Top

Numerical Algorithms

Published in:

07-01-2020 | Original Paper

Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating

Authors: Xiaoping Wang, Huanying Xu, Haitao Qi

Published in: Numerical Algorithms | Issue 4/2020

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model. Firstly, the time fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short-pulse laser heating is built. Laplace and Fourier cosine transforms are performed to derive the semi-analytical expression of temperature distribution in the Laplace domain. Then, by the L1 approximation for the Caputo derivative, the finite difference algorithm is developed for the short-pulse laser heating problem. The solvability, stability, and convergence of this algorithm are also examined. Meanwhile, the efficiency and accuracy of this method have been verified by using three numerical examples. Finally, based on numerical analysis, we study the non-Fourier heat conduction behavior and discuss the effect of variability of parameters, such as fractional parameter and the ratio between the relaxation and retardation times, on the temperature distribution graphically. We believe that this analysis, besides benefiting the laser heating applications, will also provide a deep theoretical insight to interpret the anomalous heat transport process.

previous article B-method approach to blow-up solutions of fourth-order semilinear parabolic equations

next article Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the transformation

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Yilbas, B. S.: Laser heating application. Elsevier, Amsterdam (2012)

Yilbas, B. S., Al-Dweik, A. Y., Al-Aqeeli, N., Al-Qahtani, H. M.: Laser pulse heating of surfaces and thermal stress analysis. Springer, Heidelberg (2014)

Wang, L. Q., Zhou, X. S., Wei, X. H.: Heat conduction: mathematical models and analytical solutions. Springer, Berlin (2008)MATH

Tzou, D. Y.: Macro- to micro-scale heat transfer: the lagging behavior, 2nd edn. Wiley, Chichester (2015)

Qi, H. T., Jiang, X. Y.: Solutions of the space-time fractional Cattaneo diffusion equation. Physica A 390, 1876–1883 (2011)MathSciNetMATH

Jiang, F. M., Liu, D. Y., Zhou, J. H.: Non-Fourier heat conduction phenomena in porous material heated by microsecond laser pulse. Microscale. Therm Eng. 6, 331–346 (2002)

Tzou, D. Y.: Experiment support for the lagging behavior in heat propagation. J. Thermophys Heat Transfer 9, 686–693 (1995)

Antaki, P. J.: Solution for non-Fourier dual phase lag heat conduction in a semi-infinite slab with surface heat flux. Int. J. Heat Mass Transfer 41, 2253–2258 (1998)MATH

Tang, D. W., Araki, N.: Non-Fourier heat condution behavior in finite mediums under pulse surface heating. Mat. Sci. Eng. A 292, 173–178 (2000)

10.

Tzou, D. Y., Chiu, K. S.: Temperature-dependent thermal lagging in ultrafast laser heating. Int. J. Heat Mass Transfer. 44, 1725–1734 (2001)MATH

11.

Shen, B., Zhang, P.: Notable physical anomalies manifested in non-Fourier heat conduction under the dual-phase-lag model. Int. J. Heat Mass Transfer. 51, 1713–1727 (2008)MATH

12.

Ramadan, K., Tyfour, W. R., Al-Nimr, M. A.: On the analysis of short-pulse laser heating of metals using the dual phase lag heat conduction model. J. Heat Transfer. 131, 111301 (2009)

13.

Lee, H. L., Chen, W. L., Chang, W. J., Wei, E. J., Yang, Y. C.: Analysis of dual-phase-lag heat conduction in short-pulse laser heating of metals with a hybrid method. Appl. Therm. Eng. 52, 275–283 (2013)

14.

Majchrzak, E., Mochnacki, B.: Dual-phase lag model of thermal processes in a multi-layered microdomain subjected to a strong laser pulse using the implicit scheme of FDM. Int. J. Therm. Sci. 133, 240–251 (2018)

15.

Fan, Q. M., Lu, W. Q.: A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium. Int. J. Heat Mass Transfer 45, 2815–2821 (2002)MATH

16.

Dai, W. Z., Han, F., Sun, Z. Z.: Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction. Int. J. Heat Mass Transfer 64, 966–975 (2013)

17.

Zhou, J. H., Zhang, Y. W., Chen, J. K.: An axisymmetric dual-phase-lag bioheat model for laser heating of living tissues. Int. J. Therm. Sci. 48, 1477–1485 (2009)

18.

Afrin, N., Zhou, J. H., Zhang, Y. W., Tzou, D. Y., Chen, J. K.: Numerical simulation of thermal damage to living biological tissues induced by laser irradiation based on a generalized dual phase lag model. Numer. Heat Transfer Part A Appl. 61, 483–501 (2012)

19.

Liu, K. C., Chen, Y. S.: Analysis of heat transfer and burn damage in a laser irradiated living tissue with the generalized dual-phase-lag model. Int. J. Therm. Sci. 103, 1–9 (2016)

20.

Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)MATH

21.

Magin, R. L.: Fractional calculus in bioengineering connecticut: Begell House (2006)

22.

Mainardi, F.: Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London (2010)

23.

Monje, C. A., Chen, Y. Q., Vinagre, B. M., Xue, D. Y., Feliu, V.: Fractional-order systems and controls: Fundamentals and applications. Springer, London (2010)MATH

24.

Qi, H. T., Xu, H. Y., Guo, X. W.: The Cattaneo-type time fractional heat conduction equation for laser heating. Comput. Math. Appl. 66, 824–831 (2013)MathSciNetMATH

25.

Xu, H. Y., Qi, H. T., Jiang, X. Y.: Fraction Cattaneo heat equation in a semi-infinite medium. Chinese Phys. B 22, 338–343 (2013)

26.

