- Open Access
Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method
© Su et al.; licensee Springer 2013
- Received: 23 January 2013
- Accepted: 23 March 2013
- Published: 10 April 2013
In this paper, the local fractional variational iteration method is given to handle the damped wave equation and dissipative wave equation in fractal strings. The approximation solutions show that the methodology of local fractional variational iteration method is an efficient and simple tool for solving mathematical problems arising in fractal wave motions.
MSC:74H10, 35L05, 28A80.
- local fractional variational iteration method
- damped wave equation
- dissipative wave equation
- local fractional operators
- fractal strings
The variational iteration method was effectively applied in various fields of science and engineering [1–15] and the references therein. It is in some cases, more powerful than the existing techniques, e.g., the fractional variational iteration method [6, 16, 17], the homotopy perturbation method [18, 19], the exp-function method [20, 21], the decomposition method [22–24], the homotopy analysis method [25, 26] and others . The wave equation was investigated within some differential methods [7–15, 18–26] and the references therein.
As it is known, the quantum behavior of microphysics in terms of a non-differentiable space-time continuum possesses and has fractal property. Also, it was shown by many authors that a time-space structure of microphysics is non-differentiable. The relativistic quantum mechanics in fractal time space was suggested in . It was pointed out that, while the zero set represents the Cantor point-like quantum particle, the empty set was the basic mathematical representation of the quantum wave . The exact solutions for a class of fractal time random walks were researched in . The questions of a philosophical nature about fractal spacetime and its implications for phenomenology and ontology were shown in . The fractal time-space structure for dealing with the non-differentiability and infinities of fractals derived from local fractional operators was presented in [32–34] and the references therein. A solution of the wave equation in fractal vibrating string by using the local fractional Fourier series was discussed in . The diffusion equation on Cantor time-space was reported in  while the diffusion problems on fractal space were suggested in . The heat conduction problem by local fractional variational iteration method was investigated in . The heat conduction equation in fractal time space was structured in . A relaxation equation in fractal space was set up in . The anomalous diffusion equation in the fractal time-space fabric was pointed out in . The Fokker-Planck equation in fractal time was considered in .
More recently, the local fractional variational iteration method, which was structured in , was applied to solve heat conduction equation on Cantor sets  and the local fractional Laplace equation . The purpose of this paper is to present the solutions of the damped wave equation and the dissipative wave equation in fractal strings equipped with fractal initial conditions.
if is satisfied the conditions and .
if is satisfied the conditions and .
with given conditions for .
for given condition .
where is a general fractal Lagrange’s multiplier.
and so on.
In this manuscript, utilizing the local fractional differential operators, we investigated the damped and the dissipative wave equations in fractal strings. Based on the local fractional variational iteration method, the solutions of the damped and dissipative wave equations were presented. The iteration functions, which is local fractional continuous, is obtained easily within the fractal Lagrange multipliers, which can be optimally determined by the local fractional variational theory . It is shown that the local fractional variational iteration method is an efficient and simple tool for handling partial differential equations with local fractional differential operator.
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the editor and the referees for their useful comments and remarks. The work is supported by the Natural Science Foundation of Tianjin, China (No. 10JCZDJC25100).
