### Approximate Solution for Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator

Hossein Jafari , Hassan Kamil Jassim
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.

Abstract

In this paper, the nonlinear gas dynamic and coupled KdV equations within local
fractional operator are discussed. The approximate solutions are obtained by using the
local fractional variational iteration method (LFVIM). This method is able to solve large
class of linear and nonlinear equations effectively, more easily and accurately; and thus
the method has been widely applicable to solve any class of equations in sciences and
engineering.

Key Words:
Nonlinear gas dynamic equation; Coupled Korteweg-de Vries Equations; Local fractional variational iteration method.

References

[1] H. Aminikhah and A. Jamalian, Numerical Approximation for Nonlinear Gas Dynamic Equation, International Journal of Partial Differential Equations, vol. 2013, Article ID 846749, pp. 1-7, (2013).

[2] 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, pp. 1-7, (2014).

[3] 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, pp. 1-7, (2014).

[4] D. Baleanu, J.A.T. Machado, C. Cattani, M. C. Baleanu and X.J. Yang, Local fractional variational iteration and decomposition methods for wave equation on Cantor sets, Abstract and Applied Analysis, Article ID 535048, pp.1-6, (2014).

[5] H. Jafari, and H. K. Jassim, Local Fractional Series Expansion Method for Solving Laplace and Schrodinger Equations on Cantor Sets within Local Fractional Operators, International Journal of Mathematics and Computer Research, Vol. 2, No. 11 ,736-744, (2014).

[6] A. M. Yang, Z. S. Chen, X. J. Yang, Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Operators, Abstract and Applied Analysis, Article ID 259125, pp. 1-6, (2013).

[7] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, (2012).

[8] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, China, (2011).

[9] M. S. Hu, R. P. Agarwal, and X. J. Yang, Local fractional Fourier series with application to wave equation in fractal vibrating string, Abstract and Applied Analysis, Article ID 567401, pp. 1-15, (2012).

[10] H. Jafari, H. K. Jassim, Numerical Solutions of Telegraph and Laplace Equations on Cantor Sets Using Local Fractional Laplace Decomposition Method , International Journal of Advances in Applied Mathematics and Mechanics, Vol. 2, No. 3, pp. 1-8, (2015).

[11] H. K. Jassim, Local Fractional Laplace Decomposition Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow with Local Fractional Derivative, International Journal of Advances in Applied Mathematics and Mechanics, Vol. 2, No. 7, pp. 1-7, (2015).

[12] H. Jafari, and H. K. Jassim, Local Fractional Laplace Variational Iteration Method for Solving Nonlinear Partial Differential Equations on Cantor Sets within Local Fractional Operators, Journal of Zankoy Sulaimani-Part A, vol. 16, no. 4, pp. 49-57, (2014).

[13] H. K. Jassim, C. Ünlü, S. P. Moshokoa, C. M. Khalique, Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators, Mathematical Problems in Engineering, Vol. 2015, Article ID 309870, pp. 1-7, (2015).

[14] C. F. Liu, S. S. Kong, and S. J. Yuan, Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem, Thermal Science, vol. 17, no. 3, pp. 715–721, (2013).

[15] X. J. Yang and D. Baleanu, Local fractional variational iteration method for Fokker-Planck equation on a Cantor set, Acta Universitaria, Vol. 23, No. 2, pp. 3-8, (2013).

[16] X. J. Yang and D. Baleanu, Fractal heat conduction problem solved by local fractional variation iteration method. Thermal Science. Vol. 17, no. 2, pp. 625–628, (2013).