Preface

This tutorial was made solely for the purpose of education. It is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the information for your needs, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

The title of this section is a limit/dream that author wants to achieve. Therefore, please consider this web page as an approximation to the dream value. If you find that some references are missing, please do not hesitate to send this information to the author at <Vladimir_Dobrushkin@brown.edu>

Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340

Glossary

Complete List of Publications for the Vatiational Iteration Method

Monographs and textbooks

  1. Variational Methods in the Mechanics of Solids, Proceedings of the IUTAM Symposium held at Northwestern University, 1978, Pergamon Press, Oxford.

A bibliography of the theory and applications of the VIM

Reference List for VIM, 1978

  1. M. Inokuti, H. Sekine, and T. Mura, "General use of the Lagrange multiplier in non-linear mathematical physics," in: S, Nemat-Nasser (Ed.), Variational Method in the Mechanics of solids, pergamon Press, Oxford. 1978, pp. 156--162.

Reference List for VIM, 1999

  1. Ji-Huan He, "Variational iteration method---a kind of non-linear analytical technique: some examples," Iternational Journal of Non-linear mechanics 1999, Vl. 34, pp. 699--708.
  2. He, J.-H., 1999a. Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parameterized perturbation technique. Communications in Nonlinear Science and Numerical Simulation, 1999, Vol. 4, pp.81–82.

Reference List for VIM, 2000

  1. Ji-Huan He, "Variational iteration method for autonomous ordinary differential systems," Applied mathematics and Computations, 2000, Vol. 114, pp. 115--123.
  2. He, J.-H., 2000. A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Science and Numerical Simulation, 2000, Vol. 1, pp. 51–70.

Reference List for VIM, 2001

  1. J.H. He, Variational theory for linear magneto-electro-elasticity Int. J. Nonlinear Sci. Numer. Simul., 2 (4) (2001), pp. 309-316

Reference List for VIM, 2002

Reference List for VIM, 2003

  1. J.H. He Variational principle for Nano thin film lubrication Int. J. Nonlinear Sci. Numer. Simul., 4 (3) (2003), pp. 313-314

Reference List for VIM, 2004

  1. J.H. He Variational principle for some nonlinear partial differential equations with variable coefficients Chaos, Solitons and Fractals, 19 (4) (2004), pp. 847-851

Reference List for VIM, 2005

Reference List for VIM, 2006

  1. D’Acunto, M., Determination of limit cycles for a modified van der Pol oscillator, Mechanics Research Communications, 2006, Vol. 33, No. 1, pp. 93--98; doi: 10.1016/j.mechrescom.2005.06.012
  2. Ganji, D.D., Rafei, M., 2006. Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Physics Letter A, 356:131–137. [doi:10.1016/j.physleta.2006.03.039]
  3. Ganji, D.D., Rajabi, A., 2006. Assessment of homotopy-perturbation and perturbation methods in heat radiation equations. International Communications in Heat and Mass Transfer, 33:391–400. [doi:10.1016/j.icheatmasstransfer.2005.11.001]
  4. Ganji, D.D. Sadighi, A., 2006. Application of He’s methods to nonlinear coupled systems of reactions. International Journal of Nonlinear Science and Numerical Simulation, 7(4):411–418.
  5. Momani, S. and Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos Solitons Fractals, 27 (5) (2006), pp. 1119--1123

