Volume 2013, Issue 1 351057

Research Article

Open Access

Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Ai-Min Yang,

Corresponding Author

Ai-Min Yang

[email protected]

College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China ysu.edu.cn

College of Science, Hebei United University, Tangshan 063009, China heuu.edu.cn

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Xiao-Jun Yang,

Xiao-Jun Yang

Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China cumtb.edu.cn

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Zheng-Biao Li,

Zheng-Biao Li

College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China qjnu.edu.cn

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Ai-Min Yang,

Corresponding Author

Ai-Min Yang

[email protected]

College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China ysu.edu.cn

College of Science, Hebei United University, Tangshan 063009, China heuu.edu.cn

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Xiao-Jun Yang,

Xiao-Jun Yang

Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China cumtb.edu.cn

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Zheng-Biao Li,

Zheng-Biao Li

College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China qjnu.edu.cn

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First published: 04 June 2013

https://doi.org/10.1155/2013/351057

Citations: 32

Academic Editor: Dumitru Baleanu

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Abstract

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

1. Introduction

Fractional calculus theory [1–3] has been applied to a wide class of complex problems encompassing physics, biology, mechanics, and interdisciplinary areas [4–9]. Various methods, for example, the Adomian decomposition method [10], the Rach-Adomian-Meyers modified decomposition method [11], the variational iteration method [12, 13], the homotopy perturbation method [13, 14], the fractal Laplace and Fourier transforms [15], the homotopy analysis method [16], the heat-balance integral method [17–19], the fractional variational iteration method [20–22], the fractional subequation method [23, 24], and the generalized Exp-function method [25], have been utilized to solve fractional differential equations [3, 15].

The characteristics of fractal materials have local and fractal behaviors well described by nondifferential functions. However, the classic fractional calculus is not valid for differential equation on Cantor sets due to its no-local nature. In contrast, the local fractional calculus is one of the best candidates for dealing with such problems [26–44]. The local fractional calculus theory has played crucial applications in several fields, such as theoretical physics, transport problems in fractal media described by nondifferential functions. There are some versions of the local fractional calculus where different approaches in definition of the local fractional derivative exist, among them the local fractional derivative of Kolwankar et al. [32–38], the fractal derivative of Chen et al. [39, 40], the fractal derivative of Parvate et al. [41, 42], the modified Riemann-Liouville of Jumarie [43, 44], and versions described in [45–52].

In order to deal with local fractional ordinary and partial differential equations, there are some developed technologies, for example, the local fractional variational iteration method [45, 46], the local fractional Fourier series method [47, 48], the Cantor-type cylindrical-coordinate method [49], the Yang-Fourier transform [50, 51], and the Yang-Laplace transform [52].

The local fractional derivative is defined as follows [26–31, 45–52]:

()

where Δ^α(f(x) − f(x₀))≅Γ(1 + α)Δ(f(x) − f(x₀)), and f(x) is satisfied with the condition [26, 47]

()

so that [26–31]

()

with U : |x − x₀| < δ, for ε, δ > 0 and ε, δ ∈ R.

The main idea of this paper is to present the local fractional series expansion method for effective solutions of wave and diffusion equations on Cantor sets involving local fractional derivatives. The paper has been organized as follows. Section 2 gives a local fractional series expansion method. Some illustrative examples are shown in Section 3. The conclusions are presented in Section 4.

2. Analysis of the Method

Let us consider the local fractional differential equation

()

where L is a linear local operator with respect to x, n ∈ {1,2}.

In accordance with the results in [28, 47], there are multiterm separated functions of independent variables t and x, namely,

()

where T_i(t) and X_i(x) are local fractional continuous functions.

Moreover, there is a nondifferential series term

()

where p_i is a coefficient.

In view of (6), we may present the solution in the form

()

Then, following (7), we have

()

Hence,

()

In view of (9), we have

()

Hence, from (10) we can obtain a recursion; namely,

()

with n = 1; we arrive at the following relation:

()

with n = 2; we may rewrite (11) as

()

By the recursion formulas, we can obtain the solution of (4) as

()

The convergent condition is

()

This approach is termed the local fractional series expansion method (LFSEM)

3. Applications to Wave and Diffusion Equations on Cantor Sets

In this section, four examples for wave and diffusion equations on Cantor sets will demonstrate the efficiency of LFSEM.

Example 1. Let us consider the diffusion equation on Cantor set

()

with the initial condition

()

Following (12), we have recursive formula

()

Hence, we get

()

and so on.

Therefore, through (19) we get the solution

()

Example 2. Let us consider the diffusion equation on Cantor set

()

with the initial condition

()

Following (12), we get

()

By using the recursive formula (23), we get consequently

()

As a direct result of these recursive calculations, we arrive at

()

Example 3. Let us consider the following wave equation on Cantor sets:

()

with the initial condition

()

In view of (14), we obtain

()

Hence, using the relations (29), the recursive calculations yield

()

and so on.

