Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

• Authors: Yuriy Povstenko
• Languages of publication: EN

Abstracts

EN

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

Keywords

EN

non-Fourier heat conduction; non-Fickian diffusion; fractional calculus; Mittag-Leffler function; Mainardi function

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1284-1294

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

Yuriy Povstenko

• Institute of Mathematics and Computer Science, Jan Długosz University in Częstochowa, al. Armii Krajowej 13/15, 42-200, Częstochowa, Poland

References

• [1] Y. Z. Povstenko, J. Therm. Stresses 228, 83 (2005)
• [2] Y. Povstenko, Math. Meth. Phys.-Mech. Fields 51, 239 (2008)
• [3] Y. Povstenko, J. Math. Sci. 162, 296 (2009) http://dx.doi.org/10.1007/s10958-009-9636-3[Crossref]
• [4] Y. Povstenko, Phys. Scr. T 136, 014017 (2009) http://dx.doi.org/10.1088/0031-8949/2009/T136/014017[Crossref]
• [5] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications (Gordon and Breach, Amsterdam, 1993)
• [6] R. Gorenflo, F. Mainardi, In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics (Springer-Verlag, New York, 1997) 223
• [7] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
• [8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
• [9] R. L. Magin, Fractional Calculus in Bioengineering (Begell House Publishers, Inc, Connecticut, 2006)
• [10] Y. Povstenko, Fract. Calc. Appl. Anal. 14, 418 (2011)
• [11] W. Wyss, J. Math. Phys. 27, 2782 (1986) http://dx.doi.org/10.1063/1.527251[Crossref]
• [12] W. R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989) http://dx.doi.org/10.1063/1.528578[Crossref]
• [13] Y. Fujita, Osaka J. Math. 27, 309 (1990)
• [14] F. Mainardi, Appl. Math. Lett. 9, 23 (1996) http://dx.doi.org/10.1016/0893-9659(96)00089-4[Crossref]
• [15] F. Mainardi, Chaos Soliton. Fract. 7, 1461 (1996) http://dx.doi.org/10.1016/0960-0779(95)00125-5[Crossref]
• [16] Y. Povstenko, In: I. Petraš, I. Podlubny, K. Kostúr, J. Kacúr, A. Mojžišová (Eds.), 13th International Carpathian Control Conference, May 28–31, 2012, Podbanské, Slovak Republic (Institute of Electrical and Electronics Engineers (IEEE), 2012)
• [17] Y. Povstenko, J. Therm. Stresses 36, 351 (2013) http://dx.doi.org/10.1080/01495739.2013.770693[Crossref]
• [18] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, vol. 3 (McGraw-Hill, New York, 1955)
• [19] J. Mikusinski, Stud. Math. 12, 208 (1951)
• [20] J. Mikusinski, Stud. Math. 18, 191 (1959)
• [21] B. Stankovic, Publ. Inst. Math. 10(24), 113 (1970)
• [22] L. Gajic, B. Stankovic, Publ. Inst. Math. 20(34), 91 (1976)
• [23] R. Gorenflo, Y. Luchko, F. Mainardi, Fract. Calc. Appl. Anal. 2, 383 (1999)
• [24] I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980)
• [25] Y. Povstenko, Int. J. Engrg Sci. 43, 977 (2005) http://dx.doi.org/10.1016/j.ijengsci.2005.03.004[Crossref]

Identifiers

DOI

10.2478/s11534-013-0272-7