Exact Solutions for Nonclassical Stefan Problems

Briozzo, Adriana C.; Tarzia, Domingo A.

doi:10.1155/2010/868059

Cited by 14 publications

(17 citation statements)

References 10 publications

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“…where λ 0 > is a given constant. This problem was studied in [4]. The problem is also considered with another condition at the fixed face x = 0: the convective condition [31].…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Integral balance methods applied to non-classical Stefan problems

et al. 2020

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In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type and the other (with a convective boundary condition at the fixed face) is presented in this work. Due to the complexity of the exact solution it is of interest to obtain some kind of approximate solution. For the above reason, the exact solution of each problem is compared with approximate solutions obtained by applying the heat balance integral method and the refined heat balance integral method, assuming a quadratic temperature profile in space. In all cases, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods.

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“…where λ 0 > is a given constant. This problem was studied in [4]. The problem is also considered with another condition at the fixed face x = 0: the convective condition [31].…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

Integral balance methods applied to non-classical Stefan problems

et al. 2020

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“…Some references on the subject are [8] where F(V ) = F (V ), [5,15,27,28] where the following semi-one-dimension of this nonlinear problem, have been considered. The nonclassical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [9,10,11,25]. More references on the subject can be found in [13,18,19,21,22].…”

Section: Introductionmentioning

confidence: 99%

Non-classical heat conduction problem with nonlocal source

Boukrouche

Tarzia

2017

Bound Value Probl

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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 integral function in the time variable of the heat flux on the boundary S = ∂D, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable t with a parameter in R n-1 . The solution to this Volterra equation is the heat flux (y, s) → V(y, t) = u x (0, y, t) on S, which is an additional unknown of the considered problem. We show that a unique local solution, which can be extended globally in time, exists. Finally a one-dimensional case is studied with some simplifications. We obtain the solution explicitly by using the Adomian method, and we derive its properties. MSC: 35C15; 35K05; 35K20; 35K60; 45D05; 45E10; 80A20Keywords: non-classical n-dimensional heat equation; nonlocal sources; Volterra integral equation; existence and uniqueness of solution; integral representation of solution; explicit solution in one-dimensional case and its properties

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“…Problem P for the slab 0 < x < 1 has been studied in [22]. Recently, free boundary problems (Stefan problems) for the non-classical heat equation have been studied in [3,4,5,6,10,17], where some explicit solutions are also given, and a first study of non-classical heat conduction problem for a n-dimensional material has been given in [2]. Numerical schemes for Problem P when a non-homogeneous boundary condition is considered have been studied in [16] and numerical solutions have been given for two particular choices of data function corresponding to problems with known explicit solutions.…”

Section: Introductionmentioning

confidence: 99%

Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source

2015

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A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition is studied. It is not a standard heat conduction problem because a non-uniform heat source dependent on the heat flux at the boundary is considered. The purpose of this article is to find explicit solutions and analyze how to control their asymptotic temporal behavior through the source term.Explicit solutions independent of the space or temporal variables, solutions with separated variables and solutions by an integral representation depending on the heat flux at the boundary are given. The controlling effects of the source term are analyzed by comparing the asymptotic temporal behavior of solutions corresponding to the same problem with and without source term. Finally, a relationship between the problem considered here with another non-classical problem for the heat equation is established, and explicit solutions for this second problem are also obtained.In this article, we give explicit solutions and analyze how to control them through the source term for several non-classical heat equation problems. In addition, our results enable us to compute the asymptotic temporal behavior of the heat flux at the boundary for each explicit solution obtained. As a consequence of our study, several solved non-classical problems for the heat equation that can be used for testing new numerical methods are given.MSC: 35C05; 35C15; 35C20; 35K55; 45D05; 80A20

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Exact Solutions for Nonclassical Stefan Problems

Cited by 14 publications

References 10 publications

Integral balance methods applied to non-classical Stefan problems

Integral balance methods applied to non-classical Stefan problems

Non-classical heat conduction problem with nonlocal source

Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source

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