An application of integral equation method to fractional Stefan problem

M. V. Krasnoschok

doi:10.15330/ms.44.1.56-66

Fractional Stefan problem is reduced to a nonlinear integral equation of the Volterra type. Solvability of this equation is established by the contraction mapping principle.

1. Friedman A. Partial differential equations of parabolic type. – R.E. Krieger Publishing Company, Malabar, Florida, 1983.

2. Meirmanov A.M. The Stefan Problem. – De Gruyter, Berlin, 1992. Russian edition: Nauka, Novosibirsk, 1986.

3. Bazaliy B.V., Degtyarev S.P. On the classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid// Math. USSR Sb. – 1988. – V.60, №1. – P. 11–17.

4. Bizhanova G.I., Solonnikov V.A. On free boundary problems for the second order parabolic equations// Algebra Anal. – 2000. – V.12, №6. – P. 98–139.

5. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations. North Holland, Mathematical studies, 204, Elsevier Science B.V., Amsterdam, 2006.

6. Podlubny I. Fractional differential equations. – Academic Press, San-Diego, 1999.

7. Fromberg D. Reaction kinetics under anomalous diffusion. – Ph. D. Thesis, Humboldt-Universitat zu Berlin, 2011.

8. Kochubei A.N. Fractional-order diffusion// Differential Equations. – 1990. – V.26. – P. 485–492.

9. Kochubei A.N. Fractional parabolic systems// Potential analysis. – 2012. – V.37. – P. 1–30.

10. Pskhu A.V. A fundamental solution for a fractional diffusion wave equation// Izvestia RAN. – 2009. – V.73. – P. 141–181. (in Russian)

11. Pskhu A.V. Partial differential equation equations of the fractional order. – Nauka, Moscow, 2005. (in Russian)

12. Cle’ment Ph., Londen S.-O., Simonett G. Quasilinear evolutionary equations and continuous interpolation spaces// J. Diff. Eqs. – 2004. – V.196, №2. – P. 418–447.

13. Lopushans’ka H.P., Lopushans’kyi A. O. Space-time fractional Cauchy problem in spaces of generalized functions// Ukrainian Math. J. – 2013. – V.64, №8. – P. 1215–1230.

14. Kemppainen J. Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition// Abstract and Applied Analysis. – 2011. – Article ID 321903, 11 p.

15. Kemppainen J., Ruotsalainen K. Boundary integral solution of the time-fractional diffusion equation// Integr. equ. oper. theory. – 2009. – V.64. – P. 239–249.

16. Liu J., Xu M. An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices// Z. Angew. Math. Mech. – 2004. – V.84, №1. – P. 22–28.

17. Voller V.R., Falcini F., Garra R. Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects// Physical Review E. – 2013. – V.87. – P. 5622–5625.

18. Li X.C. Fractional moving boundary problems and some of its applications to controlled release system of Drug// Ph. D. Thesis, Shandong University, Jinan, 2009.

19. Voller V.R. An exact solution of a limit case Stefan problem governed by a fractional diffusion equation// Internat. J. Heat and Mass Transf. – 2010. – №53. – P. 5622–5625.

20. Liu J., Xu M. Some exact solutions to Stefan problems with fractional differential equations// J. Math. Anal. Appl. – 2009. – V.351. – P. 536–542.

21. Atkinson C. Moving boundary problems for time fractional and composition dependent diffusion// Frac. Calcul. and Appl. Analysis. – 2012. – V.15, №2. – P. 207–221.

22. Roscani R., Marcus E.S. Two equivalent Stefan’s problems for the time fractional diffusion equation// Fractional Calculus and Applied Analysis. – 2013. – V.16. – P. 802–815.

