Anomalous diffusion equation and diffusive stresses

• Autorzy: Povstenko J.
• Języki publikacji: EN

Abstrakty

EN

Essentials of the Riemann-Liouville fractional calculus are recalled. Nonlocal generalizations of the Fourier law of the classical theory of heat conduction relating the heat flux vector to the temperature gradient and of the Fick law of the classical theory of diffusion relating the matter flux vector to the concentration gradient lead to non-classical theories. The time-nonlocal dependence between the flux vectors and corresponding gradients with “long-tale” power kernel can be interpreted in terms of fractional integrals and derivatives and yields the time-fractional diffusion equation.

Słowa kluczowe

EN

fractional calculus; Fourier law; energy flux vector

PL

rachunek ułamkowy; prawo Fouriera; wektor strumienia energii

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2007
• Tom: Vol. 12
• Strony: 97--100
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Povstenko J.

• Institute of Mathematics and Computer Science Jan Długosz University of Częstochowa Armii Krajowej 13/15, 42-201 Częstochowa, Poland

Bibliografia

• [1] S.G. Samko, A.A. Kilbas, O.I. Marichev. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam, 1993.
• [2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
• [3] M. Caputo. Linear models of dissipation whose Q is almoust frequency independent, Part II. Geophys. J. Roy. Astron. Soc. 13, 529539, 1967.
• [4] R. Goreno, F. Mainardi. Fractional calculus and stable probability distributions. Arch. Mech. 50, 377388, 1998.
• [5] Y.Z. Povstenko. Fractional heat conduction equation and associated thermal stress. J. Thermal Stresses 28, 83102, 2005.
• [6] Y.Z. Povstenko. Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation. Int. J. Engng Sci. 43, 977991, 2005.
• [7] Y.Z. Povstenko. Two-dimensional axisymmentric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Int. J. Solids Struct. 44, 2324-2348, 2007.