Finally, we obtain

()

Example 4. Let us consider the wave equation on Cantor sets [26, 30]

()

where c is a constant.

The initial condition is

()

By using (14) we have

()

Then, through the iterative relations (35), we have

()

Therefore, we obtain

()

where

()

For more details concerning (38), we refer to [26–28].

4. Conclusions

In this work, the local fractional series expansion method is demonstrated as an effective method for solutions of a wide class of problems. Analytical solutions of the wave and diffusion equations on Cantor sets involving local fractional derivatives are successfully developed by recurrence relations resulting in convergent series solutions. In this context, the suggested method is a potential tool for development of approximate solutions of local fractional differential equations with fractal initial value conditions, which, of course, draws new problems beyond the scope of the present work.

Acknowledgments

The first author was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 11126213 and no. 61170317), and the National Natural Science Foundation of Hebei Province (no. E2013209123). The third author is supported in part by NSF11061028 of China and Yunnan Province NSF Grant no. 2011FB090.

References

1 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier, Amsterdam, The Netherlands, MR2218073, https://doi.org/10.1016/S0304-0208(06)80001-0, ZBL1206.26007.
10.1016/S0304-0208(06)80001-0
Google Scholar
2 Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity, 2010, Imperial College Press, London, UK, https://doi.org/10.1142/9781848163300, MR2676137, ZBL1222.26015.
10.1142/p614
Google Scholar
3 Podlubny I., Fractional Differential Equations, 1999, Academic Press, San Diego, Calif, USA, MR1658022, ZBL1056.93542.
Web of Science® Google Scholar
4 Magin R. L., Fractional Calculus in Bioengineering,, 2006, Begerll House, West Redding, Conn, USA.
Google Scholar
5 Klafter J., Lim S. C., and Metzler R., Fractional Dynamics in Physics: Recent Advances, 2011, World Scientific, Singapore.
10.1142/8087
Google Scholar
6 Zaslavsky G. M., Hamiltonian Chaos and Fractional Dynamics, 2008, Oxford University Press, Oxford, UK, MR2451371, ZBL1257.65071.
Google Scholar
7 West B. J., Bologna M., and Grigolini P., Physics of Fractal Operators, 2003, Springer, New York, NY, USA, https://doi.org/10.1007/978-0-387-21746-8, MR1988873, ZBL1256.37039.
10.1007/978-0-387-21746-8
Google Scholar
8 Tarasov V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, 2011, Springer, Berlin, Germany, https://doi.org/10.1007/978-3-642-14003-7, MR2796453, ZBL1255.70019.
Google Scholar
9 Tenreiro Machado J. A., Luo A. C. J., and Baleanu D., Nonlinear Dynamics of Complex Systems: Applications in Physical, Biological and Financial Systems, 2011, Springer, New York, NY, USA.
10.1007/978-1-4614-0231-2
Google Scholar
10 Duan J. S., Chaolu T., Rach R., and Lu L., The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations, Computers & Mathematics With Applications. (2013) https://doi.org/10.1016/j.camwa.2013.01.019.
10.1016/j.camwa.2013.01.019
PubMed Web of Science® Google Scholar
11 Duan J. S., Chaolu T., and 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, no. 17, 8370–8392, https://doi.org/10.1016/j.amc.2012.01.063, MR2921332, ZBL1245.65087.
10.1016/j.amc.2012.01.063
Web of Science® Google Scholar
12 Das S., Analytical solution of a fractional diffusion equation by variational iteration method, Computers & Mathematics with Applications. (2009) 57, no. 3, 483–487, https://doi.org/10.1016/j.camwa.2008.09.045, MR2488620, ZBL1165.35398.
10.1016/j.camwa.2008.09.045
Web of Science® Google Scholar
13 Momani S. and 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, no. 7-8, 910–919, https://doi.org/10.1016/j.camwa.2006.12.037, MR2395628, ZBL1141.65398.
10.1016/j.camwa.2006.12.037
Web of Science® Google Scholar
14 Momani S. and Odibat Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A. (2007) 365, no. 5-6, 345–350, https://doi.org/10.1016/j.physleta.2007.01.046, MR2308776, ZBL1203.65212.
10.1016/j.physleta.2007.01.046
CAS Web of Science® Google Scholar
15 Baleanu D., Diethelm K., Scalas E., and Trujillo J. J., Fractional Calculus Models and Numerical Methods, 2012, World Scientific, Boston, Mass, USA, Series on Complexity, Nonlinearity and Chaos, https://doi.org/10.1142/9789814355216, MR2894576.