- He JH: Variational iteration method - a kind of nonlinear analytical technique: some examples. Int. J. Non-Linear Mech. 1999, 34: 699–708. 10.1016/S0020-7462(98)00048-1View ArticleGoogle Scholar
- He JH: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 2006, 20: 1141–1199. 10.1142/S0217979206033796View ArticleGoogle Scholar
- He JH, Wu XH: Variational iteration method: new development and applications. Comput. Math. Appl. 2007, 54: 881–894. 10.1016/j.camwa.2006.12.083MathSciNetView ArticleGoogle Scholar
- He JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 1998, 167(1–2):57–68. 10.1016/S0045-7825(98)00108-XView ArticleGoogle Scholar
- He JH: Comment on ‘Variational iteration method for fractional calculus using He’s polynomials’. Abstr. Appl. Anal. 2012., 2012: Article ID 964974Google Scholar
- He JH: Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. 2012., 2012: Article ID 916793Google Scholar
- Barari A, Ghotbi AR, Farrokhzad F, Ganji DD: Variational iteration method and homotopy-perturbation method for solving different types of wave equations. J. Appl. Sci. 2008, 8: 120–126.View ArticleGoogle Scholar
- Wazwaz AM: The variational iteration method: a reliable analytic tool for solving linear and nonlinear wave equations. Comput. Math. Appl. 2007, 54: 926–932. 10.1016/j.camwa.2006.12.038MathSciNetView ArticleGoogle Scholar
- Momani S, Abusaad S: Application of He’s variational-iteration method to Helmholtz equation. Chaos Solitons Fractals 2005, 27: 1119–1123.View ArticleGoogle Scholar
- Abdou MA, Soliman AA: Variational iteration method for solving Burgers’ and coupled Burgers’ equation. J. Comput. Appl. Math. 2005, 181: 245–251. 10.1016/j.cam.2004.11.032MathSciNetView ArticleGoogle Scholar
- Abbasbandy S: Numerical method for non-linear wave and diffusion equations by the variational iteration method. Int. J. Numer. Methods Eng. 2008, 73: 1836–1843. 10.1002/nme.2150MathSciNetView ArticleGoogle Scholar
- Molliq Y, Noorani RMS, Hashim MI: Variational iteration method for fractional heat-and wave-like equations. Nonlinear Anal., Real World Appl. 2009, 10: 1854–1869. 10.1016/j.nonrwa.2008.02.026MathSciNetView ArticleGoogle Scholar
- Hemeda AA: Variational iteration method for solving wave equation. Comput. Math. Appl. 2008, 56: 1948–1953. 10.1016/j.camwa.2008.04.010MathSciNetView ArticleGoogle Scholar
- Batiha B, Noorani MSM, Hashim I: Application of variational iteration method to heat-and wave-like equations. Phys. Lett. A 2007, 369: 55–61. 10.1016/j.physleta.2007.04.069MathSciNetView ArticleGoogle Scholar
- Biazar J, Ghazvini H: An analytical approximation to the solution of a wave equation by a variational iteration method. Appl. Math. Lett. 2008, 21: 780–785. 10.1016/j.aml.2007.08.004MathSciNetView ArticleGoogle Scholar
- Wu GC, Lee EWM: Fractional variational iteration method and its application. Phys. Lett. A 2010, 374(25):2506–2509. 10.1016/j.physleta.2010.04.034MathSciNetView ArticleGoogle Scholar
- He JH: A short remark on fractional variational iteration method. Phys. Lett. A 2011, 375(38):3362–3364. 10.1016/j.physleta.2011.07.033MathSciNetView ArticleGoogle Scholar
- He JH: Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 2005, 26: 695–700. 10.1016/j.chaos.2005.03.006View ArticleGoogle Scholar
- Jafari H, Momani S: Solving fractional diffusion and wave equations by modified homotopy perturbation method. Phys. Lett. A 2007, 370: 388–396. 10.1016/j.physleta.2007.05.118MathSciNetView ArticleGoogle Scholar
- He JH, Wu XH: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 2006, 30: 700–708. 10.1016/j.chaos.2006.03.020MathSciNetView ArticleGoogle Scholar
- Zhang S: Application of Exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 2007, 365: 448–453. 10.1016/j.physleta.2007.02.004MathSciNetView ArticleGoogle Scholar
- Odibat ZM, Momani S: Approximate solutions for boundary value problems of time-fractional wave equation. Appl. Math. Comput. 2006, 181: 767–774. 10.1016/j.amc.2006.02.004MathSciNetView ArticleGoogle Scholar
