Time-fractional heat conduction in a finite composite cylinder with heat source

• Autorzy: Kukla Stanisław , Siedlecka Urszula

Identyfikatory

DOI

10.17512/jamcm.2020.2.07

• Języki publikacji: EN

Abstrakty

EN

In this paper, the effect of the fractional order of the Caputo time-derivative occurring in heat conduction models on the temperature distribution in a finite cylinder consisting of an inner solid cylinder and an outer concentric layer is investigated. The inner cylinder (core) and the cylindrical layer are in perfect thermal contact. The Robin boundary condition on the outer surface and the Neumann conditions on the ends of the cylinder are assumed. An internal heat source is represented in the mathematical model by taking into account in the heat conduction equation of a function which depends on the space and time variable. An analytical solution of the problem is derived in the form of the double series of eigenfunctions. Numerical examples are presented.

Słowa kluczowe

EN

ractional heat conduction; Caputo derivative; composite cylinder

PL

pochodna Caputo; cylinder kompozytowy; przewodzenie ciepła

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2020
• Tom: Vol. 19, nr 2
• Strony: 85--94
• Opis fizyczny: Bibliogr. 24 poz., rys.

Twórcy

autor

Kukla Stanisław

• Department of Mathematics, Czestochowa University of Technology Czestochowa, Poland

autor

Siedlecka Urszula

• Department of Mathematics, Czestochowa University of Technology Czestochowa, Poland

Bibliografia

• [1] Machado, J.T., Kiryakova, V., & Mainardi, F. (2011). Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 16(3), 1140-1153.
• [2] Valerio, D., Machado, J., & Kiryakova, V. (2014). Some pioneers of the applications of fractional calculus. Fractional Calculus and Applied Analysis, 17(2), 552-578.
• [3] Podlubny, I. (1999). Fractional Differential Equations. San Diego: Academic Press.
• [4] Diethelm, K. (2010). The Analysis of Fractional Differential Equations. Berlin/Heidelberg: Springer-Verlag.
• [5] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier.
• [6] Ciesielski, M., & Blaszczyk, T. (2018). An exact solution of the second order differential equation with the fractional/generalised boundary conditions. Advances in Mathematical Physics, 2018, 7283518.
• [7] Siedlecki, J., Ciesielski, M., & Blaszczyk, T. (2015). Transformation of the second order boundary value problem into integral form - different approaches and a numerical solution. Journal of Applied Mathematics and Computational Mechanics, 14(3), 103-108.
• [8] Ozisik, M.N., & Tzou, D.Y. (1994). On the wave theory in heat conduction. Journal of Heat Transfer, 116(3), 526-535.
• [9] Tzou, D.Y. (1995). The generalized lagging response in small-scale and high-rate heating. International Journal of Heat and Mass Transfer, 38(17), 3231-3240.
• [10] Povstenko, Y., & Klekot, J. (2019). Time-fractional heat conduction in two joint half-planes. Symmetry, 11(6), 800.
• [11] Povstenko, Y. (2016). Time-fractional heat conduction in a two-layer composite slab. Fractional Calculus and Applied Analysis, 19(4), 940-953.
• [12] Ma, J., Sun, Y., & Yang, J. (2018). Analytical solution of dual-phase-lag heat conduction in a finite medium subjected to a moving heat source. International Journal of Thermal Sciences, 125, 34-43.
• [13] Xu, G., & Wang, J. (2018). Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux. Applied Mathematics and Mechanics, 39(10), 1465-1476.
• [14] Zhang, X.-Y., & Li, X.-F. (2019). Transient response of a functionally graded thermoelastic plate with a crack via fractional heat conduction. Theoretical and Applied Fracture Mechanics, 104, 102318.
• [15] Kukla, S., & Siedlecka, U. (2017). An analytical solution to the problem of time-fractional heat conduction in a composite sphere. Bulletin of the Polish Academy of Sciences: Technical Sciences, 65(2), 179-186.
• [16] Kukla, S., & Siedlecka, U. (2018). Fractional heat conduction in a sphere under mathematical and physical Robin conditions. Journal of Theoretical and Applied Mechanics, 56(2), 339-349.
• [17] Datsko, B., Podlubny, I., & Povstenko, Y. (2019). Time-fractional diffusion-wave equation with mass absorption in a sphere under harmonic impact. Mathematics, 7(5), 433.
• [18] Ning, T.-H., & Jiang, X.-Y. (2011). Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation. Acta Mechanica Sinica, 27, 994-1000.
• [19] Yu, B., & Jiang, X. (2019). Temperature prediction by a fractional heat conduction model for the bi-layered spherical tissue in the hyperthermia experiment. International Journal of Thermal Sciences, 145, 105990.
• [20] Jiang, X., & Xu, M. (2010). The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems. Physica A, 389, 3368-3374.
• [21] Povstenko, Y. (2012). Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Archive of Applied Mechanics, 82, 345-362.
• [22] Povstenko, Y. (2014). Axisymmetric solution to time-fractional heat conduction equation in an infinite cylinder under local heating and associated thermal stresses. International Journal of Mechanics, 8(1), 383-390.
• [23] Ezzat, M.A., & El-Bary, A.A. (2016). Effects of variable thermal conductivity and fractional order of heat transfer on a perfect conducting infinitely long hollow cylinder. International Journal of Thermal Sciences, 108, 62-69.
• [24] Blasiak, S. (2018). Time-fractional Fourier law in a finite hollow cylinder under Gaussiandistributed heat flux. EPJ Web of Conferences, 180, 02008

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).