10.1142/8180
Google Scholar
16 Jafari H. and Seifi S., Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 5, 2006–2012, https://doi.org/10.1016/j.cnsns.2008.05.008, MR2474460, ZBL1221.65278.
10.1016/j.cnsns.2008.05.008
Web of Science® Google Scholar
17 Hristov J., Heat-balance integral to fractional (half-time) heat diffusion sub-model, Thermal Science. (2010) 14, no. 2, 291–316, 2-s2.0-77955895114, https://doi.org/10.2298/TSCI1002291H.
10.2298/TSCI1002291H
Web of Science® Google Scholar
18 Hristov J., Integral-balance solution to the stokes′ first problem of a viscoelastic generalized second grade fluid, Thermal Science. (2012) 16, no. 2, 395–410.
10.2298/TSCI110401077H
Web of Science® Google Scholar
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, no. 6, 802–809.
Google Scholar
20 Wu G. C. and Lee E. W. M., Fractional variational iteration method and its application, Physics Letters A. (2010) 374, no. 25, 2506–2509, https://doi.org/10.1016/j.physleta.2010.04.034, MR2640023, ZBL1237.34007.
10.1016/j.physleta.2010.04.034
CAS Web of Science® Google Scholar
21 Khan Y., Faraz N., Yildirim A., and 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, no. 5, 2273–2278, https://doi.org/10.1016/j.camwa.2011.07.014, MR2831688, ZBL1231.35288.
10.1016/j.camwa.2011.07.014
Web of Science® Google Scholar
22 Wu G. C., A fractional variational iteration method for solving fractional nonlinear differential equations, Computers & Mathematics with Applications. (2011) 61, no. 8, 2186–2190, https://doi.org/10.1016/j.camwa.2010.09.010, MR2785581, ZBL1219.65085.
10.1016/j.camwa.2010.09.010
Web of Science® Google Scholar
23 Zhang S. and Zhang H.-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A. (2011) 375, no. 7, 1069–1073, https://doi.org/10.1016/j.physleta.2011.01.029, MR2765013, ZBL1242.35217.
10.1016/j.physleta.2011.01.029
CAS Web of Science® Google Scholar
24 Jafari H., Tajadodi H., Kadkhoda N., and Baleanu D., Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations, Abstract and Applied Analysis. (2013) 2013, 5, 587179, https://doi.org/10.1155/2013/587179.
10.1155/2013/587179
Google Scholar
25 Zhang S., Zong Q. A., Liu D., and Gao Q., A generalized expfunction method for fractional Riccati differential equations, Communications in Fractional Calculus. (2010) 1, 48–52.
PubMed Web of Science® Google Scholar
26 Yang X. J., Advanced Local Fractional Calculus and Its Applications, 2012, World Science, New York, NY, USA.
Google Scholar
27 Yang X. J., Local fractional integral transforms, Progress in Nonlinear Science. (2011) 4, 1–225.
Google Scholar
28 Yang X. J., Local Fractional Functional Analysis and Its Applications, 2011, Asian Academic, Hong Kong.
Google Scholar
29 Zhong W. P., Yang X. J., and Gao F., A Cauchy problem for some local fractional abstract differential equation with fractal conditions, Journal of Applied Functional Analysis. (2013) 8, no. 1, 92–99.
Google Scholar
30 Su W. H., Yang X. J., Jafari H., and Baleanu D., Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator, Advances in Difference Equations. (2013) 2013, no. 1, 97–107, https://doi.org/10.1186/1687-1847-2013-97, MR3049970.
10.1186/1687-1847-2013-97
Google Scholar
31 Hu M. S., Baleanu D., and Yang X. J., One-phase problems for discontinuous heat transfer in fractal media, Mathematical Problems in Engineering. (2013) 2013, 3, 358473, https://doi.org/10.1155/2013/358473.
10.1155/2013/358473
Web of Science® Google Scholar
32 Kolwankar K. M. and Gangal A. D., Local fractional Fokker-Planck equation, Physical Review Letters. (1998) 80, no. 2, 214–217, https://doi.org/10.1103/PhysRevLett.80.214, MR1604435, ZBL0945.82005.
10.1103/PhysRevLett.80.214
CAS Web of Science® Google Scholar
33 Carpinteri A. and Sapora A., Diffusion problems in fractal media defined on Cantor sets, ZAMM Journal of Applied Mathematics and Mechanics. (2010) 90, no. 3, 203–210, 2-s2.0-77049111800, https://doi.org/10.1002/zamm.200900376.
10.1002/zamm.200900376
Web of Science® Google Scholar
34 Carpinteri A. and Cornetti P., A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos, Solitons and Fractals. (2002) 13, no. 1, 85–94, 2-s2.0-0036028181, https://doi.org/10.1016/S0960-0779(00)00238-1.
10.1016/S0960-0779(00)00238-1
Web of Science® Google Scholar
