Existence of an exact solution for a one-phase Stefan problem with nonlinear thermal coefficients from Tirskii’s method

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.

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“…Then y is a solution to problem (5) if and only if y is a fixed point of the operator τ from K to X given by:…”

Section: Existence and Uniqueness Of The Gme Functionmentioning

confidence: 99%

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

Ceretani

Salva

Tarzia

2018

Nonlinear Analysis: Real World Applications

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“…A generalization of Neumann solution is given in (Briozzo et al, 2004(Briozzo et al, & 2007b for particular sources in the heat equations for both phases. A study of the behaviour of the Neumann solution when the latent heat goes to zero is given in (Tarzia & Villa, 1991).…”

Section: Tx T T I F X S T T T X T T If X S T T Txt T I Fs T X Tmentioning

confidence: 99%

Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface

Tarzia¹

2011

Advanced Topics in Mass Transfer

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“…According to the above result, we proceed to show that there exists a unique solution to problem (18)- (19). Proof.…”

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confidence: 94%

“…Theorem 2.4. The Stefan problem governed by (1)-(5) has a unique similarity type solution given by (11)-(12) where (y, λ) is the unique solution to the functional problem (18)- (19).…”

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confidence: 99%

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Existence and uniqueness of solution for two one-phase Stefan problems with variable thermal coefficients

Bollati

Natale

Semitiel

et al. 2020

Nonlinear Analysis: Real World Applications

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One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet or a Robin type condition at fixed face x = 0. Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solution of the problem with the Dirichlet condition at the fixed face. Computational examples are provided.

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Existence of an exact solution for a one-phase Stefan problem with nonlinear thermal coefficients from Tirskii’s method

Cited by 33 publications

References 16 publications

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface

Existence and uniqueness of solution for two one-phase Stefan problems with variable thermal coefficients

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