Modeling of freezing processes by an one-dimensional thermal conductivity equation with fractional differentiation operators

V. D. Beybalaev ; A. A. Aliverdiev ; R. A. Magomedov ; R. R. Meilanov ; E. N. Akhmedov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 2, pp. 376-387.

Résumé

We have studied the Stefan problem with Caputo fractional order time derivatives. The difference scheme is built. The algorithm and the program for a numerical solution of the Stefan problem with fractional differentiation operator are created. For the given entry conditions and freezing ground parameters we have obtained the space-time temperature dependences for different values of parameter $\alpha $. The functional dependences of the interface motion for the generalized Stefan conditions depending on the value of $\alpha $ are estimated. Finally we have found that the freezing process is slowed down during the transition to fractional derivatives.

Détail

Keywords: Caputo fractional derivative, Stefan problem, the memory effect, difference scheme, heat conductivity, phase boundary.
Mots-clés : fractal structure, phase transition

Bibliographie

[1] Liu Junyi, Xu Mingyu, “Some exact solutions to Stefan problems with fractional differential equations”, J. Math. Anal. Appl., 351:2 (2009), 536–542 | DOI | Zbl

[2] Meilanov R. P., Beybalaev V. D., Shakhbanova M. R., Prikladnye aspekty drobnogo ischisleniia [Applied aspects of fractional calculus], Palmarium Academic Publishing, Saarbrücken, 2012, 135 pp. (In Russian)

[3] Alkhasov A. B., Meilanov R. P., Shabanova M. R., “Heat conduction equation in fractional-order derivatives”, Journal of Engineering Physics and Thermophysics, 84:2 (2011), 332–341 | DOI

[4] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poriadka i nekotorye ikh prilozheniia [Integrals and derivatives of fractional order and some of their applications], Nauka i tekhnika, Minsk, 1987, 688 pp. (In Russian)

[5] Nakhushev A. M., Drobnoe ischislenie i ego primenenie [Fractional calculus and its applications], Fizmatlit, Moscow, 2003, 272 pp. (In Russian)

[6] Tadjeran C., Meerschaert M. M., Scheeffler H.-P., “A second-order accurate numerical approximation for the fractional diffusion equation”, Journal of Computational Physics, 213:1 (2006), 205–213 | DOI | Zbl

[7] Meerschaert M. M., Tadjeran C., “Finite difference approximations for two-sided space-fractional partial differential equations”, Applied Numerical Mathematics, 56:1 (2006), 80–90 | DOI | Zbl

[8] Lafisheva M. M., Shkhanukov-Lafishev M. Kh., “Locally one-dimensional difference schemes for the fractional order diffusion equation”, Comput. Math. Math. Phys., 48:10 (2008), 1875–1884 | DOI | MR | Zbl

[9] Alikhanov A. A., “Difference methods of boundary value problems solution for wave equation equipped with fractional time derivative”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2008, no. 2(17), 13–20 (In Russian) | DOI

[10] Beybalaev V. D., “Numerical method of solution of the problem on transposition of two-sided derivative of the fractional order”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2009, no. 1(18), 267–270 (In Russian) | DOI

[11] Beybalayev V. D., “Mathematical model of heat transfer in fractal structure mediums”, Math. Models Comput. Simul., 2:1 (2010), 91–97 | DOI | MR

[12] Taukenova F. I., Shkhanukov-Lafishev M. Kh., “Difference methods for solving boundary value problems for fractional differential equations”, Comput. Math. Math. Phys., 46:10 (2006), 1785–1795 | DOI | MR

[13] Goloviznin V. M., Korotkin I. A., “Numerical methods for some one-dimensional equations with fractional derivatives”, Differ. Equ., 42:7 (2006), 967–973 | DOI | MR | Zbl

[14] Beybalaev V. D., Abdulaev I. A., Navruzova K. A., Gadgieva T. Yu., “Difference methods of solving the Cauchy problem for usual differential equation with operator of fractional differentiation”, Vestnik Dagestanskogo gosudarstvennogo universiteta. Ser. 1. Estestvennye nauki, 2014, no. 6, 53–61 (In Russian)

[15] Kuznetsov G. V., Sheremet M. A., Raznostnye metody resheniia zadach teploprovodnosti [Difference methods for solving heat conduction problems], Tomsk Politechn. Univ., Tomsk, 2007, 172 pp. (In Russian)