Quintanilla²

2001

Quart. Appl. Math.

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“…This result was established in [3] using arguments based on differential inequalities for quadratic functionals. A similar result can be obtained on employing the maximum principle [4], Our purpose in the present paper is to employ qualitative methods to establish stronger results, analogous to those of [7,9], namely that the spatial decay of end effects at time t in the transient problem is faster than that for the steady-state case. For simplicity of presentation, we confine attention to the initial boundary-value problem of classical linear heat conduction in a three-dimensional semi-infinite cylinder.…”

supporting

confidence: 67%

“…Introduction. The spatial decay of solutions of parabolic differential equations has been the subject of much recent attention [1][2][3][4][5]. These studies were motivated by a desire to establish, for parabolic equations, decay estimates analogous to those obtained for elliptic equations in the investigation of Saint-Venant's principle in elasticity theory.…”

mentioning

confidence: 99%

Spatial decay estimates in transient heat conduction

Horgan¹,

Payne²,

Wheeler³

1984

Quart. Appl. Math.

101

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Abstract. The spatial decay of solutions to initial-boundary value problems for the heat equation in a three-dimensional cylinder, subject to non-zero boundary conditions only on the ends, is investigated. It is shown that the spatial decay of end effects in the transient problem is faster than that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. The explicit spatial decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation. The results give rise to versions of Saint-Venant's principle in transient heat conduction.1. Introduction. The spatial decay of solutions of parabolic differential equations has been the subject of much recent attention [1][2][3][4][5]. These studies were motivated by a desire to establish, for parabolic equations, decay estimates analogous to those obtained for elliptic equations in the investigation of Saint-Venant's principle in elasticity theory. (See [6] for a review of recent work on Saint-Venant's principle.)Boley [7][8][9][10] was apparently the first to raise the issue of the validity of a Saint-Venant principle for transient heat conduction. In [8,10] versions of Saint-Venant's principle for non-cylindrical domains, similar to those of von Mises and Sternberg for elasticity, were considered. Using explicit upper bounds for solutions of half-space problems obtained from appropriate fundamental solutions, Boley [10] observed that the spatial influence of transient effects was even more localized than that of the steady-state. The validity of a Saint-Venant principle of the more traditional type, for cylindrical domains subject to non-zero boundary conditions on the ends only, was considered in [7,9], An illustrative example involving the explicit solution of a specific initial-boundary value problem for a semi-infinite rectangular strip is discussed in [9], pp. 186-187. In this example, it is seen that the spatial decay of end effects at any time t in the transient problem is faster than that for the steady-state case [9],

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“…Finally, in Section 8, order estimates for general shells of revolution are discussed. * See also [16] for consideration of parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Maximum principles and pointwise error estimates for torsion of shells of revolution

Horgan

Wheeler

1977

J Elasticity

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This paper is concerned with the question of assessing the quality of approximate thin shell solutions for the problem of axisymmetric torsion of elastic shells of revolution, an issue previously considered by Ho and Knowles [1]. In both works, the objective is to obtain pointwise estimates, based on the threedimensional theory of elasticity, for the errors involved in using an approximate shell theory.The mathematical problem is that of obtaining explicit pointwise gradient estimates for the solutions of homogeneous and nonhomogeneous second-order uniformly elliptic equations, an issue of interest beyond the present context. In contrast to arguments using energy inequalities, here we apply a technique based on maximum principles for such equations.The particular examples of cylindrical, spherical and conical shells of revolution are discussed in detail. In the former two cases, it is shown that the shell theory solution is in fact an exact elasticity solution in the Saint-Venant sense. Asymptotic error estimates for general shells of revolution are also given. RI~SUMIEL'6tude qi-dessous traite de la qualit6 de l'approximation des coques minces appliquEe aux probl~mes de torsion asym6trique des coques 61astiques de r6volution, probl6me d6j~ consid6r6 par Ho et Knowles [1].L'objectif des deux 6tudes est d'6valuer en chaque point, par la th6orie de l'61asticit6 en trois dimensions, les erreurs cons6quentes h l'utilisation de la th6orie de l'approximation des coques.Le probl~me math6matique consiste h obtenir une estimation explicite en chaque point des gradients pour les solutions d'6quations elliptiques uniformes du deuxi~me ordre, homog~nes et non-homog6nes, probl~me d'un inter& qui d6passe le pr6sent contexte. Nous utilisons ici une technique bas6e sur le principe du maximum pour de telles 6quations, par contraste avec les raissonnements utilisant des in6galit6s 6nerg6tiques.Les exemples particuliers des coques de r6volution cylindriques, sph6riques et coniques sont discut6s en d6tail. Dans les deux premiers cas, nous montrons que la solution par la th6orie des coques est en fait une solution exacte dans le sens de Saint-Venant. Des estimations asymptotiques des erreurs dans le cas de coques de r6volution plus g6n6rales sont 6galement donn6es.

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Spatial decay estimates for the heat equation via the maximum principle

Cited by 23 publications

References 8 publications

Spatial decay of transient end effects in functionally graded heat conducting materials

Spatial decay of transient end effects in functionally graded heat conducting materials

Spatial decay estimates in transient heat conduction

Maximum principles and pointwise error estimates for torsion of shells of revolution

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