Fractional heat conduction in an infinite rod with heat absorption proportional to temperature

• Autorzy: Povstenko Y. , Klekot J.

Identyfikatory

DOI

10.16926/m.2017.22.05

• Języki publikacji: EN

Abstrakty

EN

The one-dimensional time-fractional heat conduction equation with heat absorption (heat release) proportional to temperature is considered. The Caputo time-fractional derivative is utilyzed. The fundamental solutions to the Cauchy and source problems are obtained using the Laplace transform with respect to time and the exponential Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.

Słowa kluczowe

EN

heat conduction; Fourier law; heat absorption

PL

przewodnictwo cieplne; prawo Fouriera; absorpcja ciepła

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2017
• Tom: Vol. 22
• Strony: 61--71
• Opis fizyczny: Bibliogr. 18 poz., rys.

Twórcy

autor

Povstenko Y.

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

autor

Klekot J.

• Czestochowa University of Technology, Faculty of Mechanical Engineering and Computer Science, Institute of Mathematics, al. Armii Krajowej 21, 42-200, Czestochowa, Poland

Bibliografia

• [1] A. G. Butkovskiy, Characteristics of Systems with Distributed Parameters. Nauka, Moscow, 1979 (In Russian).
• [2] G. Doetsch, Anleitung zum praktischer Gebrauch der Laplace-Transformation und der Z-Transformation. Springer, München, 1967.
• [3] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien, 1997.
• [4] A. E. Green, P. M. Naghdi, Thermoelasticity without energy dissipation. J. Elast. 31 (1993), 189–208.
• [5] M. E. Gurtin, A. C. Pipkin, A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31 (1968), 113–126.
• [6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
• [7] R. R. Nigmatullin, To the theoretical explanation of the “universal response”. Phys. Stat. Sol. (b) 123 (1984), 739–745.
• [8] R. R. Nigmatullin, On the theory of relaxation with “remnant” temperature. Phys. Stat. Sol. (b) 124 (1984), 389–393.
• [9] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.
• [10] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002.
• [11] Y. Povstenko, Fractional heat conduction equation and associated thermal stresses, J. Thermal Stresses 28 (2005), 83–102.
• [12] Y. Povstenko, Thermoelasticity which uses fractional heat conduction equation, J. Math. Sci. 162 (2009), 296–305.
• [13] Y. Povstenko, Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder, Fract. Calc. Appl. Anal. 14 (2011), 418–435.
• [14] Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhäser, New York, 2015.
• [15] Y. Povstenko, Fractional Thermoelasticity. Springer, New York, 2015.
• [16] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions. Gordon and Breach Science Publishers, Amsterdam, 1986.
• [17] I. N. Sneddon, The Use of Integral Transforms. McGraw-Hill, New York, 1972.
• [18] K. K. Tamma, X. Zhou, Macroscale and microscale thermal transport and thermomechanical interaction: some noteworthy perspectives. J. Thermal Stresses 21 (1998), 405–449.