Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8605056, 7 pages
http://dx.doi.org/10.1155/2016/8605056
Research Article

Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

Institute of Mathematics and Computer Science, Faculty of Mathematics and Natural Sciences, Jan Dlugosz University in Czestochowa, Armii Krajowej 13/15, 42-200 Czestochowa, Poland

Received 23 August 2016; Accepted 23 October 2016

Academic Editor: Filippo de Monte

Copyright © 2016 Yuriy Povstenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.

1. Introduction

In the paper [1] and later on in the book [2] Nowacki studied the classical parabolic heat conduction equation with a heat source term varying harmonically as a function of timein the domain . Here is the thermal diffusivity coefficient, is the Dirac delta function, and denotes the frequency.

Nowacki’s solution of (1) is based on the assumption that temperature can be expressed as a product of the auxiliary function and the time harmonic termIn this case, there are no initial and boundary conditions (excepting the zero condition at ), and the problem is reduced to solving the corresponding equation for the auxiliary function The final result reads The square root of the imaginary unit is defined as

If the heat conduction equationis considered in a half-line domain , then the boundary condition at should be imposed. For example, we can assume the Dirichlet boundary condition varying harmonically in time:Similar analysis can be also carried out in the case of the boundary value of heat flux varying harmonically in time (the physical Neumann boundary condition). Boundary conditions varying harmonically in time describe various situations, in particular, thermal processing of materials using pulsed lasers or collection of solar energy [3].

Under Nowacki’s assumption (2), there is no initial condition, and for the auxiliary function we obtainTo compare with the subsequent results it is worthwhile to solve (6) under boundary conditions (7) and (8) using the sin-Fourier transform with respect to the spatial coordinate . The solution has a form Hence,If the surface temperature is described by the dependencethen the solution becomes [3]

Many experimental and theoretical investigations testify that in media with complex internal structure the standard heat conduction equation is no longer sufficiently accurate. This results in formulation of nonclassical theories, in which the parabolic heat conduction equation is replaced by more general one (see [412] and the references therein).

For example, Green and Naghdi [7] proposed the theory of thermoelasticity without energy dissipation based on the wave equation for temperature. In the framework of this theory, the following boundary value problem can be studied:Under the assumption (2), using the sin-Fourier transform, we get the solution (see (A.3) from appendix)

2. Time-Fractional Heat Conduction

The time-nonlocal generalization of the Fourier law with the “long-tail” power kernel [11, 1315] can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and results in the time-fractional heat conduction equationwith the Caputo fractional derivative of order defined as [1618]and having the following Laplace transform rule:where the asterisk denotes the transform, is the transform variable, and is the gamma function. The Caputo fractional derivative is a regularization in the time origin for the Riemann-Liouville fractional derivative by incorporating the relevant initial conditions [19]. The major utility of the Caputo fractional derivative is caused by the treatment of differential equations of fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer (not fractional) order, even if the governing equation is of fractional order [17, 20]. Additional discussion on the use of the Caputo and Riemann-Liouville fractional derivatives can be found in [21] (see Section 3.4 “Which type of fractional derivative? Caputo or Riemann-Liouville?” in this book).

Equations with fractional derivatives describe many important physical phenomena in amorphous, colloid, glassy, and porous materials, in fractals, comb structures, polymers, and random and disordered materials, in viscoelasticity and hereditary mechanics of solids, in biological systems, and in geophysical and geological processes (see, e.g., [2230] and the references therein). Important applications of fractional calculus can be found in such fields as fractional dynamics [3135], fractional kinetics [3638], and fractional thermoelasticity [11, 12, 3941].

Equation (15) describes the whole spectrum from localized heat conduction (the Helmholtz equation for ) through the standard heat conduction () to the ballistic heat conduction (the wave equation when ).

The interested reader is referred to the book [15], which systematically presents solutions to different initial and boundary value problems for the time-fractional diffusion-wave equation (15) in Cartesian, cylindrical, and spherical coordinates. In [42, 43], this equation was considered in unbounded domains with the source term varying harmonically in time.

In the present paper, we study the Dirichlet problem for the time-fractional heat conduction equation in a half-line domain with the surface value of temperature varying harmonically in time. The integral transform technique is used. Different formulations of the considered problem for the classical heat conduction equation () and for the wave equation describing ballistic heat conduction () are discussed.

3. Formulation of the Problem

We consider the time-fractional heat conduction equation in a half-line:under the harmonic boundary conditionand zero condition at infinity

For the Caputo derivative of the exponential function we haveSubstituting (with ) gives the final resultwhere is the incomplete gamma function [44]:

Hence, for fractional (noninteger) values of the order of derivative,Therefore, the Nowacki assumption (2) cannot be used for the time-fractional heat conduction equation and the corresponding initial conditions should be imposed (see also [42, 43]). In the present paper we assume zero initial conditions:

4. Solution to the Problem

Application of the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate to (18) under the initial conditions (25) and the boundary conditions (19), (20) giveswhere the tilde denotes the sin-Fourier transform and is the transform variable.

