New applications of the variational iteration method - from differential equations to q-fractional difference equations
© Wu and Baleanu; licensee Springer 2013
Received: 11 September 2012
Accepted: 5 January 2013
Published: 24 January 2013
The non-classical calculi such as q-calculus, fractional calculus and q-fractional calculus have been hot topics in both applied and pure sciences. Then some new linear and nonlinear models have appeared. This study mainly concentrates on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem.
MSC: 39A13, 74H10.
Recently, q-fractional calculus has been paid much attention to [1–7], i.e., q-factional modeling, linear q-fractional systems, q-special functions etc. As is well known, both fractional calculus (FC) and q-calculus (QC) are not new as they appeared in 1695 and about 1920s, respectively. Fractional q-calculus (FQC) serves as a bridge between FC and QC. The early developments of q-fractional calculus can be found in [8–10]. Now, various q-fractional initial value problems are proposed in [3, 11–16].
The variational iteration method (VIM) [17–20] has been one of the often used nonlinear methods in initial boundary value problems of differential equations. In this study, the extension of the method into FQC is undertaken and the Caputo q-fractional initial value problems are investigated. Our study is organized as follows. In Section 2, the basic idea of the VIM is illustrated. In Section 3, the VIM is extended to q-difference equations, and the Lagrange multipliers of the method are presented for the equations of high-order q-derivatives. In Section 4, recent development of the method in fractional calculus is introduced. Following Section 4, the application of the VIM in q-fractional calculus is considered. Then the method is applied to the Caputo q-fractional initial value problem.
2 The VIM in ordinary calculus
where , R is a linear operator, N is a nonlinear operator, is a given continuous function and is the term of the highest-order derivative.
where is called the Lagrange multiplier which can be identified optimally by variational calculus and is the n th term approximate solution.
Following the above steps, we can design a Maple-program which contains three parameters: ICs, Eqs and n. ICs reads the value of initial points. Eqs contains information of the linear terms, the nonlinear terms and the interval functions. n means the approximate solution’s truncated order.
Comparisons between , and the exact solution
3 The VIM in q-calculus
Definition 3.1 (q-calculus)
Lemma 3.4 ()
one of the Lagrange multipliers is .
Lemma 3.5 ()
where denotes the q-factorial and for the integer k.
More generally, one can derive the following Theorem 3.6.
where for the integer m.
Here the initial iteration value can be determined via the q-Taylor series .
Recall that the limit is an exact solution of (22). Here is one of the q-exponential functions.
4 The VIM in fractional calculus
Let be a real-valued function defined on a closed interval .
One can check the formula (26) results in a poor convergence even for a linear FDE. Such difficulty can be overcome by the Laplace transform [31–33]. The following iteration formula is initially proposed in [34, 35]. Let us revisit the proof.
where the function is a Lagrange multiplier for any order α.
Setting the Lagrange multiplier , Eq. (30) can be considered as a convolution of the function and the term .
This completes the proof. □
For , tends to which is an exact solution of (41).
Remarks Our simplest iteration formula (34) can reduce to the Volterra integral equation. See the analysis of the convergence and existence in  and the references therein. However, regarding Eq. (36), the VIM transforms it into a more general Volterra integral equation from which one can obtain approximate solutions of higher accuracies.
FDEs have been proven to be a useful tool to describe the nonlocal behaviors or long range interactions of dynamical systems. The previous applications of the VIM just ‘guessed’ the Lagrange multipliers or directly used the one in ordinary differential equations. In this study, various Lagrange multipliers are identified more explicitly and the variational approach for FDEs is systematically developed now.
5 The Caputo q-fractional initial value problem
where Z is the set of integers.
where and .
Now, we introduce the q-Laplace transform and some properties.
where , .
Lemma 5.4 ()
Lemma 5.6 ()
The existence and uniqueness of the solutions of the Caputo q-initial value problems have been discussed in .
One of the Lagrange multipliers can be identified as .
From Lemma 5.6, we set , where . Then the Lagrange multiplier is ‘good’ enough so that the product of and is similar as the function in (51b) and becomes a convolution (51b).
where is the Laplace transform of some function.
Readers are referred to the recent development in the application of the VIM for solving fuzzy equations [41–43] and the calculus of variations on time scales [44–48]. Since this study only concentrates on the applications of the VIM, other numerical methods in FC can be found in [49–51].
