Variational iteration method for fractional calculus - a universal approach by Laplace transform

Springer Science and Business Media LLC - Tập 2013 Số 1 - 2013
Guo‐Cheng Wu1,2, Dumitru Băleanu3,4
1College of Mathematics and Information Science, Neijiang Normal University, Neijiang, China
2College of Water Resources and Hydropower, Sichuan University, Chengdu, Sichuan, China
3Department of Mathematics and Computer Sciences, Cankaya University, Balgat, Ankara, Turkey
4Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Tóm tắt

Từ khóa


Tài liệu tham khảo

Inokuti M, Sekine H, Mura T: General use of the Lagrange multiplier in nonlinear mathematical physics. In Variational Methods in the Mechanics of Solids. Edited by: Nemat-Nasser S. Pregman Press, New York; 1978:156-162.

He JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 1998, 167(1-2):57-68. 10.1016/S0045-7825(98)00108-X

He JH: Variational iteration method - a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 1999, 34(4):699-708. 10.1016/S0020-7462(98)00048-1

He JH: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 2008, 22(21):3487-3578. 10.1142/S0217979208048668

Abbasbandy S: A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. J. Comput. Appl. Math. 2007, 207(1):59-63. 10.1016/j.cam.2006.07.012

Noor MA, Mohyud-Din ST: Variational iteration technique for solving higher order boundary value problems. Appl. Math. Comput. 2007, 189(2):1929-1942. 10.1016/j.amc.2006.12.071

Noor MA, Mohyud-Din ST: Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials. Int. J. Nonlinear Sci. Numer. Simul. 2008, 9(2):141-156.

Yusufoglu E: The variational iteration method for studying the Klein-Gordon equation. Appl. Math. Lett. 2008, 21(7):669-674. 10.1016/j.aml.2007.07.023

Yıldırım A, Öziş T: Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Anal., Theory Methods Appl. 2009, 70(6):2480-2484. 10.1016/j.na.2008.03.012

Wu GC: New trends in variational iteration method. Commun. Fract. Calc. 2011, 2(2):59-75.

Wu GC, Wu KT: Variational approach for fractional diffusion-wave equations on Cantor sets. Chin. Phys. Lett. 2012., 29(6): Article ID 060505

Wu GC: Variational iteration method for q -difference equations of second order. J. Appl. Math. 2012., 2012: Article ID 102850

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(2):2186-2192. 10.1016/j.asoc.2010.07.018

Jafari H, Khalique CM: Homotopy perturbation and variational iteration methods for solving fuzzy differential equations. Commun. Fract. Calc. 2012, 3(1):38-48.

Jafari H, Saeidy M, Baleanu D: The variational iteration method for solving n -th order fuzzy differential equations. Cent. Eur. J. Phys. 2012, 10(1):76-85. 10.2478/s11534-011-0083-7

He JH, Wu XH: Variational iteration method: new development and applications. Comput. Math. Appl. 2007, 54(7-8):881-894. 10.1016/j.camwa.2006.12.083

Shawagfeh NT: Analytical approximate solutions for nonlinear fractional differential equations. Appl. Math. Comput. 2002, 131(2-3):517-529. 10.1016/S0096-3003(01)00167-9

Ray SS, Bera RK: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 2005, 167(1):561-571. 10.1016/j.amc.2004.07.020

Duan JS, Rach R, Baleanu D, Wazwaz AM: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 2012, 3(2):73-99.

Wang Q: Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Solitons Fractals 2008, 35(5):843-850. 10.1016/j.chaos.2006.05.074

Momani S, Odibat Z: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 2007, 365(5-6):345-350. 10.1016/j.physleta.2007.01.046

Kadem A, Baleanu D: Homotopy perturbation method for the coupled fractional Lotka-Volterra equations. Rom. J. Phys. 2011, 56(3-4):332-338.

Tsai PY, Chen CK: An approximate analytic solution of the nonlinear Riccati differential equation. J. Franklin Inst. 2010, 347(10):1850-1862. 10.1016/j.jfranklin.2010.10.005

Zeng DQ, Qin YM: The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives. Commun. Fract. Calc. 2012, 3(1):26-29.

Javidi M, Raji MA: Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations. Commun. Fract. Calc. 2012, 3(1):10-19.

Wazwaz AM: The variational iteration method for analytic treatment for linear and nonlinear ODEs. Appl. Math. Comput. 2009, 212(1):120-134. 10.1016/j.amc.2009.02.003

Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.

Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

Kilbas AA, Srivastav HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, New York; 2006.

Mainardi F: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 1996, 7(9):1461-1477. 10.1016/0960-0779(95)00125-5

Das S: Analytical solution of a fractional diffusion equation by variational iteration method. Comput. Math. Appl. 2009, 57(3):483-487. 10.1016/j.camwa.2008.09.045

Hristov J: Heat-balance integral to fractional (half-time) heat diffusion sub-model. Therm. Sci. 2010, 14(2):291-316. 10.2298/TSCI1002291H

Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.

Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore; 2012.

Wu GC: Challenge in the variational iteration method - a new approach to the identification of the Lagrange multipliers. J. King Saud Univ., Sci. 2012. doi:10.1016/j.jksus.2012.12.002 (in press)

Hristov J: An exercise with the He’s variation iteration method to a fractional Bernoulli equation arising in transient conduction with non-linear heat flux at the boundary. Int. Rev. Chem. Eng. 2012, 4(5):489-497.

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)

Wu GC: Variational iteration method for solving the time-fractional diffusion equations in porous medium. Chin. Phys. B 2012., 21(12): Article ID 120504