Reference List for VIM, 2007

  1. Ganji, D.D., Nourollahi, M., Rostamian, M., 2007. A comparison of variational iteration method with Adomian’s Decomposition Method in some highly nonlinear equations. International Journal of Science and Technology, 2(2):179–188.
  2. Gorji, M., Ganji, D.D., Soleimani, S., 2007. New application of He’s homotopy perturbation method. International Journal of Nonlinear Science and Numerical Simulation, 8(3):319–328.
  3. Ji-Huan He, Variational iteration method—Some recent results and new interpretations, Journal of Computational and Applied Mathematics, Volume 207, Issue 1, 1 October 2007, Pages 3--17; doi: https://doi.org/10.1016/j.cam.2006.07.009
  4. He, J.H., Variational iteration method—Some recent results and new interpretations J. Comput. Appl. Math., 207 (1) (2007), pp. 3-17
  5. He, J.H., Wu, X.H., Variational iteration method: New development and applications Comput. Math. Appl., 54 (7–8) (2007), pp. 881-894
  6. Noor, M.A., and Mohyud-Din, S.T., Variational Iteration Decomposition Method for Solving Eighth-Order Boundary Value Problems, Hindawi Publishing Corporation, Differential Equations and Nonlinear Mechanics, 2007, Volume 2007, Article ID 19529, 16 pages; doi:10.1155/2007/19529
  7. H. Ozer, Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics Int. J. Nonlinear Sci. Numer. Simul., 8 (4) (2007), pp. 513-518
  8. Rafei, M., Ganji, D.D., Daniali, H., Pashaei, H., 2007. The variational iteration method for nonlinear oscillators with discontinuities. Journal of Sound and Vibration, 2007, 305:614–620. [doi:10.1016/j.jsv.2007.04.020]
  9. Varedi, S.M., Hosseini, M.J., Rahimi, M., Ganji, D.D., 2007. He’s variational iteration method for solving a semilinear inverse parabolic Equation, Physics Letters A, 370:275–280. [doi:10.1016/j.physleta.2007.05.100]
  10. Abdul-Majid Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Computers & Mathematics with Applications, 2007, Volume 54, Issues 7–8, pp. 895--902; doi: https://doi.org/10.1016/j.camwa.2006.12.059
  11. Wazwaz, A.M., The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations Comput. Math. Appl., 54 (7–8) (2007), pp. 933-939

Reference List for VIM, 2008

  1. Bormashenko, E., Whyman, G., 2008. Variational approach to wetting problems: calculation of a shape of sessile liquid drop deposited on a solid substrate in external field. Chemical Physics Letters, 463(1–3):103–105. [doi:10.1016/j.cplett.2008.08.049]
  2. Jafari, H., Yazdani, A., Vahidi, J., Ganji, D.D., Application of He's variational method for solving seventh order Sawada-Kotera equations, Applied Mathematical Sciences, 2008, Vol. 2, No. 10, pp. 471--477.
  3. Odibat, Z., Reliable approaches of variational iteration method for nonlinear operators Math. Comput. Model., 48 (1–2) (2008), pp. 222-231
  4. Saberi-Nadjafi, J., Tamamgar, M., The variational iteration method: A highly promising method for solving the system of integro-differential equations, Computers & Mathematics with Applications, 2008, Volume 56, Issue 2, July 2008, Pages 346--351. https://doi.org/10.1016/j.camwa.2007.12.014

Reference List for VIM, 2009

  1. Belal Mohammed Batiha, Application of Variational Iteration Method to Linear Partial Differential Equations, Applied Mathematical Sciences, Vol. 3, 2009, no. 50, 2491 - 2498; doi:
  2. Ganji, D.D., Gorji, M., Soleimani, S., Esmaeilpour, M., Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method, Journal of Zhejiang University-SCIENCE A, 2009, Volume 10, Issue 9, pp 1263–1268; https://doi.org/10.1631/jzus.A082065
  3. Hemeda, A.A., Variational iteration method for solving non-linearpartial differential equations, Chaos, Solitons and Fractals, 2009, 39, pp. 1297--1303.
  4. Zaid Odibata and Shaher Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications, Volume 58, Issues 11–12, December 2009, Pages 2199--2208; doi: https://doi.org/10.1016/j.camwa.2009.03.009
  5. Rashidi, M.M., Shahmohamadi, H., Analytical solution of three-dimensional Navier–Stokes equations forthe flow near an infinite rotating disk, Communications in Nonlinear Science and Numerical Simulation, 2009, Vol. 14, pp. 2999–3006.
  6. Soliman, A.A., On the solution of two-dimensional coupled Burgers’ equations by variational iteration method, Chaos, Solitons and Fractals, 2009, 40, pp. 1146–1155.
  7. Soltanian, F., Karbassi, S.M., Hosseini, M.M., Application of He’s variational iteration method for solution of differential-algebraic equations, Chaos, Solitons and Fractals, Vol. 41, 2009, pp. 436--445.
  8. Yildirim, A. and Ozis, T., Solutions of singular IVPs of Lane--Emden type by the variational iteration method, Nonlinear. Anal 70 (2009) 2480.

