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**Variational iteration method: New development and applications.**
*(English)*
Zbl 1141.65372

Summary: The variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. The variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions.

This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

34A34 | Nonlinear ordinary differential equations and systems |

45J05 | Integro-ordinary differential equations |

65R20 | Numerical methods for integral equations |

45G10 | Other nonlinear integral equations |

### Keywords:

variational iteration method; convergence; nonlinear equation; fractional differential equation; wave equation; nonlinear oscillator; survey paper; convergence; numerical examples
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\textit{J.-H. He} and \textit{X.-H. Wu}, Comput. Math. Appl. 54, No. 7--8, 881--894 (2007; Zbl 1141.65372)

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### References:

[1] | He, J. H., Variational iteration method — a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34, 4, 699-708 (1999) · Zbl 1342.34005 |

[2] | He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167, 1-2, 57-68 (1998) · Zbl 0942.76077 |

[3] | He, J. H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering, 167, 1-2, 69-73 (1998) · Zbl 0932.65143 |

[4] | He, J. H., Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 118, 2-3, 115-123 (2000) · Zbl 1027.34009 |

[5] | He, J. H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 10, 1141-1199 (2006) · Zbl 1102.34039 |

[6] | He, J. H., New Interpretation of Homotopy perturbation method, International Journal of Modern Physics B, 20, 18, 2561-2568 (2006) |

[7] | He, J. H.; Wu, X. H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals, 29, 1, 108-113 (2006) · Zbl 1147.35338 |

[8] | He, J. H., Non-perturbative methods for strongly nonlinear problems (2006), dissertation.de-Verlag im Internet GmbH: dissertation.de-Verlag im Internet GmbH Berlin |

[9] | He, J. H., Variational iteration method — some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207, 1, 3-17 (2007) · Zbl 1119.65049 |

[10] | Abbasbandy, S., An approximation solution of a nonlinear equation with Riemann-Liouville’s fractional derivatives by He’s variational iteration method, Journal of Computational and Applied Mathematics, 207, 1, 53-58 (2007) · Zbl 1120.65133 |

[11] | Draganescu, G. E.; Cofan, N.; Rujan, D. L., Nonlinear vibration of a nano sized sensor with fractional damping, Journal of Optoelectronics and Advanced Materials, 7, 2, 877-884 (2005) |

[12] | Momani, S.; Odibat, Z., Numerical simulation of systems of differential equations of fractional order, Journal of Computational and Applied Mathematics, 207, 1, 96-110 (2007) · Zbl 1119.65127 |

[13] | Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Physics Letters A, 355, 4-5, 271-279 (2006) · Zbl 1378.76084 |

[14] | Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons and Fractals, 31, 5, 1248-1255 (2007) · Zbl 1137.65450 |

[15] | Odibat, Z. M.; Momani, S., Application of variational iteration method to Nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 27-34 (2006) · Zbl 1401.65087 |

[16] | Abbasbandy, S., A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Journal of Computational and Applied Mathematics, 207, 1, 59-63 (2007) · Zbl 1120.65083 |

[17] | Abdou, M. A.; Soliman, A. A., New applications of variational iteration method, Physica D, 211, 1-2, 1-8 (2005) · Zbl 1084.35539 |

[18] | Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burger’s and coupled Burger’s equations, Journal of Computational and Applied Mathematics, 181, 2, 245-251 (2005) · Zbl 1072.65127 |

[19] | Ganji, D. D.; Jannatabadi, M.; Mohseni, E., Application of He’s variational iteration method to nonlinear Jaulent-Miodek equations and comparing it with ADM, Journal of Computational and Applied Mathematics, 207, 1, 35-45 (2007) · Zbl 1120.65107 |

[20] | Ganji, D. D.; Sadighi, A., Application of He’s Homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 4, 411-418 (2006) |

[21] | Inc, M., Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method, Chaos, Solitons and Fractals, 34, 4, 1075-1081 (2007) · Zbl 1142.35572 |

[22] | Moghimi, M.; Hejazi, F. S.A., Variational iteration method for solving generalized Burger-Fisher and Burger equations, Chaos, Solitons and Fractals, 33, 5, 1756-1761 (2007) · Zbl 1138.35398 |

[23] | Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos, Solitons and Fractals, 27, 5, 1119-1123 (2006) · Zbl 1086.65113 |

[24] | Soliman, A. A., A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations, Chaos, Solitons and Fractals, 29, 2, 294-302 (2006) · Zbl 1099.35521 |

[25] | Soliman, A. A., Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method, Mathematics and Computers in Simulation, 70, 2, 119-124 (2005) · Zbl 1152.65467 |

[26] | Soliman, A. A.; Abdou, M. A., Numerical solutions of nonlinear evolution equations using variational iteration method, Journal of Computational and Applied Mathematics, 207, 1, 111-120 (2007) · Zbl 1120.65111 |

[27] | Tian, L. X.; Yin, J. L., Shock-peakon and shock-compacton solutions for \(K(p, q)\) equation by variational iteration method, Journal of Computational and Applied Mathematics, 207, 1, 46-52 (2007) · Zbl 1119.65099 |