35 Carpinteri A., Chiaia B., and Cornetti P., The elastic problem for fractal media: basic theory and finite element formulation, Computers and Structures. (2004) 82, no. 6, 499–508, 2-s2.0-0742324870, https://doi.org/10.1016/j.compstruc.2003.10.014.
10.1016/j.compstruc.2003.10.014
Google Scholar
36 Babakhani A. and Daftardar-Gejji V., On calculus of local fractional derivatives, Journal of Mathematical Analysis and Applications. (2002) 270, no. 1, 66–79, https://doi.org/10.1016/S0022-247X(02)00048-3, MR1911751, ZBL1005.26002.
10.1016/S0022-247X(02)00048-3
Web of Science® Google Scholar
37 Ben Adda F. and Cresson J., About non-differentiable functions, Journal of Mathematical Analysis and Applications. (2001) 263, no. 2, 721–737, https://doi.org/10.1006/jmaa.2001.7656, MR1866075, ZBL0995.26006.
10.1006/jmaa.2001.7656
Web of Science® Google Scholar
38 Chen Y., Yan Y., and Zhang K., On the local fractional derivative, Journal of Mathematical Analysis and Applications. (2010) 362, no. 1, 17–33, https://doi.org/10.1016/j.jmaa.2009.08.014, MR2557664, ZBL1196.26011.
10.1016/j.jmaa.2009.08.014
Web of Science® Google Scholar
39 Chen W., Time-space fabric underlying anomalous diffusion, Chaos, Solitons and Fractals. (2006) 28, no. 4, 923–925, 2-s2.0-27744450698, https://doi.org/10.1016/j.chaos.2005.08.199.
10.1016/j.chaos.2005.08.199
Web of Science® Google Scholar
40 Chen W., Sun H., Zhang X., and Korošak D., Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications. (2010) 59, no. 5, 1754–1758, https://doi.org/10.1016/j.camwa.2009.08.020, MR2595948, ZBL1189.35355.
10.1016/j.camwa.2009.08.020
Web of Science® Google Scholar
41 Golmankhaneh A. K., Fazlollahi V., and Baleanu D., Newtonian mechanics on fractals subset of real-line, Romania Reports in Physics. (2013) 65, 84–93.
Web of Science® Google Scholar
42 Parvate A. and Gangal A. D., Fractal differential equations and fractal-time dynamical systems, Pramana. (2005) 64, no. 3, 389–409, 2-s2.0-15744373583.
10.1007/BF02704566
Web of Science® Google Scholar
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, no. 3, 1428–1448, https://doi.org/10.1016/j.chaos.2007.09.028, MR2526363, ZBL1197.60039.
10.1016/j.chaos.2007.09.028
Web of Science® Google Scholar
44 Jumarie G., Laplace′s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Applied Mathematics Letters. (2009) 22, no. 11, 1659–1664, https://doi.org/10.1016/j.aml.2009.05.011, MR2569059, ZBL1181.44001.
10.1016/j.aml.2009.05.011
Web of Science® Google Scholar
45 Yang X. J. and Baleanu D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science. (2013) 17, no. 2, 625–628.
10.2298/TSCI121124216Y
Web of Science® Google Scholar
46 Su W. H., Baleanu D., Yang X.-J., and 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, no. 1, 89–102, https://doi.org/10.1186/1687-1812-2013-89, MR3053815.
10.1186/1687-1812-2013-89
Google Scholar
47 Hu M. S., Agarwal R. P., and 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, https://doi.org/10.1155/2012/567401.
10.1155/2012/567401
Google Scholar
48 Anastassiou G. A. and Duman O., Advances in Applied Mathematics and Approximation Theory, 2013, Springer, New York, NY, USA.
10.1007/978-1-4614-6393-1
Google Scholar
49 Yang X. J., Srivastava H. M., He J. H., and Baleanu D., Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Physics Letters A. (2013) 377, no. 28–30, 1696–1700.
10.1016/j.physleta.2013.04.012
CAS Web of Science® Google Scholar
50 Yang X., Liao M., and 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, https://doi.org/10.1016/j.proeng.2012.01.420.
10.1016/j.proeng.2012.01.420
Google Scholar
51 Zhong W., Gao F., and 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, https://doi.org/10.4028/www.scientific.net/AMR.461.306.
10.4028/www.scientific.net/AMR.461.306
Google Scholar
52 Zhong W. P. and 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.
Google Scholar

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Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Abstract

1. Introduction

2. Analysis of the Method

3. Applications to Wave and Diffusion Equations on Cantor Sets

4. Conclusions

Acknowledgments

References

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Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Abstract

1. Introduction

2. Analysis of the Method

3. Applications to Wave and Diffusion Equations on Cantor Sets

4. Conclusions

Acknowledgments

References

Citing Literature

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