Determination of One Unknown Thermal Coefficient through the One-Phase Fractional Lamé-Clapeyron-Stefan Problem

DOI: 10.4236/am.2015.613191   PDF   HTML   XML   3,087 Downloads   3,490 Views   Citations


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 .

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.


[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.
[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.
[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.
[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.
[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.
[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.
[26] Voller, V.R. (2014) Fractional Stefan Problems. International Journal of Heat and Mass Transfer, 74, 269-277.
[27] Caputo, M. (1967) A Model of Dissipation Whose Q Is Almost Frequency Independent—II. Geophysical Journal International, 13, 529-539.
[28] Wright, E.M. (1933) On the Coefficients of Power Series Having Exponential Singularities. Journal of the London Mathematical Society, 8, 71-79.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.