[28] | Wazwaz, A. M., The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations, Journal of Computational and Applied Mathematics, 207, 1, 18-23 (2007) · Zbl 1119.65102 |

[29] | Wazwaz, A. M., Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations, Chaos, Solitons and Fractals, 28, 4, 1005-1013 (2006) · Zbl 1099.35125 |

[30] | Wazwaz, A. M., Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos, Solitons and Fractals, 28, 2, 454-462 (2006) · Zbl 1084.35079 |

[31] | Ye, J. F.; Zheng, C. L.; Xie, L. S., Exact solutions and localized excitations of general Nizhnik-Novikov-Veselov system in (2+1)-dimensions via a projective approach, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 2, 203-208 (2006) |

[32] | Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A., The solution of nonlinear coagulation problem with mass loss, Chaos, Solitons and Fractals, 29, 2, 313-330 (2006) · Zbl 1101.82018 |

[33] | Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A., Nonlinear fluid flows in pipe-like domain problem using variational-iteration method, Chaos, Solitons and Fractals, 32, 4, 1384-1397 (2007) · Zbl 1128.76019 |

[34] | Ariel, P. D.; Hayat, T.; Asghar, S., Homotopy perturbation method and axisymmetric flow over a stretching sheet, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 4, 399-406 (2006) |

[35] | D’Acunto, M., Determination of limit cycles for a modified van der Pol oscillator, Mechanics Research Communications, 33, 1, 93-98 (2006) · Zbl 1192.70026 |

[36] | D’Acunto, M., Self-excited systems: Analytical determination of limit cycles, Chaos, Solitons and Fractals, 30, 3, 719-724 (2006) · Zbl 1142.70010 |

[37] | Xu, L., He’s Parameter-expanding methods for strongly nonlinear oscillators, Journal of Computational and Applied Mathematics, 207, 1, 148-154 (2007) · Zbl 1120.65084 |

[38] | Draganescu, G. E.; Capalnasan, V., Nonlinear relaxation phenomena in polycrystalline solids, International Journal of Nonlinear Sciences and Numerical Simulation, 4, 3, 219-225 (2003) |

[39] | Ganji, D. D.; Sadighi, A., Application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics, 207, 1, 24-34 (2007) · Zbl 1120.65108 |

[40] | Marinca, V., An approximate solution for one-dimensional weakly nonlinear oscillations, International Journal of Nonlinear Sciences and Numerical Simulation, 3, 2, 107-120 (2002) · Zbl 1079.34028 |

[41] | Lu, J. F., Variational iteration method for solving two-point boundary value problems, Journal of Computational and Applied Mathematics, 207, 1, 92-95 (2007) · Zbl 1119.65068 |

[42] | Sweilam, N. H., Harmonic wave generation in non-linear thermoelasticity by variational iteration method and Adomian’s method, Journal of Computational and Applied Mathematics, 207, 1, 64-72 (2007) · Zbl 1115.74028 |

[43] | Sweilam, N. H., Variational iteration method for solving cubic nonlinear Schrödinger equation, Journal of Computational and Applied Mathematics, 207, 1, 155-163 (2007) · Zbl 1119.65098 |

[44] | Sweilam, N. H.; Khader, M. M., Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos, Solitons and Fractals, 32, 1, 145-149 (2007) · Zbl 1131.74018 |

[45] | Tatari, M.; Dehghan, M., He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, Solitons and Fractals, 33, 2, 671-677 (2007) · Zbl 1131.65084 |

[46] | Liu, H. M., Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos, Solitons and Fractals, 23, 2, 573-576 (2005) · Zbl 1135.76597 |

[47] | Liu, H. M., Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method, Chaos, Solitons and Fractals, 23, 2, 577-579 (2005) · Zbl 1078.34509 |

[48] | Siddiqui, A. M.; Ahmed, M.; Ghori, Q. K., Couette and Poiseuille flows for non-Newtonian fluids, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 15-26 (2006) · Zbl 1401.76018 |

[49] | Tatari, M.; Dehghan, M., On the convergence of He’s variational iteration method, Journal of Computational and Applied Mathematics, 207, 1, 121-128 (2007) · Zbl 1120.65112 |

[50] | Wazwaz, A. M., A comparison between the variational iteration method and Adomian decomposition method, Journal of Computational and Applied Mathematics, 207, 1, 129-136 (2007) · Zbl 1119.65103 |

[51] | Abassy, T. A.; El-Tawil, M. A.; El Zoheiry, H., Solving nonlinear partial differential equations using the modified variational iteration-Padé technique, Journal of Computational and Applied Mathematics, 207, 1, 73-91 (2007) · Zbl 1119.65095 |

[52] | Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 65-70 (2006) · Zbl 1401.35010 |

[53] | Abassy, T. A.; El-Tawil, M. A.; El Zoheiry, H., Toward a modified variational iteration method, Journal of Computational and Applied Mathematics, 207, 1, 137-147 (2007) · Zbl 1119.65096 |

[54] | Hashim, I., Adomian decomposition method for solving BVPs for fourth-order-differential equations, Journal of Computational and Applied Mathematics, 193, 658-664 (2006) · Zbl 1093.65122 |

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