AM  Vol.6 No.13 , November 2015
Determination of One Unknown Thermal Coefficient through the One-Phase Fractional Lamé-Clapeyron-Stefan Problem
ABSTRACT
We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815; 2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .

Cite this paper
Tarzia, D. (2015) Determination of One Unknown Thermal Coefficient through the One-Phase Fractional Lamé-Clapeyron-Stefan Problem. Applied Mathematics, 6, 2182-2191. doi: 10.4236/am.2015.613191.
References
[1]   Alexiades, V. and Solomon, A.D. (1993) Mathematical Modelling of Melting and Freezing Processes. Hemisphere-Taylor & Francis, Washington DC.

[2]   Cannon, J.R. (1984) The One-Dimensional Heat Equation. Addison-Wesley, Menlo Park.
http://dx.doi.org/10.1017/CBO9781139086967

[3]   Carslaw, H.S. and Jaeger, J.C. (1959) Conduction of Heat in Solids. Clarendon Press, Oxford.

[4]   Crank, J. (1984) Free and Moving Boundary Problem. Clarendon Press, Oxford.

[5]   Fasano, A. (2005) Mathematical Models of Some Diffusive Processes with Free Boundary. MAT-Series A, 11, 1-128.

[6]   Gupta, S.C. (2003) The Classical Stefan Problem. Basic Concepts, Modelling and Analysis. Elsevier, Amsterdam.

[7]   Lunardini, V.J. (1991) Heat Transfer with Freezing and Thawing. Elsevier, London.

[8]   Rubinstein, L.I. (1971) The Stefan Problem. American Mathematical Society, Providence.

[9]   Tarzia, D.A. (2000) A Bibliography on Moving-Free Boundary Problems for the Heat-Diffusion Equation. The Stefan and Related Problems. MAT-Series A, 2, 1-297.

[10]   Tarzia, D.A. (2011) Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface. In: El-Amin, M., Ed., Advanced Topics in Mass Transfer, InTech Open Access Publisher, Rijeka, 439-484.

[11]   Tarzia, D.A. (1981) An Inequality for the Coefficient of the Free Boundary of the Neumann Solution for the Two-Phase Stefan Problem. Quarterly of Applied Mathematics, 39, 491-497.

[12]   Tarzia, D.A. (1982) Determination of the Unknown Coefficients in the Lamé-Clapeyron-Stefan Problem (Or One-Phase Stefan Problem. Advances in Applied Mathematics, 3, 74-82.
http://dx.doi.org/10.1016/S0196-8858(82)80006-7

[13]   Kilbas, A., Srivastava, H. and Trujillo, H. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.

[14]   Mainardi, F. (2010) Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London.

[15]   Podlubny, S.I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[16]   Gorenflo, R., Luchko, Y. and Mainardi, F. (1999) Analytical Properties and Applications of the Wright Function. Fractional Calculus and Applied Analysis, 2, 383-414.

[17]   Luchko, Y. (2010) Some Uniqueness and Existence Results for the Initial-Boundary-Value Problems for the Generalized Time-Fractional Diffusion Equation. Computers & Mathematics with Applications, 59, 1766-1772.
http://dx.doi.org/10.1016/j.camwa.2009.08.015

[18]   Mainardi, F., Luchko, Y. and Pagnini, G. (2001) The Fundamental Solution of the Space-Time Fractional Diffusion Equation. Fractional Calculus and Applied Analysis, 4, 153-192.

[19]   Mainardi, F., Mura, F. and Pagnini, G. (2010) The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. International Journal of Differential Equations, 2010, Article ID: 104505.

[20]   Falcini, F., Garra, V. and Voller, V.R. (2013) Fractional Stefan Problems Exhibiting Lumped and Distributed Latent-Heat Memory Effects. Physical Review E, 87, Article ID: 042401.

[21]   Jinyi, L. and Mingyu, X. (2009) Some Exact Solutions to Stefan Problems with Fractional Differential Equations. Journal of Mathematical Analysis and Applications, 351, 536-542.
http://dx.doi.org/10.1016/j.jmaa.2008.10.042

[22]   Kholpanov, L.P., Zaklev, Z.E. and Fedotov, V.A. (2003) Neumann-Lamé-Clapeyron-Stefan Problem and Its Solution Using Fractional Differential-Integral Calculus. Theoretical Foundations of Chemical Engineering, 37, 113-121.

[23]   Roscani, S. and Marcus, E. (2013) Two Equivalent Stefan’s Problems for the Time-Fractional Diffusion Equation. Fractional Calculus and Applied Analysis, 16, 802-815.
http://dx.doi.org/10.2478/s13540-013-0050-7

[24]   Roscani, S. and Tarzia, D.A. (2014) A Generalized Neumann Solution for the Two-Phase Fractional Lamé-Clapeyron-Stefan Problem. Advances in Mathematical Sciences and Applications, 24, 237-249.

[25]   Voller, V.R. (2010) An Exact Solution of a Limit Case Stefan Problem Governed by a Fractional Diffusion Equation. International Journal of Heat and Mass Transfer, 53, 5622-5625.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.07.038

[26]   Voller, V.R. (2014) Fractional Stefan Problems. International Journal of Heat and Mass Transfer, 74, 269-277.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.008

[27]   Caputo, M. (1967) A Model of Dissipation Whose Q Is Almost Frequency Independent—II. Geophysical Journal International, 13, 529-539.
http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.x

[28]   Wright, E.M. (1933) On the Coefficients of Power Series Having Exponential Singularities. Journal of the London Mathematical Society, 8, 71-79.
http://dx.doi.org/10.1112/jlms/s1-8.1.71

 
 
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