Fractional complex transform method for wave equations on cantor sets within local fractional differential operator

Advances in Difference Equations

Volumn 2013, Issue , 2013, Pages

  Su, Wei Hua ^a,b   Yang, Xiao Jun ^c,d   Jafari, H ^e   Baleanu, Dumitru ^f,g,h  

^a TIANJIN UNIVERSITY   (China)

^b BEIJING INSTITUTE OF MICROBIOLOGY AND EPIDEMIOLOGY   (China)

^c CHINA UNIVERSITY OF MINING AND TECHNOLOGY   (China)

^d ZHENGZHOU NORMAL UNIVERSITY   (China)

^e UNIVERSITY OF MAZANDARAN   (Iran)

^f CANKAYA UNIVERSITY   (Turkey)

^g KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^h INSTITUTE OF SPACE SCIENCES   (Romania)

Author keywords

Cantor sets; Fractional complex transform method; Local fractional differential operators; Wave equations

Indexed keywords

EID: 84879315907     PISSN: 16871839     EISSN: 16871847     Source Type: Journal    
DOI: 10.1186/1687-1847-2013-97     Document Type: Article

References (39)

1. Kolwankar, KM, Gangal, AD: Hölder exponents of irregular signals and local fractional derivatives. Pramana J. Phys. 48(1), 49-68 (1997) (Pubitemid 127651272)
2. Kolwankar, KM, Gangal, AD: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 80, 214-217 (1998) (Pubitemid 128621921)
3. Yang, XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012)
4. Yang, XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher Limited, Hong Kong (2011)
5. Yang, XJ: Local fractional integral transforms. Prog. Nonlinear Sci. 4, 1-225 (2011)
6. Jumarie, G: The Minkowski's space-time is consistent with differential geometry of fractional order. Phys. Lett. A 363, 5-11 (2007)
7. Jumarie, G: Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results. Comput. Math. Appl. 51, 1137-1376 (2006)
8. Parvate, A, Gangal, AD: Calculus on fractal subsets of real line - I: formulation. Fractals 17(1), 53-81 (2009)
9. Parvate, A, Gangal, AD: Fractal differential equations and fractal-time dynamical systems. Pramana J. Phys. 64, 389-409 (2005) (Pubitemid 40407933)
10. Chen, W: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 28, 923-929 (2006)
11. Chen, W, Zhang, XD, Korosak, D: Investigation on fractional and fractal derivative relaxation-oscillation models. Int. J. Nonlinear Sci. Numer. Simul. 11, 3-9 (2010)
12. Chen, W, Sun, HG, Zhang, XD, Koroak, D: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59, 1754-1758 (2010)
13. Adda, FB, Cresson, J: About non-differentiable functions. J. Math. Anal. Appl. 263, 721-737 (2001) (Pubitemid 33132693)
14. He, JH: A new fractal derivation. Therm. Sci. 15(1), 145-147 (2011)
15. He, JH: Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. 2012, 916793 (2012)
16. Fan, J, He, JH: Fractal derivative model for air permeability in hierarchic porous media. Abstr. Appl. Anal. 2012, 354701 (2012)
17. Hu, MS, Baleanu, D, Yang, XJ: One-phase problems for discontinuous heat transfer in fractal media. Math. Probl. Eng. 2013, 358473 (2013)
18. Yang, XJ: Local fractional calculus and its applications. In: Proceedings of the 5th IFAC Workshop Fractional Differentiation and Its Applications, (FDA '12), pp. 1-8, Nanjing, China (2012)
19. Yang, XJ: Research on fractal mathematics and some applications in mechanics. M.S. thesis, China University of Mining and Technology (2009) (in Chinese)
20. Yang, XJ: A short note on local fractional calculus of function of one variable. J. Appl. Libr. Inf. Sci. 1(1), 1-13 (2012)
21. Zhong, WP, Gao, F, Shen, XM: Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral. Adv. Math. Res. 461, 306-310 (2012)
22. Yang, XJ: Local fractional variational iteration method and its algorithms. Adv. Comput. Math. Appl. 1(3), 139-145 (2012)
23. Yang, XJ: Local fractional kernel transform in fractal space and its applications. Adv. Comput. Math. Appl. 1(2), 86-93 (2012)
24. Carpinteri, A, Sapora, A: Diffusion problems in fractal media defined on Cantor sets. Z. Angew. Math. Mech. 90, 203-210 (2010)
25. Carpinteri, A, Chiaia, B, Cornetti, P: Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 191, 3-19 (2001) (Pubitemid 33105419)
26. Balankin, AS, Elizarraraz, BE: Map of fluid flow in fractal porous medium into fractal continuum flow. Phys. Rev. E 85(5), 056314 (2012)
27. Yang, XJ, Baleanu, D, Zhong, WP: Approximation solution to diffusion equation on Cantor time-space. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. (2013, in press)
28. Hu, MS, Agarwal, RP, Yang, XJ: Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstr. Appl. Anal. 2012, 567401 (2012)
29. Li, ZB, He, JH: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 15(5), 970-973 (2010)
30. Li, ZB, He, JH: Application of the fractional complex transform to fractional differential equations. Nonlinear Sci. Lett. A, Math. Phys. Mech. 2(3), 121-126 (2011)
31. Li, ZB: An extended fractional complex transform. Int. J. Nonlinear Sci. Numer. Simul. 11, 0335-0337 (2010)
32. Li, ZB, Zhu, WH, He, JH: Exact solutions of time-fractional heat conduction equation by the fractional complex transform. Therm. Sci. 16(2), 335-338 (2012)
33. He, JH: A short remark on fractional variational iteration method. Phys. Lett. A 375(38), 3362-3364 (2011)
34. He, JH, Elagan, SK, Li, ZB: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376(4), 257-259 (2012)
35. Yang, XJ: The zero-mass renormalization group differential equations and limit cycles in non-smooth initial value problems. Prespacetime J. 3(9), 913-923 (2012)
36. Ibrahim, RW: Fractional complex transforms for fractional differential equations. Adv. Differ. Equ. 2012, 192-203 (2012)
37. Ibrahim, RW: Complex transforms for systems of fractional differential equations. Abstr. Appl. Anal. 2012, 814759 (2012)
38. Ghazanfari, B, Ghazanfari, AG: Solving system of fractional differential equations by fractional complex transform method. Asian J. Appl. Sci. (2012). doi:10.3923/ajaps.2012
39. Liangprom, A, Nonlaopon, K: On the convolution equation related to the diamond Klein-Gordon operator. Abstr. Appl. Anal. 2011, 908491 (2011)