Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform

Abstract

Analytical solution of transient heat conduction through a semi-infinite fractal medium is developed. The solution focuses on application of a local fractional derivative operator to model the heat transfer process and a solution through the Yang-Laplace transform.

Keywords

Dates

  • Submission Date2013-09-01
  • Revision Date2013-11-08
  • Acceptance Date2013-11-08
  • Online Date2013-11-16

DOI Reference

10.2298/TSCI130901152Z

References

  1. Özişik, M.N., Boundary value problems of heat conduction, (Dover, New York, 1989).
  2. Widder, D. V., The heat equation, (Academic Press, New York, 1976).
  3. Chapko, R., Kress, R., Yoon, J. R., An inverse boundary value problem for the heat equation: the Neumann condition, Inverse Problems, 15(1999), 4, pp.1033.
  4. Wei, T., Yamamoto, M., Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Problems in Science and Engineering, 17(2009), 4, pp.551-567.
  5. Douglas, J., Rachford, H. H., On the numerical solution of heat conduction problems in two and three space variables, Transactions of the American mathematical Society, 82(1956), 2, pp.421-439.
  6. Pruess, K., Calore, C., Celati, R., Wu, Y. S., An analytical solution for heat transfer at a boiling front moving through a porous medium, International Journal of Heat and Mass Transfer, 30(1987), 12, pp.2595-2602.
  7. G. -C. Wu, Laplace transform overcoming princeple drawbacks in application of the variational iteration method to fractional heat equations, Thermal Science, 16(2012), 4, pp.1257-1261.
  8. Pamuk, S., An application for linear and nonlinear heat equations by Adomian's decomposition method, Applied Mathematics and Computation, 163(2005), 1, pp.89-96.
  9. Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A, 360(2006). 1, pp.109-113.
  10. Agarwal, R. P., Benchohra, M., Hamani, S., Boundary value problems for fractional differential equations, Georgian Mathematical Journal, 16(2009), 3, pp.401-411.
  11. Benchohra, M., Hamani, S., Ntouyas, S. K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications, 71(2009), 7, pp.2391-2396.
  12. Q. L. Wang, J. H. He, Z. B. Li, Fractional model for heat conduction in polar bear hairs, Thermal Science, 16 (2012), 2, pp.339-342.
  13. J. Fan, J. H. He, Biomimic design of multi-scale fabric with efficient heat transfer property, Thermal Science, 16 (2012), 5, pp.1349-1352.
  14. Z. B. Li, W. H. Zhu, J.H. He, Exact solutions of time-fractional heat conduction equation by the fractional complex transform, Thermal Science, 16 (2012), 2, pp.335-338.
  15. Metzler, R., Klafter, J., Boundary value problems for fractional diffusion equations, Physica A: Statistical Mechanics and its Applications, 278(2000), 1, pp.107-125.
  16. Luchko, Y., Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 374(2011), 2, pp.538-548.
  17. Povstenko, Y. Z., Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses, Mechanics Research Communications, 37(2010), 4, pp.436-440.
  18. Scherer, R., Kalla, S. L., Boyadjiev, L., Al-Saqabi, B., Numerical treatment of fractional heat equations, Applied Numerical Mathematics, 58(2008), 8, pp.1212-1223.
  19. Hristov, J., Approximate solutions to fractional sub-diffusion equations: The heat-balance integral method, The European Physical Journal Special Topics, 193 (2011), 4, pp. 229-243.
  20. Hristov, J., Heat-balance integral to fractional (half-time) heat diffusion sub-model, Thermal Science, 14 (2010), 2, pp. 291-316.
  21. Hristov, J., Transient Flow of a Generalized Second Grade Fluid Due to a Constant Surface Shear Stress: an approximate Integral-Balance Solution, Int. Rev. Chem. Eng, 3 (2011), 6, pp. 802-809.
  22. Yang, X. -J., Baleanu, D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17(2013), 2, pp.625-628.
  23. He, J. -H., Liu, F. -J., Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy, Nonlinear Science Letters A, 4(2013), 1, pp.15-20.
  24. Zhang, Y. -Z, Yang, A. -M, Yang, X. -J., 1-D heat conduction in a fractal medium: A solution by the local fractional Fourier series method, Thermal Science, 17(2013), 3, 953-956.
  25. Yang, X.-J., Heat transfer in discontinuous media, Advances in Mechanical Engineering and its Applications, 1(2012), 3, pp.47-53.
  26. Yang, X.-J., Advanced Local Fractional Calculus and Its Applications, (World Science Publisher, New York, 2012).
  27. Liu, C. F., Kong, S. S., Yuan, S. J., Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Thermal Science, 17(2013), 3, pp.715-721.
  28. Yang, X.-J., Local Fractional Functional Analysis and Its Applications, (Asian Academic publisher Limited, Hong Kong, 2011).
  29. He, J. -H., Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012 (2012), Article ID 916793.
  30. Yang, A.-M., Yang, X.-J., Li, Z.-B., Local fractional series expansion method for solving wave and diffusion equations on Cantor sets, Abstract and Applied Analysis, 2013(2013), Article ID 351057, 8 pages.
Volume 18, Issue 2, Pages667 -675