23. Roscani S., Marcus E.A.S. A new equivalence of Stefan’s problems for the time-fractional-diffusion equation, http://arxiv.org/abs/

	An application of integral equation method to fractional Stefan problem(in Ukrainian)
Author	M. V. Krasnoschok iamm012@ukr.net Division of Applied Problems of Modern Analysis, Institute of Mathematics of NASU
Abstract	Fractional Stefan problem is reduced to a nonlinear integral equation of the Volterra type. Solvability of this equation is established by the contraction mapping principle.
Keywords	Stefan problem; free boundary; fractional integral; fractional derivative; Volterra integral equation
DOI	doi:10.15330/ms.44.1.56-66
Reference	1. Friedman A. Partial differential equations of parabolic type. – R.E. Krieger Publishing Company, Malabar, Florida, 1983. 2. Meirmanov A.M. The Stefan Problem. – De Gruyter, Berlin, 1992. Russian edition: Nauka, Novosibirsk, 1986. 3. Bazaliy B.V., Degtyarev S.P. On the classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid// Math. USSR Sb. – 1988. – V.60, №1. – P. 11–17. 4. Bizhanova G.I., Solonnikov V.A. On free boundary problems for the second order parabolic equations// Algebra Anal. – 2000. – V.12, №6. – P. 98–139. 5. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations. North Holland, Mathematical studies, 204, Elsevier Science B.V., Amsterdam, 2006. 6. Podlubny I. Fractional differential equations. – Academic Press, San-Diego, 1999. 7. Fromberg D. Reaction kinetics under anomalous diffusion. – Ph. D. Thesis, Humboldt-Universitat zu Berlin, 2011. 8. Kochubei A.N. Fractional-order diffusion// Differential Equations. – 1990. – V.26. – P. 485–492. 9. Kochubei A.N. Fractional parabolic systems// Potential analysis. – 2012. – V.37. – P. 1–30. 10. Pskhu A.V. A fundamental solution for a fractional diffusion wave equation// Izvestia RAN. – 2009. – V.73. – P. 141–181. (in Russian) 11. Pskhu A.V. Partial differential equation equations of the fractional order. – Nauka, Moscow, 2005. (in Russian) 12. Cle’ment Ph., Londen S.-O., Simonett G. Quasilinear evolutionary equations and continuous interpolation spaces// J. Diff. Eqs. – 2004. – V.196, №2. – P. 418–447. 13. Lopushans’ka H.P., Lopushans’kyi A. O. Space-time fractional Cauchy problem in spaces of generalized functions// Ukrainian Math. J. – 2013. – V.64, №8. – P. 1215–1230. 14. Kemppainen J. Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition// Abstract and Applied Analysis. – 2011. – Article ID 321903, 11 p. 15. Kemppainen J., Ruotsalainen K. Boundary integral solution of the time-fractional diffusion equation// Integr. equ. oper. theory. – 2009. – V.64. – P. 239–249. 16. Liu J., Xu M. An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices// Z. Angew. Math. Mech. – 2004. – V.84, №1. – P. 22–28. 17. Voller V.R., Falcini F., Garra R. Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects// Physical Review E. – 2013. – V.87. – P. 5622–5625. 18. Li X.C. Fractional moving boundary problems and some of its applications to controlled release system of Drug// Ph. D. Thesis, Shandong University, Jinan, 2009. 19. Voller V.R. An exact solution of a limit case Stefan problem governed by a fractional diffusion equation// Internat. J. Heat and Mass Transf. – 2010. – №53. – P. 5622–5625. 20. Liu J., Xu M. Some exact solutions to Stefan problems with fractional differential equations// J. Math. Anal. Appl. – 2009. – V.351. – P. 536–542. 21. Atkinson C. Moving boundary problems for time fractional and composition dependent diffusion// Frac. Calcul. and Appl. Analysis. – 2012. – V.15, №2. – P. 207–221. 22. Roscani R., Marcus E.S. Two equivalent Stefan’s problems for the time fractional diffusion equation// Fractional Calculus and Applied Analysis. – 2013. – V.16. – P. 802–815. 23. Roscani S., Marcus E.A.S. A new equivalence of Stefan’s problems for the time-fractional-diffusion equation, http://arxiv.org/abs/
Pages	56-66
Volume	44
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

An application of integral equation method to fractional Stefan problem(in Ukrainian)