Time-Fractional Heat Conduction with Heat Absorption in a Half-Line Domain Due to Boundary Value of the Heat Flux Varying Harmonically in Time

  • Yuriy PovstenkoEmail author
  • Tamara Kyrylych
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)


The time-fractional heat conduction equation with heat absorption is considered in a half-line domain under the mathematical and physical Neumann boundary conditions varying harmonically in time. The Caputo derivative is employed. The Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate are used. The solutions are obtained in terms of integrals with integrands being the Mittag-Leffler functions. The numerical results are illustrated graphically.


Fractional calculus Caputo derivative Harmonic impact Mittag-Leffler function 


  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Abuteen, E., Freihat, A., Al-Smadi, M., Khalil, H., Khan, R.A.: Approximate series solution of nonlinear, fractional Klein-Gordon equations using fractional reduced differential transform method. J. Math. Stat. 12, 23–33 (2016). Scholar
  3. 3.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Oxford University Press, Oxford (1959)zbMATHGoogle Scholar
  4. 4.
    Cui, Z., Chen, G., Zhang, R.: Analytical solution for the time-fractional Pennes bioheat transfer equation on skin tissue. Adv. Mater. Res. 1049–1050, 1471–1474 (2014). Scholar
  5. 5.
    Crank, J.: The Mathematics of Diffusion, 2nd edn. Clarendon Press, Oxford (1975)zbMATHGoogle Scholar
  6. 6.
    Damor, R.S., Kumar, S., Shukla, A.K.: Solution of fractional bioheat equation in terms of Fox’s H-function. SpringerPlus 111, 1–10 (2016). Scholar
  7. 7.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of Integral Transforms, vol. 1. McGraw-Hill, New York (1954)zbMATHGoogle Scholar
  8. 8.
    Ezzat, M.A., AlSowayan, N.S., Al-Muhiameed, Z.I.A., Ezzat, S.M.: Fractional modeling of Pennes’ bioheat transfer equation. Heat Mass Transf. 50, 907–914 (2014). Scholar
  9. 9.
    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). Scholar
  10. 10.
    Gabbiani, F., Cox, S.J.: Mathematics for Neuroscientists, 2nd edn. Academic Press, Amsterdam (2017)zbMATHGoogle Scholar
  11. 11.
    Gafiychuk, V.V., Lubashevsky, I.A., Datsko, B.Y.: Fast heat propagation in living tissue caused by branching artery network. Phys. Rev. E 72, 051920 (2005). Scholar
  12. 12.
    Golmankhaneh, A.K., Golmankhaneh, A.K., Baleanu, D.: On nolinear fractional Klein-Gordon equation. Sig. Process. 91, 446–451 (2011). Scholar
  13. 13.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences (Courses and Lectures), vol. 378, pp. 223–276. Springer, Wien (1997). Scholar
  14. 14.
    Gravel, P., Gauthier, C.: Classical applications of the Klein-Gordon equation. Am. J. Phys. 79, 447–453 (2011). Scholar
  15. 15.
    Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31, 113–126 (1968). Scholar
  16. 16.
    Holmes, W.R.: Cable equation. In: Jaeger, D., Jung, R. (eds.) Encyclopedia of Computational Neuroscience, pp. 471–482. Springer, New York (2015).
  17. 17.
    Kheiri, H., Shahi, S., Mojaver, A.: Analytical solutions for the fractional Klein-Gordon equation. Comput. Meth. Diff. Equat. 2, 99–114 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  19. 19.
    Liu, J., Xu, L.X.: Estimation of blood perfusion using phase shift in temperature response to sinusoidal heating the skin surface. IEEE Trans. Biomed. Eng. 46, 1037–1043 (1999). Scholar
  20. 20.
    Liu, J., Zhou, Y.X., Deng, Z.S.: Sinusoidal heating method to noninvasively measure tissue perfusion. IEEE Trans. Biomed. Eng. 49, 867–877 (2002). Scholar
  21. 21.
    Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996). Scholar
  22. 22.
    Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996). Scholar