- Datta BK: A new approach to the wave equation - an application of the decomposition method. J. Math. Anal. Appl. 1989, 142: 6–12. 10.1016/0022-247X(89)90158-3MathSciNetView ArticleGoogle Scholar
- Momani S: Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl. Math. Comput. 2005, 165: 459–472. 10.1016/j.amc.2004.06.025MathSciNetView ArticleGoogle Scholar
- Liao SJ: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 2009, 4: 983–997.View ArticleGoogle Scholar
- Jafari H, Seifi S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 2006–2012. 10.1016/j.cnsns.2008.05.008MathSciNetView ArticleGoogle Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Ord GN: Fractal space-time: a geometric analogue of relativistic quantum mechanics. J. Phys. A 1999, 16: 1869.MathSciNetView ArticleGoogle Scholar
- Marek-Crnjac L: Polypseudologarithms and their applications to quantum ideal gas and the quantum wave collapse. Fractal Spacetime Noncommut. Geom. Quantum High Energy Phys. 2012, 2: 15–21.Google Scholar
- Hilfer R: Exact solutions for a class of fractal time random walks. Fractals 1995, 3: 211–216. 10.1142/S0218348X95000163MathSciNetView ArticleGoogle Scholar
- Vrobel S: Fractal time and fractal spacetime: phenomenology vs ontology. Fractal Spacetime Noncommut. Geom. Quantum High Energy Phys. 2011, 1: 41–44.Google Scholar
- Yang XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York; 2012.Google Scholar
- Yang XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher Limited, Hong Kong; 2011.Google Scholar
- Yang XJ: Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4: 1–225.Google Scholar
- Hu MS, Agarwal RP, Yang XJ: Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstr. Appl. Anal. 2012., 2012: Article ID 567401Google Scholar
- Yang, XJ, Baleanu, D, Zhong, WP: Approximation solution to diffusion equation on Cantor time-space. Proc. Rom. Acad., Ser. A (2013, in press)Google Scholar
- Carpinteri A, Sapora A: Diffusion problems in fractal media defined on Cantor sets. Z. Angew. Math. Mech. 2010, 90: 203–210. 10.1002/zamm.200900376View ArticleGoogle Scholar
- Yang XJ, Baleanu D: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 2012. doi:10.2298/TSCI121124216YGoogle Scholar
- He JH: A new fractal derivation. Therm. Sci. 2011, 15: 145–147.Google Scholar
- Chen W: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 2006, 28: 923–929. 10.1016/j.chaos.2005.08.199View ArticleGoogle Scholar
- Kolwankar KM, Gangal AD: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 1998, 80: 214–217. 10.1103/PhysRevLett.80.214MathSciNetView ArticleGoogle Scholar
- Machado JAT, Kiryakova V, Mainardi F: A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 2010, 13(3):329–334.MathSciNetGoogle Scholar
- Baleanu D, Guvenç ZB, Machado JAT: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin; 2009.Google Scholar
- Baleanu D, Machado JAT, Luo ACJ: Fractional Dynamics and Control. Springer, New York; 2011.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Sabatier J, Agrawal OP, Machado JAT: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, New York; 2007.View ArticleGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
- Mainardi F: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London; 2010.View ArticleGoogle Scholar
- Yang XJ: Local fractional variational iteration method and its algorithms. Adv. Comput. Math. Appl. 2012, 1: 139–145.Google Scholar
- Yang YJ, Baleanu D, Yang XJ: A local fractional variational iteration method for Laplace equation within local fractional operators. Abstr. Appl. Anal. 2013., 2013: Article ID 202650Google Scholar
- Yang XJ: The zero-mass renormalization group differential equations and limit cycles in non-smooth initial value problems. Prespacetime J. 2012, 3(9):913–923.Google Scholar
- Hu MS, Baleanu D, Yang XJ: One-phase problems for discontinuous heat transfer in fractal media. Math. Probl. Eng. 2013., 2013: Article ID 358473Google Scholar
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