Ezzat, M. A., El-Karamany, A. S., Fayik, M. A.: Fractional ultrafast laser-induced thermo-elastic behavior in metal films. J. Therm. Stresses 35, 637–651 (2012)

27.

Ezzat, M. A., El Karamany, A. S., Fayik, M. A.: Fractional order theory in thermoelastic solid with three-phase lag heat transfer. Arch. Appl. Mech. 82, 557–572 (2012)MATH

28.

Ezzat, M. A., El-Bary, A. A., Fayik, M. A.: Fractional Fourier law with three-phase lag of thermoelasticity. Mech. Adv. Mater. Struc. 20, 593–602 (2013)

29.

Ferrás, L. L., Ford, N. J., Morgado, M. L., Nóbrega, J. M., Rebelo, M. S.: Fractional Pennes’ bioheat equation: theoretical and numerical studies. Fract. Calc. Appl. Anal. 18, 1080–1106 (2015)MathSciNetMATH

30.

Xu, H. Y., Jiang, X. Y.: Time fractional dual-phase-lag heat conduction equation. Chinese Phys. B 24, 034401 (2015)

31.

Kumar, D., Rai, K. N.: Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy. J. Therm. Biol. 67, 49–58 (2017)

32.

Mishra, T. N., Rai, K. N.: Numerical solution of FSPL heat conduction equation for analysis of thermal propagation. Appl. Math. Comput. 273, 1006–1017 (2016)MathSciNetMATH

33.

Ji, C. C., Dai, W. Z., Sun, Z. Z.: Numerical method for solving the time-fractional dual-phase-lagging heat conduction equation with the temperature-jump boundary condition. J. Sci. Comput. 75, 1307–1336 (2018)MathSciNetMATH

34.

Zhang, X.Y., Cheng, Z.T., Li, X.F.: Non-Fourier fractional heat conduction in two bonded dissimilar materials with a penny-shaped interface crack. Int. J. Therm. Sci. 140, 319–328 (2019)

35.

Kumar, D., Singh, J., Baleanu, D., Rathore, S.: Analysis of a fractional model of Ambartsumian equation. Eur. Phys. J. Plus 133, 259 (2018)

36.

Goswami, A., Singh, J., Kumar, D.: Sushila: An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A 524, 563–575 (2019)MathSciNet

37.

Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y. Q., Vinagre Jara, B. M.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228, 3137–3153 (2009)MathSciNetMATH

38.

Li, X. J., Xu, C. J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)MathSciNetMATH

39.

Li, C. P., Zeng, F. H., Liu, F. W.: Spectral approximations to the fractional integral and derivative. Frac. Calc. Appl. Anal. 15, 383–406 (2012)MathSciNetMATH

40.

Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38, 3871–3878 (2014)MathSciNetMATH

41.

Li, J., Liu, F., Feng, L., Turner, I.: A novel finite volume method for the Riesz space distributed-order advection-diffusion equation. Appl. Math. Model. 46, 536–553 (2017)MathSciNetMATH

42.

Feng, L. B., Zhuang, P., Liu, F., Turner, I., Gu, Y. T.: Finite element method for space-time fractional diffusion equation. Numer. Algorit. 72, 749–767 (2016)MathSciNetMATH

43.

Li, C. P., Zeng, F. H.: Numerical methods for fractional calculus. CRC Press, Boca Raton (2015)MATH

44.

Alikhanov, A. A.: Stability and convergence of difference schemes approximating a two-parameter non-local boundary value problem for time-fractional diffusion equation. Comput. Math. Model. 26, 252–272 (2015)MathSciNetMATH

45.

Feng, L.B., Liu, F.W., Turner, I., Zheng, L.C.: Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD couette flow of a generalized oldroyd-B fluid. Fract. Calc. Appl. Anal. 21, 1073–1103 (2018)MathSciNetMATH

46.

Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Analysis of a meshless method for the time fractional diffusion-wave equation. Numer. algor. 73, 445–476 (2016)MathSciNetMATH

47.

Tayebi, A., Shekari, Y., Heydari, M. H.: A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys. 340, 655–669 (2017)MathSciNetMATH

48.

Kumar, A., Bhardwaj, A., Rathish Kumar, B. V.: A meshless local collocation method for time fractional diffusion wave equation. Comput. Math. Appl. 78, 1851–1861 (2019)MathSciNetMATH

49.

Kumar, D., Singh, J., Baleanu, D.: A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses. Nonlinear Dyn. 91, 307–317 (2018)MathSciNetMATH

50.

Kumar, D., Agarwal, R.P., Singh, J.: A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math. 339, 405–413 (2018)MathSciNetMATH

51.

Odibat, Z. M., Shawagfeh, N. T.: Generalized Taylor’s formula. Appl. Math. Comput. 186, 286–293 (2007)MathSciNetMATH

52.

Jiang, X.Y., Qi, H.T.: Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative. J. Phys. A: Math. Theor. 45, 485101 (2012)MathSciNetMATH

53.

Qiu, T. Q., Tien, C. L.: Heat transfer mechanisms during short-pulse laser heating of metals. J. Heat Transfer 115, 835–841 (1993)

54.

Debnath, M., Bhatta, D.: Integral transforms and their applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)MATH

55.

Sun, Z. Z.: Numerical methods of partial differential equations, 2nd edn. Science Press, Beijing (2012). (in Chinese)

Title: Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating
Authors: Xiaoping Wang
Huanying Xu
Haitao Qi
Publication date: 07-01-2020
Publisher: Springer US
Published in: Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-019-00869-6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 4/2020

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

Numerical solution of separable nonlinear equations with a singular matrix at the solution

Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization

Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems

Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

Premium Partner