  23. 23.
    Mandelis, A.: Diffusion waves and their uses. Phys. Today 53, 29–33 (2000). Scholar
  24. 24.
    Mandelis, A.: Diffusion-Wave Fields: Mathematical Methods and Green Functions. Springer, New York (2001)CrossRefGoogle Scholar
  25. 25.
    Monai, H., Omori, T., Okada, M., Inoue, M., Miyakawa, H., Aonishi, T.: An analytic solution of the cable equation predicts frequency preference of a passive shunt-end cylindrical cable in response to extracellular oscillating electric fields. Biophys. J. 98, 524–533 (2010). Scholar
  26. 26.
    Nigmatullin, R.R.: To the theoretical explanation of the universal response. Phys. Stat. Sol. (b) 123, 739–745 (1984). Scholar
  27. 27.
    Nigmatullin, R.R.: On the theory of relaxation for systems with “remnant” memory. Phys. Stat. Sol. (b) 124, 389–393 (1984). Scholar
  28. 28.
    Nowacki, W.: State of stress in an elastic space due to a source of heat varying harmonically as function of time. Bull. Acad. Polon. Sci. Sér. Sci. Techn. 5, 145–154 (1957)Google Scholar
  29. 29.
    Nowacki, W.: Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Warsaw and Pergamon Press (1986)zbMATHGoogle Scholar
  30. 30.
    Pennes, H.H.: Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1, 93–122 (1948). Scholar
  31. 31.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  32. 32.
    Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, Boca Raton (2002)zbMATHGoogle Scholar
  33. 33.
    Povstenko, Y.: Fractional heat conduction equation and associated thermal stress. J. Thermal Stresses 28, 83–102 (2005). Scholar
  34. 34.
    Povstenko, Y.: Thermoelasticity that uses fractional heat conduction equation. J. Math. Sci. 162, 296–305 (2009). Scholar
  35. 35.
    Povstenko, Y.: Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr. T 136, 014017 (2009). Scholar
  36. 36.
    Povstenko, Y.: Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, 418–435 (2011). Scholar
  37. 37.
    Povstenko, Y.: Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhäuser, New York (2015)CrossRefGoogle Scholar
  38. 38.
    Povstenko, Y.: Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses. J. Thermal Stresses 39, 1442–1450 (2016). Scholar
  39. 39.
    Povstenko, Y., Kyrylych, T.: Time-fractional diffusion with mass absorption under harmonic impact. Fract. Calc. Appl. Anal. 21, 118–133 (2018). Scholar
  40. 40.
    Povstenko, Y., Kyrylych, T.: Time-fractional diffusion with mass absorption in a half-line domain due to boundary value of concentration varying harmonically in time. Entropy 20, 346 (2018). Scholar
  41. 41.
    Qin, Y., Wu, K.: Numerical solution of fractional bioheat equation by quadratic spline collocation method. J. Nonlinear Sci. Appl. 9, 5061–5072 (2016). Scholar
  42. 42.
    Shih, T.-C., Yuan, P., Lin, W.-L., Koe, H.S.: Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med. Eng. Phys. 29, 946–953 (2007)CrossRefGoogle Scholar
  43. 43.
    Vitali, S., Castellani, G., Mainardi, F.: Time fractional cable equation and applications in neurophysiology. Chaos Solitons Fractals 102, 467–472 (2017). Scholar
  44. 44.
    Vrentas, J.S., Vrentas, C.M.: Diffusion and Mass Transfer. CRC Press, Boca Raton (2013)zbMATHGoogle Scholar
  45. 45.
    Wazwaz, A.-M.: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Springer, Beijing, Berlin (2009)CrossRefGoogle Scholar
  46. 46.
    Zolfaghari, A., Maerefat, M.: Bioheat transfer. In: dos Santos Bernardes, M.A. (ed.) Developments in Heat Transfer, pp. 153–170. InTech (2011). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Dlugosz University in CzestochowaCzestochowaPoland
  2. 2.Institute of Law, Administration and ManagementJan Dlugosz University in CzestochowaCzestochowaPoland

Personalised recommendations