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