We aim at some new applications of the VIM from differential equations to q-fractional difference equations, and the following main contributions of this study are obtained:
(a) Designing a maple program of the VIM for differential equations. Now, there is no need for one to obtain approximate solutions of high order by hand. The efficiency and accuracy are improved;
(b) Correcting the popularly used variational iteration formulae in FC and explicitly identifying some new Lagrange multipliers from the Laplace transform. The FDEs are transformed into generalized Volterra integral equations;
(c) Applying the VIM in q-difference equations and identifying a Lagrange multiplier of q-difference equations of m th order;
(d) Extending the VIM to FQC and investigating the initial value problems analytically. The obtained variational iteration formula in FQC can reduce to those in FC and QC.
Due to the rapid development of advanced applied sciences, non-classical tools of calculus, i.e., fractional calculus, q-calculus, etc., have been becoming more active and have been found useful in describing important physical phenomena. This study discusses some new applications of the VIM and provides a potential tool to analytically investigate such models. There is still some other work needed to consider, i.e., maple-packages or the symbolic computation of the VIM in FC even in FQC, other numerical methods based on the VIM, etc. The authors believe, in not far future, the VIM can play the same crucial role as that in ordinary calculus.
The authors would like to express their deep gratitude to the referees for their valuable suggestions and comments. The work is financially supported by the NSFC (11061028) and the key program of the NSFC (51134018).
- Saxena R, Yadav R, Purohit S: Kober fractional q -integral operator of the basic analogue of the H -function. Rev. Téc. Fac. Ing., Univ. Zulia 2005, 28: 154-158.MathSciNetGoogle Scholar
- Rajković PM, Marinković SD, Stanković MS: Fractional integrals and derivatives in q -calculus. Appl. Anal. Discrete Math. 2007, 1: 311-323. 10.2298/AADM0701311RMathSciNetView ArticleGoogle Scholar
- Mansour Z: Linear sequential q -difference equations of fractional order. Fract. Calc. Appl. Anal. 2009, 12: 159-178.MathSciNetGoogle Scholar
- Atici FM, Senguel S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369: 1-9. 10.1016/j.jmaa.2010.02.009MathSciNetView ArticleGoogle Scholar
- Herrmann R: Common aspects of q -deformed Lie algebras and fractional calculus. Physica A 2010, 389: 4613-4622. 10.1016/j.physa.2010.07.004MathSciNetView ArticleGoogle Scholar
- Purohit SD, Yadav RK: On generalized fractional q -integral operators involving the q -Gauss hypergeometric function. Bull. Math. Anal. Appl. 2010, 2: 35-44.MathSciNetGoogle Scholar
- Atici FM, Eloe PW: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 2011, 41: 353-370. 10.1216/RMJ-2011-41-2-353MathSciNetView ArticleGoogle Scholar
- Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966, 15: 135-140. 10.1017/S0013091500011469MathSciNetView ArticleGoogle Scholar
- Agarwal R: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060View ArticleGoogle Scholar
- Al-Salam W, Verma A: A fractional Leibniz q -formula. Pac. J. Math. 1975, 60: 1-9. 10.2140/pjm.1975.60.1MathSciNetView ArticleGoogle Scholar
- Mozyrska D, Pawłuszewicz E: Observability of linear q -difference fractional-order systems with finite initial memory. Bull. Pol. Acad. Sci. 2010, 58: 601-605.Google Scholar
- Abdeljawad T, Baleanu D: Caputo q -fractional initial value problems and a q -analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4682-4688. 10.1016/j.cnsns.2011.01.026MathSciNetView ArticleGoogle Scholar
- Ferreira RAC: Positive solutions for a class of boundary value problems with fractional q -differences. Comput. Math. Appl. 2011, 61: 367-373. 10.1016/j.camwa.2010.11.012MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: On nonlocal boundary value problems of nonlinear q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 81Google Scholar
- Ahmad B, Ntouyas SK, Purnaras IK: Existence results for nonlocal boundary value problems of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 140Google Scholar
- Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006MathSciNetView ArticleGoogle Scholar
- He JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 1998, 167: 57-68. 10.1016/S0045-7825(98)00108-XView ArticleGoogle Scholar
- He JH: Variational iteration method - a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 1999, 34: 699-708. 10.1016/S0020-7462(98)00048-1View ArticleGoogle Scholar
- He JH: Variational iteration method - some recent results and new interpretations. J. Comput. Appl. Math. 2007, 207: 3-17. 10.1016/j.cam.2006.07.009MathSciNetView ArticleGoogle Scholar