Reference List for VIM, 2010

  1. J. Biazar, M. Shahbala, and H. Ebrahimi, VIM for Solving the Pollution Problem of a System of Lakes, Journal of Control Science and Engineering, Volume 2010, Article ID 829152, 6 pages http://dx.doi.org/10.1155/2010/829152
  2. Jawad, A.J.M., Petković, M.D., Biswas, A., Soliton solutions of a few nonlinear wave equations, Applied Mathematics and Computation, 2010, 216, pp. 2649–2658.

Reference List for VIM, 2011

  1. Al-Saif, A.S.J., and Hattim, T.A.K., Variational Iteration Method for Solving Some Models of Nonlinear Partial Differential Equations, Int. J. Pure Appl. Sci. Technol. , 4(1) (2011), pp. 30-40
  2. Jafari, H., Chun, Ch., and Khalique, C.M., The Variational Iteration Method for Finding Exact Solution of Nonlinear Gas Dynamics Equations,
  3. Olayiwola , M.O., Akinpelu, F.O., Gbolagade, A.W., Modified Variational Iteration Method for the Solution of nonlinear Partial Differential Equation, International Journal of Scientific & Engineering Research, 2011, Volume 2, Issue 10, pp. 1--10.

Reference List for VIM, 2012

  2. Mehmet Giyas Sakara, Fevzi Erdogana, and Ahmet Yıldırım, "Variational iteration method for the time-fractional Fornberg–Whitham equation," Computers & Mathematics with Applications, 2012, Volume 63, Issue 9, pp. 1382--1388; doi: https://doi.org/10.1016/j.camwa.2012.01.031
  3. Raslan, K.R., Soliman, A.A., Ali, A.H., A comparison between the variational iteration method and Adomian decomposition method for the FitzHugh-Nagumo equations, International Journal of Physical Sciences, 2012, Vol. 7(15), pp. 2302 - 2309, doi: 10.5897/IJPS12.174

Reference List for VIM, 2013

  1. Yasir Khan, F. Naeem, and Zdeněk Šmarda, A Novel Iterative Scheme and Its Application to Differential Equations, Scientific World Journal, 2014; doi: 10.1155/2014/605376
  2. Asif Mehmood, Farah Jabeen Awan, and Syed Tauseef Mohyud-Din, Comparison of Lagrange Multipliers for Nonlinear BVPs, International Journal of Modern Mathematical Sciences, 2013, Vol. 5, No. 3, pp. 156--165; available on the web

Reference List for VIM, 2014

  2. E. Abdolmaleki and S.A. Yousefi. Application of VIM method for nonlinear porous media equations. American Journal of Numerical Analysis, 2014; 2(1):11-13. doi: 10.12691/ajna-2-1-3.
  3. Ameina S. Nuseir and Mohammad Al-Towaiq, The modified variational Iteration method for solving the impenetrable AGAR model problem, International Journal of Pure and Applied Mathematics Volume 96 No. 4 2014, 445-456; doi: http://dx.doi.org/10.12732/ijpam.v96i4.3

Reference List for VIM, 2015

  1. Emad A. Az-Zo`bi , 2015. On the Convergence of Variational Iteration Method for Solving Systems of Conservation Laws. Trends in Applied Sciences, doi: 10.3923/tasr.2015.157.165 URL: https://scialert.net/abstract/?doi=tasr.2015.157.165
  2. A.R. Gómez Plata and Edmundo Capelas de Oliveira, New Lagrange multipliers for the time fractional Burgers' equation,

Reference List for VIM, 2016

Reference List for VIM, 2017

Reference List for VIM, 2018

  1. Chamekh, M. and Elzaki, T.M., Explicit solution for some generalized fluids in laminar flowwith slip boundary conditions, Journal of Mathematics and Computer Science, 2018, Vol. 18. pp. 272--281.

Reference List for VIM, 2019