THE YANG-FOURIER TRANSFORMS TO HEAT-CONDUCTION IN A SEMI-INFINITE FRACTAL BAR

Abstract

1-D fractal heat-conduction problem in a fractal semi-infinite bar has been developed by local fractional calculus employing the analytical Yang-Fourier transforms method. The simplicity and the accuracy of the method are discussed.

Keywords

Dates

  • Submission Date2012-08-26
  • Revision Date2013-05-30
  • Acceptance Date2013-05-30
  • Online Date2013-06-16

DOI Reference

10.2298/TSCI120826074Y

References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo,J.J. Theory and Applications of Fractional Differential Equations, (Elsevier Science, Amsterdam, 2006).
  2. Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, (Imperial College Press, London, 2010).
  3. Podlubny, I., Fractional Differential Equations, (Academic Press, New York, 1999).
  4. Klafter, J., Lim, S.C., Metzler,R., Eds., Fractional Dynamics in Physics: Recent Advances, (World Scientific, Singapore, 2012).
  5. Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005).
  6. West, B., Bologna,M., Grigolini, P., Physics of Fractal Operators (Springer, New York, 2003)
  7. Carpinteri,A., Mainardi, F., (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, (Springer, Wien, 1997).
  8. Baleanu, D., Diethelm, K., Scalas, E., Trujillo,J.J., Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, (World Scientific, Singapore, 2012).
  9. Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316.
  10. Hristov, J., Integral-balance solution to the Stokes' first problem of a viscoelastic generalized second grade fluid, Thermal Science, 16 (2012), 2, pp.395-410.
  11. 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.
  12. Jafari, H., Tajadodi,H., Baleanu, D., A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials. Fractional Calculus and Applied Analysis, 16 (2013), 1, pp.109-122.
  13. Wu, G.C., Baleanu,D., Variational iteration method for fractional calculus-a universal approach by Laplace transform, Advances in Difference Equations, 2013 (2013),1, pp. 1-9.
  14. Ateş, I., Yildirim, A., Application of variational iteration method to fractional initial-value problems, Int. J. Nonl.Sci. Num.Sim., 10 (2009),7, pp.877-884.
  15. Duan,J.S., Chaolu, T., Rach R., Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method, Appl. Math.Comput., 218 (2012),17, pp.8370-8392.
  16. Momani, S., Yıldırım, A., Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J Comp. Math. 87 (2010), 5,pp. 1057-1065.
  17. Guo, S., Mei, L., Li, Y., Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation, Appl. Math.Comput., 219 (2013), 11, pp.5909-5917.
  18. Sun, H. G., Chen, W., Sze, K. Y., A semi-discrete finite element method for a class of time-fractional diffusion equations, Phil. Trans. Royal Soc. A: 371 (2013) 1990, pp.1471-2962.
  19. Jafari, H., Tajadodi,H., Kadkhoda, N., Baleanu, D., Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations, Abstract and Applied Analysis, 2013, Article ID 587179.
  20. Luchko,Y., Kiryakova, V., The Mellin integral transform in fractional calculus, Fractional Calculus and Applied Analysis, 16 (2013),2, pp.405-430.
  21. Abbasbandy,S., Hashemi, M.S., On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaestiones Mathematicae, 36 (2013),1, pp.93-105.
  22. Hashim,I., Abdulaziz,O., Momani,S., Homotopy analysis method for fractional IVPs, Comm. Nonl. Sci. Num.Sim., 14 (2009), 3, pp.674-684.
  23. Li,C., Zeng, F., The finite difference methods for fractional ordinary differential equations. Num.Func.Anal. Optim., 34 (2013), 2, pp.149-179.
  24. Demir, A., Erman, S., Özgür, B, Korkmaz, E., Analysis of fractional partial differential equations by Taylor series expansion, Boundary Value Problems, 2013 (2013), 1, pp.68-80.
  25. Kolwankar, K.M., Gangal, A.D., Local fractional Fokker-Planck equation, Physical Review Letters, 80 (1998), 2, pp.214-217.
  26. Chen, W., Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals, 28 (2006), 4, pp.923-929.
  27. Fan, J., He,J.H., Fractal derivative model for air permeability in hierarchic porous media, Abstract and Applied Analysis, vol.2012, Article ID 354701.
  28. Jumarie, G., Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions, Chaos, Solitons and Fractals, 40 (2009), 3, pp.1428-1448.
  29. Carpinteri, A., Sapora, A., Diffusion problems in fractal media defined on Cantor sets, ZAMM 90 (2010), 3, pp.203-210.
  30. Yang, X.J., Local Fractional Functional Analysis and Its Applications, (Asian Academic publisher Limited, Hong Kong, 2011).
  31. Yang, X.J., Local fractional integral transforms, Progress in Nonlinear Science, 4 (2011), pp.1-225.
  32. Yang, X.J., Advanced Local Fractional Calculus and Its Applications, (World Science Publisher, New York, 2012).
  33. Su, W.H., Baleanu,D., Yang , X.J., Jafari,H., Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator, Advances in Difference Equations, 2013 (2013),1, pp.97-103.
  34. Hu, M.S., Baleanu, D., Yang, X. J., One-phase problems for discontinuous heat transfer in fractal media, Mathematical Problems in Engineering, vol.2013, Article ID 358473, 2013.
  35. Yang, X,J., Baleanu,D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17 (2013),2, pp.625-628.
  36. Yang, Y,J., Baleanu, D., Yang, X.J., A local fractional variational iteration method for Laplace equation within local fractional operators, Abstract and Applied Analysis, 2013 (2013), Article ID 202650.
  37. Su, W.H., Baleanu, D., Yang X.J., Jafari, H., Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method, Fixed Point Theory and Applications, 2013 (2013),1, pp.89-102, 2013.
  38. Hu, M.S., Agarwal, R.P., Yang, X.J., Local fractional Fourier series with application to wave equation in fractal vibrating string, Abstract and Applied Analysis, 2012, Article ID 567401.
  39. Zhong, W.P., Gao,F., Shen,X.M., Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral, Advanced Materials Research, 461 (2012), pp.306-310.
  40. He, J.H., Asymptotic methods for solitary solutions and compactions, Abstract and Applied Analysis, 2012, Article ID 916793.
Volume 17, Issue 3, Pages707 -713