Local fractional series expansion method for solving wave and diffusion equations on cantor sets

Abstract and Applied Analysis

Volumn 2013, Issue , 2013, Pages

  Yang, Ai Min ^a,b   Yang, Xiao Jun ^c   Li, Zheng Biao ^d  

^a YANSHAN UNIVERSITY   (China)

^b HEBEI UNITED UNIVERSITY   (China)

^c CHINA UNIVERSITY OF MINING AND TECHNOLOGY   (China)

^d QUJING NORMAL UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84879302696     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2013/351057     Document Type: Article

References (52)

1. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations 2006 Amsterdam, The Netherlands Elsevier MR2218073 10.1016/S0304-0208(06)80001-0 ZBL1206.26007
2. Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity 2010 London, UK Imperial College Press 10.1142/9781848163300 MR2676137 ZBL1222.26015
3. Podlubny I., Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press MR1658022 ZBL1056.93542
4. Magin R. L., Fractional Calculus in Bioengineering, 2006 West Redding, Conn, USA Begerll House
5. Klafter J., Lim S. C., Metzler R., Fractional Dynamics in Physics: Recent Advances 2011 Singapore World Scientific
6. Zaslavsky G. M., Hamiltonian Chaos and Fractional Dynamics 2008 Oxford, UK Oxford University Press MR2451371 ZBL1257.65071
7. West B. J., Bologna M., Grigolini P., Physics of Fractal Operators 2003 New York, NY, USA Springer 10.1007/978-0-387-21746-8 MR1988873 ZBL1256.37039
8. Tarasov V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media 2011 Berlin, Germany Springer 10.1007/978-3-642-14003-7 MR2796453 ZBL1255.70019
9. Tenreiro Machado J. A., Luo A. C. J., Baleanu D., Nonlinear Dynamics of Complex Systems: Applications in Physical, Biological and Financial Systems 2011 New York, NY, USA Springer
10. Duan J. S., Chaolu T., Rach R., Lu L., The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations. Computers & Mathematics With Applications 2013 10.1016/j.camwa.2013.01.019
11. 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. Applied Mathematics and Computation 2012 218 17 8370 8392 10.1016/j.amc.2012.01.063 MR2921332 ZBL1245.65087
12. Das S., Analytical solution of a fractional diffusion equation by variational iteration method. Computers & Mathematics with Applications 2009 57 3 483 487 10.1016/j.camwa.2008.09.045 MR2488620 ZBL1165.35398
13. Momani S., Odibat Z., Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Computers & Mathematics with Applications 2007 54 7-8 910 919 10.1016/j.camwa.2006.12.037 MR2395628 ZBL1141.65398
14. Momani S., Odibat Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physics Letters A 2007 365 5-6 345 350 10.1016/j.physleta.2007.01.046 MR2308776 ZBL1203.65212
15. Baleanu D., Diethelm K., Scalas E., Trujillo J. J., Fractional Calculus Models and Numerical Methods 2012 Boston, Mass, USA World Scientific Series on Complexity, Nonlinearity and Chaos 10.1142/9789814355216 MR2894576
16. Jafari H., Seifi S., Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 2009 14 5 2006 2012 10.1016/j.cnsns.2008.05.008 MR2474460 ZBL1221.65278
17. Hristov J., Heat-balance integral to fractional (half-time) heat diffusion sub-model. Thermal Science 2010 14 2 291 316 2-s2.0-77955895114 10.2298/TSCI1002291H
18. Hristov J., Integral-balance solution to the stokes' first problem of a viscoelastic generalized second grade fluid. Thermal Science 2012 16 2 395 410
19. Hristov J., Transient flow of a generalized second grade fluid due to a constant surface shear stress: an approximate integral-balance solution. International Review of Chemical Engineering 2011 3 6 802 809
20. Wu G. C., Lee E. W. M., Fractional variational iteration method and its application. Physics Letters A 2010 374 25 2506 2509 10.1016/j.physleta.2010.04. 034 MR2640023 ZBL1237.34007
21. Khan Y., Faraz N., Yildirim A., Wu Q., Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Computers & Mathematics with Applications 2011 62 5 2273 2278 10.1016/j.camwa.2011.07.014 MR2831688 ZBL1231.35288
22. Wu G. C., A fractional variational iteration method for solving fractional nonlinear differential equations. Computers & Mathematics with Applications 2011 61 8 2186 2190 10.1016/j.camwa.2010.09.010 MR2785581 ZBL1219.65085
23. Zhang S., Zhang H.-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A 2011 375 7 1069 1073 10.1016/j.physleta.2011.01.029 MR2765013 ZBL1242.35217
24. Jafari H., Tajadodi H., Kadkhoda N., Baleanu D., Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations. Abstract and Applied Analysis 2013 2013 5 587179 10.1155/2013/587179
