Keywords:-

Keywords: fractional Laplace equation; fractional Schrodinger equation; Local fractional series expansion method; Cantor set.

Article Content:-

Abstract

In this paper, we proposed a local fractional series expansion method (LFSEM) to solve the Laplace and Schrodinger 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.

References:-

References

A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Elsevier, (1989)

Balkema, The Netherlands.W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,”, Journal of

Mathematical Physics, (1989), vol. 30, no. 1, pp.134–144.

Z. G. Zhao and C. P. Li, “Fractional difference/finite element approximations for the time-space

fractional telegraph equation,” Applied Mathematics and Computation, (2012),vol. 219, no. 6,

pp. 2975–2988.

S. Momani, Z. Debut, and A. Alawneh, “Variational iteration method for solving the space- and

time-fractional KdV equation,” Numerical Methods for Partial Differential Equations, (2008),

vol.24, no. 1, pp. 262–271.

N. Laskin, “Fractional Schrodinger equation,” Physical Review E, (2002), vol. 66, Article

ID 056108, 2002.

Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear

Analysis: Real World Applications, (2010), vol. 11, no. 5, pp. 4465–4475.

S. Momani and Z. Odibat, “Analytical solution of a time fractional Navier-Stokes equation by

Adomian decomposition method,” Applied Mathematics and Computation, (2006), vol. 177, no.

, pp. 488–494.

V. E. Tarasov, “Fractional Heisenberg equation,” Physics Letters A, (2008), vol. 372, no. 17, pp.

–2988.

A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional Klein

Gordon equation,” Signal Processing, (2011), vol. 91, no. 3, pp. 446–451.

Z. B. Li, W. H. Zhu, and L. L.Huang, “Application of fractional variational iteration method to

time-fractional Fisher equation,” Advanced Science Letters, (2001), vol.10, no. 1, pp. 610–614.

J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,” Thermal

Science, (2010), vol. 14,no. 2, pp.291–316.

D. Baleanu, K. Diethelm, E.Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical

Methods, (2012), vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific,

Boston, Mass, USA.

C. Cattani, “Harmonic wavelet solution of Poisson’s problem,” Balkan Journal of Geometry and

Its Applications, (2008), vol. 13, no. 1, pp. 27–37.

C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,” Mathematics with

Applications, (2005), vol. 50, no. 8-9, pp. 1191–1210.

A. M. Yang, X. J. Yang, and Z. B. Li, “Local fractional series expansion method for solving

wave and diffusion equations on Cantor sets,” Abstract and Applied Analysis, (2013), Article

ID 351057.X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional

variation iteration method,” Thermal Science, (2013), vol. 17, no. 2, pp. 625– 628.

W. H. Su, D. Baleanu, X. J. Yang, and H. Jafari, “Damped wave equation and dissipative wave

equation in fractal strings within the local fractional variational iteration method,” Fixed Point

Theory and Applications, (2011), Article 89.

Y. J. Yang, D. Baleanu and X. J. Yang, " Local Fractional Variational Iteration Method for

Laplace Equation within Local Fractional Operators", Abstract and Applied Analysis, (2013),

Article ID 202650.

X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on

Cantor time-space,” Proceeding of the Romanian Academy A, (2013), vol. 14, no. 2, pp. 127–

X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty

principle within local fractional Fourier analysis,” Boundary Value Problems, (2013), no. 1,pp.

– 146.

X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, (2012)

New York, NY, USA.

X. J. Yang, Local Fractional Functional Analysis and Its Applications, (2011), Asian Academic,

Hong Kong, China.

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, (2014), Article ID 176395.

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, (2014), Article ID 161580.

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Jafari, H., & Jassim, H. K. (2014). 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, 2(11), 736-744. Retrieved from http://ijmcr.in/index.php/ijmcr/article/view/183