On certain exact solutions of the Fourier equation for regions varying with time

Grinberg, G.A.; Koss, V. A.

doi:10.1016/0021-8928(71)90066-9

Cited by 4 publications

(6 citation statements)

References 3 publications

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“…Thus, we have arrived at problem (9) and (29) for G(τ, 0) = 0 or at (12) and (30) for G(r, 0) = 0 in cylindrical coordinates. It is noteworthy that functional (17) corresponding to the formulated problem in cylindrical coordinates retains its form. Therefore, we can use variational and projection methods to analyze this problem.…”

Section: Formulation Of the Problem With The Boundary Condition Of Thmentioning

confidence: 98%

“…where the function Φ(ϕ) is defined from the variational principle using functional (17) and corresponding boundary conditions, and the function P(r) must be found using boundary conditions (30). We introduce the notation μ = Φ(ψ) sin ψ and ω = Φ′(ψ) cos ψ for temporarily unknown parameters.…”

Section: Formulation Of the Problem With The Boundary Condition Of Thmentioning

confidence: 99%

“…The equation type itself is often a relative notion [22]. This circumstance differentiates the problems considered here from the classical ones [12][13][14][15][16][17][18][19][20][21] for which the methods of the present work (as [6,7,10]) should be interpreted as those approximate with certain specific properties. In the classical variants, the moving boundary is present as an initially given condition in the boundary-value problem (replacements of variables, which "arrest" boundary motion (see, e.g., [14]), with a possible change in the form of equations are not taken into account).…”

mentioning

confidence: 99%

“…It should be noted that moving-boundary problems for parabolic-type equations have a long history and have been described in many works (e.g., [12][13][14][15][16][17][18][19]). Recent results on analytical methods of solution of corresponding boundary-value problems have been reviewed in [20,21].…”

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confidence: 99%

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Application of an equivalent equation to description of heat- and mass-exchange processes determined by differential equations in the domain with a moving boundary

Moshinskii

2011

J Eng Phys Thermophy

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A mathematical model describing heat-and mass-exchange processes that are prescribed by differential equations in the domain with a boundary moving with a constant velocity has been constructed for the "plane" problem. Consideration has also been given to the "radial" problem in cylindrical coordinates where the boundary of the domain moves according to the law of the square root of time. The proposed models can be used for description of problems of desalination of soils, extraction, flushing of sediments, and other processes in porous systems.Introduction. We know of mathematical models from the literature for analysis of a number of mass-exchange processes (desalination of soils, flushing of sediments, and others) on the basis of a two-component diffusion model where the equation in the flow part of a porous material describes the process in the domain with a boundary moving according to the law 0 < x < Ut [1][2][3][4][5]. Solution of the problems on the basis of such models involves artificial techniques [3][4][5] of extension of the solution beyond the domain of its determination and of analysis of a certain (derived) integral equation, which makes it difficult to solve the problems in other cases. In so doing, when one is able to find the solution, it is expressed by three quadratures related to conversion of integral transforms with composite (nonstandard) variables. Therefore, one selects analytical solutions rather than finds them. For example, in [3-5], it becomes possible to perform two quadratures out of three in the particular case of a C U (x)-type exponential function appearing in the boundary condition on the "front" of the flushing fluid. In the models of the class in question, one actually does not analyze the process behind the flushing-fluid "front" x = Ut (despite the filtration flow present there), i.e., it is assumed by default that there are no diffusion processes behind this front (D = 0). Otherwise, by virtue of the fact that the velocity of propagation of disturbances in the diffusion processes described by linear equations is infinite for D ≠ 0, we would have to take into account that, with the start of the mass-transfer process, the concentration of the target component (TC) has been changed in the entire accessible region of a porous body (in particular, for x > 0, i.e., in infinite space). It seems likely that the problems were formulated in such a manner under the influence of hyperbolic problems in porous systems where disturbances propagate with a finite velocity. Such problems have been considered, e.g., in [4]. In actual fact, the influence of the domain x > Ut and of the kinetic passage of the TC from the flow part to stagnation zones and back is reduced to the boundary condition on the "front" [1,4].At the same time, it has been established [6, 7] that the system of equations, determined in the domain t ʦ (0, ∞), x ʦ (0, Ut),

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Section: Formulation Of the Problem With The Boundary Condition Of Thmentioning

confidence: 98%

Section: Formulation Of the Problem With The Boundary Condition Of Thmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Application of an equivalent equation to description of heat- and mass-exchange processes determined by differential equations in the domain with a moving boundary

Moshinskii

2011

J Eng Phys Thermophy

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“…−1 , and their combinations, was considered in the case of the diffusion-type equations in [74,75], where it was shown that this family admits exact solutions of the problem. )…”

Section: Classical Fields In Cavities With Moving Boundariesmentioning

confidence: 99%

Nonstationary Casimir Effect and Analytical Solutions for Quantum Fields in Cavities with Moving Boundaries

Dodonov

Advances in Chemical Physics

106

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Addendum to the paper “on certain exact solutions of the fourier equation for regions varying with time”

Grinberg

Koss

1973

Journal of Applied Mathematics and Mechanics

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On certain exact solutions of the Fourier equation for regions varying with time

Cited by 4 publications

References 3 publications

Application of an equivalent equation to description of heat- and mass-exchange processes determined by differential equations in the domain with a moving boundary

Application of an equivalent equation to description of heat- and mass-exchange processes determined by differential equations in the domain with a moving boundary

Nonstationary Casimir Effect and Analytical Solutions for Quantum Fields in Cavities with Moving Boundaries

Addendum to the paper “on certain exact solutions of the fourier equation for regions varying with time”

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