25. Zhang S., Zong Q. A., Liu D., Gao Q., A generalized expfunction method for fractional Riccati differential equations. Communications in Fractional Calculus 2010 1 48 52
26. Yang X. J., Advanced Local Fractional Calculus and Its Applications 2012 New York, NY, USA World Science
27. Yang X. J., Local fractional integral transforms. Progress in Nonlinear Science 2011 4 1 225
28. Yang X. J., Local Fractional Functional Analysis and Its Applications 2011 Hong Kong Asian Academic
29. Zhong W. P., Yang X. J., Gao F., A Cauchy problem for some local fractional abstract differential equation with fractal conditions. Journal of Applied Functional Analysis 2013 8 1 92 99
30. Su W. H., Yang X. J., Jafari H., Baleanu D., Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator. Advances in Difference Equations 2013 2013 1 97 107 10.1186/1687-1847-2013-97 MR3049970
31. Hu M. S., Baleanu D., Yang X. J., One-phase problems for discontinuous heat transfer in fractal media. Mathematical Problems in Engineering 2013 2013 3 358473 10.1155/2013/358473
32. Kolwankar K. M., Gangal A. D., Local fractional Fokker-Planck equation. Physical Review Letters 1998 80 2 214 217 10.1103/PhysRevLett.80.214 MR1604435 ZBL0945.82005
33. Carpinteri A., Sapora A., Diffusion problems in fractal media defined on Cantor sets. ZAMM Journal of Applied Mathematics and Mechanics 2010 90 3 203 210 2-s2.0-77049111800 10.1002/zamm.200900376
34. Carpinteri A., Cornetti P., A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos, Solitons and Fractals 2002 13 1 85 94 2-s2.0-0036028181 10.1016/S0960-0779(00)00238-1
35. Carpinteri A., Chiaia B., Cornetti P., The elastic problem for fractal media: basic theory and finite element formulation. Computers and Structures 2004 82 6 499 508 2-s2.0-0742324870 10.1016/j.compstruc.2003.10.014
36. Babakhani A., Daftardar-Gejji V., On calculus of local fractional derivatives. Journal of Mathematical Analysis and Applications 2002 270 1 66 79 10.1016/S0022-247X(02)00048-3 MR1911751 ZBL1005.26002
37. Ben Adda F., Cresson J., About non-differentiable functions. Journal of Mathematical Analysis and Applications 2001 263 2 721 737 10.1006/jmaa.2001.7656 MR1866075 ZBL0995.26006
38. Chen Y., Yan Y., Zhang K., On the local fractional derivative. Journal of Mathematical Analysis and Applications 2010 362 1 17 33 10.1016/j.jmaa.2009.08. 014 MR2557664 ZBL1196.26011
39. Chen W., Time-space fabric underlying anomalous diffusion. Chaos, Solitons and Fractals 2006 28 4 923 925 2-s2.0-27744450698 10.1016/j.chaos.2005. 08.199
40. Chen W., Sun H., Zhang X., Korošak D., Anomalous diffusion modeling by fractal and fractional derivatives. Computers & Mathematics with Applications 2010 59 5 1754 1758 10.1016/j.camwa.2009.08.020 MR2595948 ZBL1189.35355
41. Golmankhaneh A. K., Fazlollahi V., Baleanu D., Newtonian mechanics on fractals subset of real-line. Romania Reports in Physics 2013 65 84 93
42. Parvate A., Gangal A. D., Fractal differential equations and fractal-time dynamical systems. Pramana 2005 64 3 389 409 2-s2.0-15744373583
43. 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 2009 40 3 1428 1448 10.1016/j.chaos.2007.09.028 MR2526363 ZBL1197.60039
44. Jumarie G., Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative. Applied Mathematics Letters 2009 22 11 1659 1664 10.1016/j.aml.2009.05.011 MR2569059 ZBL1181.44001
45. Yang X. J., Baleanu D., Fractal heat conduction problem solved by local fractional variation iteration method. Thermal Science 2013 17 2 625 628
46. 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 89 102 10.1186/1687-1812-2013-89 MR3053815
47. 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 2012 15 567401 MR3004869 ZBL1257.35193 10.1155/2012/567401
48. Anastassiou G. A., Duman O., Advances in Applied Mathematics and Approximation Theory 2013 New York, NY, USA Springer
49. Yang X. J., Srivastava H. M., He J. H., Baleanu D., Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Physics Letters A 2013 377 28-30 1696 1700
50. Yang X., Liao M., Chen J., A novel approach to processing fractal signals using the Yang-Fourier transforms. 29 Proceedings of the International Workshop on Information and Electronics Engineering (IWIEE '12) March 2012 2950 2954 2-s2.0-84858387019 10.1016/j.proeng.2012.01.420
51. Zhong W., Gao F., Shen X., Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral. Advanced Materials Research 2012 461 306 310 2-s2.0-84857467321 10.4028/www.scientific.net/AMR.461.306
52. Zhong W. P., Gao F., Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with fractional derivative. Proceedings of the 3rd International Conference on Computer Technology and Development 2011 ASME 209 213