At first, we analyze the standard heat conduction equation corresponding to :The inverse Laplace transform gives (see (A.6) from appendix):Integrals (A.2) and (A.4) allow us to invert the sin-Fourier transform and to obtain the solutionThe first term in (29) coincides with solution (10) and describes the quasi-steady-state oscillations; the second and third ones describe the transient process.

Starting in (27) from the inversion of the sin-Fourier transform, we get

Taking into account (A.10) and using the convolution theorem for the Laplace transform allow us to obtain an alternative form of the solution to the standard heat conduction equation:

Another particular case of solution (26) in the transform domain corresponds to the ballistic heat conduction ():Inversion of the sin-Fourier transform (see (A.2)) results inTaking into account (A.9), we obtainIt should be noted that solution (34) describes the wavefront at

In applications there often appears the value (see investigations of diffusion on fractals [45] and comb structures [46, 47]). Using (A.7), we get

Now we return to the analysis of the time-fractional heat conduction equation and its solution in the transform domain (26). The inverse sin-Fourier transform givesThe inverse Laplace transform of is expressed in terms of the Mainardi function (see (A.11) from appendix). The solution has the form

The particular case of the Mainardi function is reduced to the exponential function (see [48, 49]): and solution (37) for coincides with solution (31).

Starting from the inverse Laplace transform of (26), we havewhere is the Mittag-Leffler function in two parameters and (see appendix). In parallel with (37), the inverse Fourier transform of (39) leads to another form of the solutionComparison of (37) and (40) allows us to establish the relation between the Mainardi function and the Mittag-Leffler function in the form of sin-Fourier transform (see also [15, 50], where the similar relations were obtained in terms of the cos-Fourier transform).

Figures 1 and 2 present the dependence of solution on distance in the case of the boundary condition for different values of the order of fractional derivative and different values of time. In numerical calculations we have used the following nondimensional quantities:To evaluate the Mittag-Leffler function , the algorithm suggested in the paper [51] has been used.

Figure 1: Dependence of temperature on distance ().
Figure 2: Dependence of temperature on distance ().

5. Concluding Remarks

We have considered the Dirichlet problem for the time-fractional heat conduction equation in a half-line with the Caputo fractional derivative and with the boundary value of temperature varying harmonically in time. The solution has been obtained using the integral transform technique.

The Caputo fractional derivative of the exponential function has more complicated form than the corresponding derivative of the integer order. Hence, the Nowacki approach based on the representation of temperature as the product of a function of the spatial coordinate and a function harmonic in time cannot be used, and the initial conditions should be taken into account.

In such a statement of the problem, the particular cases of the general solution for integer values of the order of derivative ( and ) describe both the quasi-steady-state oscillations and the transient process. It should be emphasized that in the case of the ballistic heat conduction equation () the obtained solution presents the wavefront at ( in Figures 1 and 2), which does not appear in the Nowacki-type solution.

The obtained solution may also be used in constructing solutions for boundary functions varying periodically in an arbitrary manner. Expanding the boundary function in the time-Fourier series, the solution can be obtained as a result of superposition of successive harmonic terms.

Appendix

We present integrals [52, 53] used in the paper: where is the complementary error function

Formulae for inverse Laplace transform (equations (A.6)–(A.10)) are borrowed from [54, 55]:where is the error function of an imaginary argument:

Equation (A.11) can be found in [15, 48, 49]Here is the Mainardi function [17, 48, 49], being the particular case of the Wright function:

Equation (A.13) is taken from [16, 17]where is the Mittag-Leffler function in two parameters and [16, 17, 56] described by the following series representation:

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

  1. W. Nowacki, “State of stress in an elastic space due to a source of heat varying harmonically as function of time,” Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Techniques, vol. 5, no. 3, pp. 145–154, 1957. View at Google Scholar
  2. W. Nowacki, Thermoelasticity, Poland and Pergamon Press, Oxford, UK, 2nd edition, 1986. View at MathSciNet
  3. T. L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Hoboken, NJ, USA, 7th edition, 2011.
  4. C. Cattaneo, “Sulla conduzione del calore,” Atti del Seminario Matematico e Fisico dell' Universitè di Modena, vol. 3, no. 3, pp. 83–101, 1948. View at Google Scholar
  5. M. E. Gurtin and A. C. Pipkin, “A general theory of heat conduction with finite wave speeds,” Archive for Rational Mechanics and Analysis, vol. 31, no. 2, pp. 113–126, 1968. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. D. Joseph and L. Preziosi, “Heat waves,” Reviews of Modern Physics, vol. 61, no. 1, pp. 41–73, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. E. Green and P. M. Naghdi, “Thermoelasticity without energy dissipation,” Journal of Elasticity, vol. 31, no. 3, pp. 189–208, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford University Press, Oxford, UK, 2009. View at MathSciNet
  11. Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,” Journal of Thermal Stresses, vol. 28, no. 1, pp. 83–102, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Y. Povstenko, Fractional Thermoelasticity, Springer, New York, NY, USA, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. Povstenko, “Thermoelasticity which uses fractional heat conduction equation,” Journal of Mathematical Sciences, vol. 162, no. 2, pp. 296–305, 2009. View at Publisher · View at Google Scholar
  14. Y. Povstenko, “Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder,” Fractional Calculus and Applied Analysis, vol. 14, no. 3, pp. 418–435, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkhäuser, New York, NY, USA, 2015. View at MathSciNet
  16. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997. View at Google Scholar · View at MathSciNet
  17. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  18. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  19. R. Gorenflo and F. Mainardi, “Fractional calculus and stable probability distributions,” Archives of Mechanics, vol. 50, no. 3, pp. 377–388, 1998. View at Google Scholar · View at MathSciNet
  20. F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Yu. N. Rabotnov, Creep Problems in Structural Members, North-Holland Publishing Company, Amsterdam, The Netherlands, 1969.
  23. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  24. Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997. View at Publisher · View at Google Scholar · View at Scopus
  25. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  27. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, Mass, USA, 2006.
  28. Y. A. Rossikhin and M. V. Shitikova, “Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results,” Applied Mechanics Reviews, vol. 63, no. 1, Article ID 010801, 52 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. V. Gafiychuk and B. Datsko, “Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1101–1107, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Berlin, Germany, 2013.
  31. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, New York, NY, USA, 2005. View at MathSciNet
  32. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Higher Education Press, Bejing, China; Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  33. J. S. Leszczyński, An Introduction to Fractional Mechanics, The Publishing Office of Czestochowa University of Technology, Czestochowa, Poland, 2011.
  34. D. Baleanu, J. A. Tenreiro Machado, and A. C. J. Luo, Eds., Fractional Dynamics and Control, Springer, New York, NY, USA, 2012.
  35. T. M. Atanacković, S. Pilipović, B. Stanković, and D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley & Sons, Hoboken, NJ, USA, 2014.
  36. V. Uchaikin and R. Sibatov, Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems, World Scientific, New Jersey, NJ, USA, 2013.
  37. R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2nd edition, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  38. R. Abi Zeid Daou and X. Moreau, Eds., Fractional Calculus: Applications, NOVA Science Publishers, New York, NY, USA, 2015.
  39. Y. Z. Povstenko, “Fractional Cattaneo-type equations and generalized thermoelasticity,” Journal of Thermal Stresses, vol. 34, no. 2, pp. 97–114, 2011. View at Publisher · View at Google Scholar · View at Scopus
  40. A. S. El-Karamany and M. A. Ezzat, “On fractional thermoelasticity,” Mathematics and Mechanics of Solids, vol. 16, no. 3, pp. 334–346, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. M. A. Ezzat and M. A. Fayik, “Fractional order theory of thermoelastic diffusion,” Journal of Thermal Stresses, vol. 34, no. 8, pp. 851–872, 2011. View at Publisher · View at Google Scholar · View at Scopus
  42. Y. Povstenko, “Harmonic impact in the plane problem of fractional thermoelasticity,” in Proceedings of the 11th International Congress on Thermal Stresses, pp. 227–230, Salerno, Italy, June 2016.
  43. Y. Povstenko, “Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses,” Journal of Thermal Stresses, vol. 39, no. 11, pp. 1442–1450, 2016. View at Publisher · View at Google Scholar
  44. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, New York, NY, USA, Dover, 1972.
  45. R. R. Nigmatullin, “The realization of the general transfer equation in a medium with fractal geometry,” Physica Status Solidi B, vol. 133, no. 1, pp. 425–430, 1986. View at Publisher · View at Google Scholar · View at Scopus
  46. V. E. Arkhincheev, “Anomalous diffusion and charge relaxation on comb model: Exact solutions,” Physica A: Statistical Mechanics and its Applications, vol. 280, no. 3-4, pp. 304–314, 2000. View at Publisher · View at Google Scholar · View at Scopus
  47. V. E. Arkhincheev, “Diffusion on random comb structure: effective medium approximation,” Physica A: Statistical Mechanics and Its Applications, vol. 307, no. 1-2, pp. 131–141, 2002. View at Publisher · View at Google Scholar · View at Scopus
  48. F. Mainardi, “The fundamental solutions for the fractional diffusion-wave equation,” Applied Mathematics Letters, vol. 9, no. 6, pp. 23–28, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals, vol. 7, no. 9, pp. 1461–1477, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  50. Y. Povstenko, “Generalized theory of diffusive stresses associated with the time-fractional diffusion equation and nonlocal constitutive equations for the stress tensor,” Computers & Mathematics with Applications, 2016. View at Publisher · View at Google Scholar
  51. R. Gorenflo, J. Loutchko, and Y. Luchko, “Computation of the Mittag-Leffler function and its derivatives,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 491–518, 2002. View at Google Scholar · View at MathSciNet
  52. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon and Breach Science, Amsterdam, The Netherlands, 1986. View at MathSciNet
  53. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, vol. 1, McGraw-Hill, New York, NY, USA, 1954.
  54. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 5: Inverse Laplace Transforms, vol. 5, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1992.
  55. V. A. Ditkin and A. P. Prudnikov, Reference Book on Operational Calculus, Higher School, Moscow, Russia, 1965 (Russian).
  56. R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, Germany, 2014. View at Publisher · View at Google Scholar · View at MathSciNet