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Variational Iteration Method for q-Difference Equations of Second Order
Recently, Liu extended He's variational iteration method to strongly nonlinear q-difference equations. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. The q-oscillation equation of second order is approximately solved to show the new Lagrange multiplier's validness.
Generally, applying the variational iteration method (VIM) [1, 2] in differential equations follows the three steps:(a)establishing the correction functional;(b)identifying the Lagrange multipliers;(c)determining the initial iteration.
Obviously, the step (b) is crucial and critical in the method.
In this paper, it is pointed out that the iteration formula (1.3) can be given in a more accurate way and a new Lagrange multiplier is explicitly identified.
2. Properties of -Calculus
Let be a real continuous function. The -derivative is defined as and
The partial -derivative with respect to is The corresponding -integral  is
2.2. -Leibniz Product Law
2.3. -Integration by Parts
The properties above are needed in the construction of the correction functional for -difference equations. For more results and properties in -calculus, readers are referred to the recent monographs [5–8].
3. A -Analogue of Lagrange Multiplier
In order to identify the Lagrange multipliers of the -difference equations, we first establish the correctional functional for (1.1) as
The correction functional here is different from the one in ordinary calculus since the parameter “disappears” after the integration by parts (2.5) each time. As a result, we use in the above functional.
We only need to consider the leading term when other terms are restricted variations in (1.1)
Through the integration by parts (2.5), we can have where is the variation operator and “′” denotes the -derivative with respect to . As a result, the system of the Lagrange multiplier can be obtained: the coefficient of , the coefficient of the coefficient of in the -integral : ,from which we can get instead of in . More introductions to the identification of various Lagrange multipliers of the VIM can be found in [9, 10].
We also can show the above -analogue of Lagrange multiplier’s validness. For , let be the time scale: , where is the set of positive integers. For the real continuous function , a -oscillator equation of second order is
From (3.4), the iteration formula can be given as
Starting from the initial iteration , the successive approximate solutions can be obtained as The limit is an exact solution of (3.5). Here is one of the -exponential functions.
In the past ten years, the VIM has been one of the often used nonlinear methods. The -derivative is a deformation of the classical derivative and it has played a crucial role in quantum mechanics and quantum calculus. In this study, the method is successfully extended to difference equations of second order. A -analogue of Lagrange multiplier is presented. Readers who feel interested in the initial value problems of the difference equations are referred to [11–17].
- J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
- J. H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
- F. H. Jackson, “-form of Taylor's theorem,” Messenger of Mathematics, vol. 38, pp. 62–64, 1909.
- H. K. Liu, “Application of the variational iteration method to strongly nonlinear -difference equations,” Journal of Applied Mathematics, vol. 2010, Article ID 704138, 12 pages, 2010.
- V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.
- G. Gasper and M. Rahman, Encyclopedia of Mathematics and Its Applications, Basic Hypergeometric Series, Cambridge University Press, Cambridge, UK, 1990.
- M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäauser, 2003.
- G. Bangerezako, An Introduction To q-Difference Equations, preprint, 2008.
- J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Science Letters A, vol. 1, pp. 1–30, 2011.
- G. C. Wu, “New trends in the variational iteration method,” Communications in Fractional Calculus, vol. 2, pp. 59–75, 2011.
- P. M. Rajković, M. S. Stanković, and S. D. Marinković, “On -iterative methods for solving equations and systems,” Novi Sad Journal of Mathematics, vol. 33, no. 2, pp. 127–137, 2003.
- P. M. Rajković, S. D. Marinković, and M. S. Stanković, “On -Newton-Kantorovich method for solving systems of equations,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1432–1448, 2005.
- Z. S. I. Mansour, “Linear sequential -difference equations of fractional order,” Fractional Calculus & Applied Analysis, vol. 12, no. 2, pp. 159–178, 2009.
- T. Abdeljawad and D. Baleanu, “Caputo -fractional initial value problems and a -analogue Mittag-Leffler function,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4682–4688, 2011.
- M. El-Shahed and M. Gaber, “Two-dimensional -differential transformation and its application,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9165–9172, 2011.
- K. A. Aldwoah, A. B. Malinowska, and D. F. M. Torres, “The power quantum calculus and variational problems,” Dynamics of Continuous, Discrete and Impulsive Systems Series B, vol. 19, no. 1-2, pp. 93–116, 2012.
- G. C. Wu, “Variational iteration method for q-diffusion equations on time scales,” Heat Transfer Research Accepted. In press.
Copyright © 2012 Guo-Cheng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.