Local Fractional Laplace Variational Iteration Method for Solving Nonlinear Partial Differential Equations on Cantor Sets within Local Fractional Operators

1 Hossein Jafari , 2 Hassan Kamil Jassim
1 University of Mazandaran, Faculty of Mathematical Sciences, Department of Mathematics, 2 University of Thi-Qar, Faculty of Education for Pure Sciences, Department of Mathematics

In this work, we discuss solutions of the nonlinear partial differential equations on
Cantor sets within local fractional operators. The nondifferentiable approximate
solutions are obtained by using the local fractional Laplace variational iteration method,
which is the coupling method of local fractional variational iteration method and
Laplace transform. The obtained results show the efficiency and accuracy of implements
of the present method.

Keywords: Nonlinear differential equation; Fractional Gas dynamics equation; Cantor set;
Yang-Laplace transform; local fractional variational iteration method.

[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam, The Netherlands, (2006).
[2] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College
Press, London, UK, (2010).
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, (1999).
[4] R. L. Magin, Fractional Calculus in Bioengineering, Begerll House, West Redding, Conn, USA, (2006).
[5] J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics in Physics: Recent Advances,
World Scientific, Singapore, (2011).
[6] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press,
Oxford, UK, (2008).
[7] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New
York, NY, USA, (2003).
[8] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of
Particles, Fields and Media, Springer, Berlin, Germany, (2011).
[9] J. A. Tenreiro Machado, A. C. J. Luo, and D. Baleanu, Nonlinear Dynamics of Complex
Systems: Applications in Physical, Biological and Financial Systems, Springer, New
York, NY, USA, (2011).
[10] J. S. Duan, T. Chaolu, R. Rach, and L. Lu, “The Adomian decomposition method with
convergence acceleration techniques for nonlinear fractional differential equations,”
Computers & Mathematics With Applications, (2013).
[11] J. S. Duan, T. Chaolu, and R. Rach, “Solutions of the initial value problem for nonlinear
fractional ordinary differential equations by the Rach-Adomian-Meyers modified
decomposition method,” Applied Mathematics and Computation, vol. 218, no. 17, pp.
8370–8392, (2012).
[12] S. Das, “Analytical solution of a fractional diffusion equation by variational iteration
method,” Computers & Mathematics with Appl., vol. 57, no. 3, pp. 483-487, (2009).
[13] S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and
the variational iteration method for linear fractional partial differential equations,”
Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 910–919, (2007).
[14] S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial
differential equations,” Physics Letters A, vol. 365, no. 5-6, pp. 345-350, (2007).
[15] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and
Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific,
Boston, Mass, USA, (2012).
[16] H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear
fractional diffusion-wave equation,” Communications in Nonlinear Science and
Numerical Simulation, vol. 14, no. 5, pp. 2006–2012, (2009).
[17] J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,
”Thermal Science, vol. 14,no. 2, pp.291–316, (2010).
[18] J. Hristov, “Integral-balance solution to the stokes’ first problem of a viscoelasticity
generalized second grade fluid,” Thermal Science, vol. 16, no. 2, pp. 395–410, (2012).
[19] J. Hristov, “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, vol. 3, no. 6, pp. 802–809, (2011).
[20] G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its
application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, (2010).
[21] Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “Fractional variational iteration method for
fractional initial-boundary value problems in the application of nonlinear science,
Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2273–2278, (2011).
[22] G. C. Wu, “A fractional variational iteration method for solving fractional nonlinear
differential equations,” Computers & Mathematics with Applications, vol. 61, no. 8, pp.
2186–2190, (2011).
[23] S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to
nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, (2011).
[24] H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, Abstract and Applied Analysis,
Article ID 587179, (2013).
[25] K. B. Oldham, J. Spanier, The fractional calculus, Academic Press, New York, (1999).
[26] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional
Differential Equations, John Willey and Sons, Inc., New York, (2003).
[27] M. Caputo, Linear models of dissipation whose Q is almost frequency independent: Part
II, J Roy Astronom Soc., vol. 13, pp. 529-539, (1967).
[28] H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional
derivatives, Z. Angew Math. Mech., vol. 75, pp. 623-635, (1995).
[29] W. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., vol.
30, pp. 134-144, (1989).
[30] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena,
Chaos Solitons and Fractals, vol. 7, pp. 1461-1477, (1996).
[31] F. Huang, F. Liu, The time fractional diffusion equation and fractional advection
dispersion equation, ANZIAM J. 46, pp 1-14, (2005).
[32] J.H. He, Nonlinear oscillation with fractional derivative and its applications,
International Conference on Vibrating Engineering, pp. 288–291, Dalian, Chaina, (1998).
[33] J.H. He, Approximate analytical solution for seepage flow with fractional derivative in
porous media, Comput. Methd. Appl. Mech. Eng. 167 , pp. 57-68, (1998).
[34] N. Faraz, Y. Khan, D. S. Sankar, Decomposition-transform method for fractional
differential equations, J. Nonlinear Sci. Numer. Simulation, vol. 11, pp. 305–310, (2010).
[35] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic
Publisher, Hong Kong, (2011).
[26] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science
Publisher, New York, NY,USA, (2012).
[37] S. Q. Wang, Y. J. Yang, and H. K. Jassim, " Local Fractional Function Decomposition
Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative,"
Abstract and Applied Analysis, Article ID 176395, (2014).
[38] W. P. Zhong, X. J. Yang, F. Gao, Journal of Applied Functional Analysis, pp. 92–99 (2013).
[39] M. S. Hu, Ravi P. Agarwal, X. J. Yang, Abstract Applied Analysis, Article ID 567401 (2012).
[40] S. P. Yan, H. Jafari, and H. K. Jassim, " Local Fractional Adomian Decomposition
and Function Decomposition Methods for Solving Laplace Equation within Local
Fractional Operators," Advances in Mathematical Physics, Article ID 161580, (2014).
[41] C. Cattani, D. Baleanu, and X. J. Yang, “A local fractional variational iteration and
Decomposition methods for Wave equation on Cantor set within local fractional
Operators,” Abstract and Applied Analysis, Article ID 535048, (2014).
[42] Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional
Variation Iteration Method, Thermal Science, pp. 625-628, (2013).
[43] Su,W. H., et al., Damped Wave Equation and Dissipative Wave Equation in Fractal
Strings within the Local Fractional Variational Iteration Method, Fixed Point Theory and
Applications, pp. 89-102, (2013).
[44] He, J.-H., Liu, F.-J., Local Fractional Variational Iteration Method for Fractal Heat
Transfer in Silk Cocoon Hierarchy, Nonlinear Science Letters A, pp. 15-20, (2013).
[45] A. M. Yang, X. J. Yang, and Z. B. Li, " Local Fractional Series Expansion Method for
Wave and Diffusion Equations on Cantor Sets. ,"Abstract and Applied Analysis, Article
ID 351057, (2014).