- He JH, Wu XH: Variational iteration method: new development and applications. Comput. Math. Appl. 2007, 54: 881-894. 10.1016/j.camwa.2006.12.083MathSciNetView ArticleGoogle Scholar
- El-Tawil MA, Bahnasawi AA, Abdel-Naby A: Solving Riccati differential equation using Adomian’s decomposition method. Appl. Math. Comput. 2004, 157: 503-514. 10.1016/j.amc.2003.08.049MathSciNetView ArticleGoogle Scholar
- Hull T, Enright W, Fellen B, Sedgwick A: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 1972, 9: 603-637. 10.1137/0709052MathSciNetView ArticleGoogle Scholar
- Kac VG, Cheung P: Quantum Calculus. Springer, Berlin; 2002.View ArticleGoogle Scholar
- Bohner M, Peterson AC: Advances in Dynamic Equations on Time Scales. Birkhäuser, Basel; 2003.View ArticleGoogle Scholar
- Gasper G, Rahman M: Basic Hypergeometric Series. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
- Jackson FH: q -form of Taylor’s theorem. Messenger Math. 1909, 38: 62-64.Google Scholar
- Jackson FH: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.Google Scholar
- Wu, GC: Variational iteration method for q-diffusion equations on time scales. Heat Transf. Res. (2012, in press)Google Scholar
- Wu GC: Variational iteration method for q -difference equations of second order. J. Appl. Math. 2012., 2012: Article ID 102850Google Scholar
- Kong H, Huang LL: Lagrange multipliers of q -difference equations of third order. Commun. Fract. Calc. 2012, 3: 30-33.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Kilbas AA, Srivastav HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Sheng H, Li Y, Chen YQ: Application of numerical inverse Laplace transform algorithms in fractional calculus. J. Franklin Inst. 2011, 348: 315-330. 10.1016/j.jfranklin.2010.11.009MathSciNetView ArticleGoogle Scholar
- Wu GC: Variational iteration method for the fractional diffusion equations in porous media. Chin. Phys. B 2012., 21: Article ID 120504Google Scholar
- Wu GC, Baleanu D: Variational iteration method for the Burgers’ flow with fractional derivatives - new Lagrange multipliers. Appl. Math. Model. 2012. doi:10.1016/j.apm.2012.12.018. (in press)Google Scholar
- Baleanu D, Golmankhaneh AK, Golmankhaneh AK: Solving of the nonlinear and linear Schrödinger equations by the homotopy perturbation method. Rom. J. Phys. 2010, 54: 823-832.MathSciNetGoogle Scholar
- Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
- Atici FM, Eloe PW: Fractional q -calculus on a time scale. J. Nonlinear Math. Phys. 2007, 14: 341-352. 10.2991/jnmp.2007.14.3.4MathSciNetView ArticleGoogle Scholar
- Hahn W: Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q -Differenzengleichung. Das q -Analogon der Laplace-Transformation. Math. Nachr. 1949, 2: 340-379. 10.1002/mana.19490020604MathSciNetView ArticleGoogle Scholar
- Annaby MH, Mansour ZSI: q-Fractional Calculus and Equations. Springer, Berlin; 2012.View ArticleGoogle Scholar
- Allahviranloo T, Abbasbandy S, Rouhparvar H: The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method. Appl. Soft Comput. 2011, 11: 2186-2192. 10.1016/j.asoc.2010.07.018View ArticleGoogle Scholar
- Jafari H, Saeidy M, Baleanu D: The variational iteration method for solving n -th order fuzzy differential equation. Cent. Eur. J. Phys. 2012, 10: 76-85. 10.2478/s11534-011-0083-7Google Scholar
- Jafari H, Khalique CM: Homotopy perturbation and variational iteration methods for solving fuzzy differential equations. Commun. Fract. Calc. 2012, 3: 38-48.Google Scholar
- Bangerezako G: Variational q -calculus. J. Math. Anal. Appl. 2004, 289: 650-665. 10.1016/j.jmaa.2003.09.004MathSciNetView ArticleGoogle Scholar
- Ferreira RAC, Torres DFM: Higher-order calculus of variations on time scales. Math. Control Theory Finance 2008, 2008: 149-159.MathSciNetView ArticleGoogle Scholar
- Abdeljawad T, Jarad F, Baleanu D: Variational optimal-control problems with delayed arguments on time scales. Adv. Differ. Equ. 2009., 2009: Article ID 840386Google Scholar
- Martins N, Torres DFM: Calculus of variations on time scales with nabla derivatives. Nonlinear Anal., Theory Methods Appl. 2009, 71: e763-e773. 10.1016/j.na.2008.11.035MathSciNetView ArticleGoogle Scholar
- Malinowska AB, Torres DFM: The Hahn quantum variational calculus. J. Optim. Theory Appl. 2010, 147: 419-442. 10.1007/s10957-010-9730-1MathSciNetView ArticleGoogle Scholar
- Chen W, Sun HG, Li XC: Mechanics Engineering Problems of Fractional Derivative Modeling. Science Press, Beijing; 2010.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore; 2012.Google Scholar
- Duan JS, Buleanu D, Wazwaz AM: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 2012, 3: 73-99.Google Scholar
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