Variational iteration method for q-difference equations of second order

Transcription

1 Sichuan University From the SelectedWorks of G.C. Wu Summer June 6, 1 Variational iteration method for -difference euations of second order Guo-Cheng Wu Available at:

2 Hindawi Publishing Corporation Journal of Applied Mathematics Volume 1, Article ID 185, 5 pages doi:1.1155/1/185 Letter to the Editor Variational Iteration Method for -Difference Euations of Second Order Guo-Cheng Wu Key Laboratory of Numerical Simulation of Sichuan Province, Neijiang Normal University, Sichuan 64111, China Correspondence should be addressed to Guo-Cheng Wu, wuguocheng@yahoo.com.cn Received 6 April 1; Accepted 6 June 1 Copyright 1 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. Recently, Liu extended He s variational iteration method to strongly nonlinear -difference euations Liu 1. In this study, the iteration formula and the Lagrange multiplier are given in a more accurate way. The -oscillation euation of second order is approximately solved to show the new Lagrange multiplier s validness. 1. Introduction Generally, applying the variational iteration method VIM 1, in differential euations 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. For the strongly nonlinear -difference euation, d d t x εx d d t x Ω x x, 1.1 where d /d t is the -derivative 3, Liu 4 used the Lagrange multiplier λ t, s s t, 1.

3 Journal of Applied Mathematics which results in the iteration formula see 4, 4.1 and 4.11 : ( d x n 1 x n s t d s x n εx n d ) d s x n Ω x n xn d s. 1.3 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.. Properties of -Calculus.1. -Calculus Let f x be a real continuous function. The -derivative is defined as ( ) d f x f x f x ( ), x/, <<1,.1 d x 1 x and d /d x f x x lim n f n f / n. The partial -derivative with respect to x is x f( x; y;... ) f ( ) ( ) x; y;... f x; y;... ( ).. 1 x The corresponding -integral 5 is x f t d t ( 1 ) x n f ( n x ). n.3.. -Leibniz Product Law One has d [ ] ( ) d [ ] d [ ] g x f x g x f x f x g x..4 d x d x d x.3. -Integration by Parts One has b a g ( t ) d d t f t d t f t g t b a b a f t d d t g t d t..5 The properties above are needed in the construction of the correction functional for -difference euations. For more results and properties in -calculus, readers are referred to the recent monographs 5 8.

4 Journal of Applied Mathematics 3 3. A -Analogue of Lagrange Multiplier In order to identify the Lagrange multipliers of the -difference euations, we first establish the correctional functional for 1.1 as x n 1 x n ( ) ( d λ t, s d s x n εx n d ) d s x n Ω x n xn d s. 3.1 The correction functional here is different from the one in ordinary calculus since the parameter disappears after the integration by parts.5 each time. As a result, we use λ t, s in the above functional. We only need to consider the leading term d /d t x when other terms are restricted variations in 1.1 x n 1 x n ( ) ( d λ t, s d s x n εx n d ) d s x n Ω x n xn d s. 3. Through the integration by parts.5, we can have ( δx n 1 1 λ t, s s s t ) δx n λ ( t, s ) s t δx n s λ t, s δx nd s, 3.3 where δ is the variation operator and denotes the -derivative with respect to t. Asa result, the system of the Lagrange multiplier can be obtained: the coefficient of δx n :1 / s λ t, s s t, the coefficient of δx n : λ t, s s t, the coefficient of δx n in the -integral : / s λ t, s, from which we can get λ t, s 1( s t ), 3.4 instead of λ t, s s t in 4. More introductions to the identification of various Lagrange multipliers of the VIM can be found in 9, 1. We also can show the above -analogue of Lagrange multiplier s validness. For < <1, let T be the time scale: T { n : n Z} {}, where Z is the set of positive integers. For the real continuous function u t : T R,a-oscillator euation of second order is d d t u u, u 1, d d t u 1. t 3.5

5 4 Journal of Applied Mathematics From 3.4, the iteration formula can be given as u n 1 u n 1( ) [ d ] s t d s u n s u n s d s. 3.6 Starting from the initial iteration u 1 t/ 1!, the successive approximate solutions can be obtained as u 1 t 1!, u 1 1. t 1! t! t3 3!, 3.7 u n n 1 k t k k!. The limit u lim n u n e t is an exact solution of 3.5. Heree t is one of the - exponential functions. 4. Conclusions 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 uantum mechanics and uantum calculus. In this study, the method is successfully extended to difference euations of second order. A -analogue of Lagrange multiplier is presented. Readers who feel interested in the initial value problems of the difference euations are referred to References 1 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-, pp , J. H. He, Variational iteration method a kind of non-linear analytical techniue: some examples, International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp , F. H. Jackson, -form of Taylor s theorem, Messenger of Mathematics, vol. 38, pp. 6 64, H. K. Liu, Application of the variational iteration method to strongly nonlinear -difference euations, Journal of Applied Mathematics, vol. 1, Article ID 74138, 1 pages, 1. 5 V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA,. 6 G. Gasper and M. Rahman, Encyclopedia of Mathematics and Its Applications, Basic Hypergeometric Series, Cambridge University Press, Cambridge, UK, M. Bohner and A. C. Peterson, Advances in Dynamic Euations on Time Scales,Birkhäauser, 3. 8 G. Bangerezako, An Introduction To -Difference Euations, preprint, 8. 9 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 3, G. C. Wu, New trends in the variational iteration method, Communications in Fractional Calculus, vol., pp , 11.

6 Journal of Applied Mathematics 5 11 P. M. Rajković, M. S. Stanković, and S. D. Marinković, On -iterative methods for solving euations and systems, Novi Sad Journal of Mathematics, vol. 33, no., pp , 3. 1 P. M. Rajković, S. D. Marinković, and M. S. Stanković, On -Newton-Kantorovich method for solving systems of euations, Applied Mathematics and Computation, vol. 168, no., pp , Z. S. I. Mansour, Linear seuential -difference euations of fractional order, Fractional Calculus & Applied Analysis, vol. 1, no., pp , 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. 1, pp , M. El-Shahed and M. Gaber, Two-dimensional -differential transformation and its application, Applied Mathematics and Computation, vol. 17, no., pp , K. A. Aldwoah, A. B. Malinowska, and D. F. M. Torres, The power uantum calculus and variational problems, Dynamics of Continuous, Discrete and Impulsive Systems Series B, vol. 19, no. 1-, pp , G. C. Wu, Variational iteration method for -diffusion euations on time scales, Heat Transfer Research Accepted. In press.