An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

Salva, Natalia N.; Tarzia, Domingo A.; Villa, L. T.

doi:10.1186/1687-2770-2011-4

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…In this paper, we generalize to the factional derivative in time the previous results for ordinary partial derivative [17], where the initial boundary value problem for the one-dimensional non-classical heat equation in a slab was considered and extended in [1] in the semi-space R + × R n−1 .…”

Section: Introductionmentioning

confidence: 92%

Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

Berrabah¹,

Boukrouche²,

Hedia³

2021

Z. mat. fiz. anal. geom.

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We consider the non-classical heat equation with Caputo fractional derivative with respect to the time variable in a bounded domain Ω ⊂ R + × R d−1 for which the energy supply depends on the heat flux on a part of the boundary S = {0} × R d−1 with homogeneous Dirichlet boundary condition on S, the periodicity on the other parts of the boundary and an initial condition. The problem is motivated by the modeling of the temperature regulation in the medium. The existence of the solution to the problem is based on a Volterra integral of second kind in the time variable t with a parameter in R d−1 , its solution is the heat flux (y, τ ) → V (y, t) = u x (0, y, t) on S, which is also an additional unknown of the considered problem. We establish that a unique local solution exists and can be extended globally in time.

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Section: Introductionmentioning

confidence: 92%

Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

Berrabah¹,

Boukrouche²,

Hedia³

2021

Z. mat. fiz. anal. geom.

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“…Problems of this type are related to the thermostat problem [21,29,30,31,32,34,35]. For example, we will use mathematical ideas developed for the one-dimensional case in [2,25,46,49,50] and for the n-dimensional case in [8,9,10]. The first paper connecting the nonclassical heat equation with a phase-change process (i.e.…”

Section: Introductionmentioning

confidence: 99%

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

Bollati¹,

Natale²,

Semitiel³

et al. 2022

Preprint

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A one-phase Stefan problem for a semi-infinite material is investigated for special functional forms of the thermal conductivity and specific heat depending on the temperature of the phase-change material. Using the similarity transformation technique, an explicit solution for these situations are showed. The mathematical analysis is made for two different kinds of heat source terms, and the existence and uniqueness of the solutions are proved.

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“…Some references on the subject are [6], where F 1 t t 0 u x (0, y, s) ds is replaced by F (u x (0, y, t)), or [7], where it is replaced by F t 0 u x (0, y, s) ds ; see also [4,14,23,24], where the semi-infinite case of this nonlinear problem with F (u x (0, y, t)) has been considered. The non-classical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [14,22]. See also other references on the subject: [8]- [10], [12], [16]- [19].…”

Section: Introductionmentioning

confidence: 99%

A heat conduction problem with sources depending on the average of the heat flux on the boundary

Boukrouche¹,

Tarzia²

2020

Rev. Un. Mat. Argentina

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Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain D = R n−1 × R + for which the internal energy supply depends on an average in the time variable of the heat flux (y, s) → V (y, s) = ux(0, y, s) on the boundary S = ∂D. The solution to the problem is found for an integral representation depending on the heat flux on S which is an additional unknown of the considered problem. We obtain that the heat flux V must satisfy a Volterra integral equation of the second kind in the time variable t with a parameter in R n−1. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.

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An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

Cited by 9 publications

References 8 publications

Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

Explicit solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms

A heat conduction problem with sources depending on the average of the heat flux on the boundary

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