Laplace Transform: Developing the Variational Iteration Method

Authors

  • M. JHANSI LAKSHMI S&H MATHEMATICS, St Martin's Engineering College, DhulapallyPin: 500100 Author

DOI:

https://doi.org/10.61841/8mvjbj31

Abstract

The classification of the Lagrange multiplier plays an import rule in the variational method, and the variational theory is widely used for this purpose. This paper suggests an easier approach by the Laplace transform to determining the multiplier, making the process accessible to researchers facing different nonlinear problems. A nonlinear oscillator is adopted as an illustration to elucidate the detection process and the solution process; only one iteration leads to an ultimate result. 

Downloads

Download data is not yet available.

References

1. J.H. He, Variational iteration method—a kind of non-linear analytical technique:

some examples, Int. J. Nonlinear Mech. 34 (1999) 699–708.

2. J.H. He, Approximate analytical solution for seepage flow with fractional derivatives

in porous media, Comput. MethodsAppl. Mech. Eng. 167 (1998) 57–68.

3. J.H. He, Variational iteration method—some recent results and new interpretations, J.

Comput. Appl. Math. 207 (2007) 3–17.

4. J.H. He, X.H. Wu, Variational iteration method: new development and applications,

Comput. Math. Appl. 54 (2007) 881–894.

5. J.H. He, Some asymptotic methods for strongly nonlinear equations, Internat. J.

Modern Phys. 10 (2006) 1141–1199.

6. D.D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational

iteration methods to nonlinear heat transfer and porous media equations, J. Comput.

Appl. Math. 207 (2007) 24–34.

7. T. Ozis, A. Yildirim, Traveling wave solution of the Korteweg-De Vries equation using

he’s homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 8 (2007)

239–242.

8. M.A. Noor, S.T. Mohyud-Din, Variational iteration method for solving higher-order

nonlinear boundary value problemsusing he’s polynomials, Int. J. Nonlinear Sci.

Numer. Simul. 9 (2008) 141–156.

9. J.H. He, Generalized equilibrium equations for shell derived from a generalized

variational principle, Appl. Math. Lett. 64(2017) 94–100.

10. J.H. He, An alternative approach to establishment of a variational principle for the

torsional problem of piezo elasticbeams, Appl. Math. Lett. 52 (2016) 1–3.

11. Y. Wu, J.H. He, A remark on samuelson’s variational principle in economics, Appl.

Math. Lett. 84 (2018) 143–147.

12. T.A. Abassy, M.A. El-Tawil, and H. EI-Zoheiry, Exact solutions of some nonlinear partial

differential equations using thevariational iteration method linked with laplace

transforms and the pade technique, Comput. Math. Appl. 54 (2007) 940–954.

13. R. Mokhtari, M. Mohammadi, Some remarks on the variational iteration method, Int.

J. Nonlinear Sci. Numer. Simul. 10(2009) 67–74.

14. E. Hesameddini, H. Latifizadeh, Reconstruction of variational iteration algorithms

using the laplace transform, Int. J.Nonlinear Sci. Numer. Simul. 10 (2009) 1377–

1382.

15. A. Prakash, M. Kumar, Numerical method for solving time-fractional multidimensional diffusion equations, Int. J. Comp. Sci. Math. 8 (2017) 257–267.

16. A. Prakash, M. Kumar, Numerical solution of two-dimensional time fractional order

biological population model, OpenPhys. 14 (2016) 177–186.

17. A. Prakash, M. Goyal, S. Gupta, Fractional variational iteration method for solving

time-fractional newell-whitehead-segelequation, Nonlinear Eng.-Modell. Appl.

(2018) http://dx.doi.org/10.1515/nleng-2018-0001.

18. A. Prakash, M. Kumar, Numerical method for time-fractional gas dynamic equations,

Proc. Natl. Acad. Sci. India. A (2018), http://dx.doi.org/10.1007/s40010-018-0496-4.

138 N. Anjum and J.-H. He / Applied Mathematics Letters 92 (2019) 134–138

19. A. Prakash, M. Kumar, K.K. Sharma, Numerical method for solving coupled burgers

equation, Appl. Math. Comput. 260(2015) 314–320.

20. J.H. He, A short remark on fractional variational iteration method, Phys. Lett. A. 375

(2011) 3362–3364.[21] D. Baleanu, H.K. Jassim, H. Khan, A modification fractional

variational iteration method for solving non-linear gasdynamic and coupled kdv

equations involving local fractional operators, Therm. Sci. 22 (2018) S165–S175.

21. D.D. Durgun, A. Konuralp, Fractional variational iteration method for time-fractional

non-linear functional partialdifferential equation having proportional delays, Therm.

Sci. 22 (2018) S33–S46.

22. J.H. He, Fractal calculus and its geometrical explanation, Res. Phy. 10 (2018) 272–

276.

23. X.X. Li, D. Tian, C.H. He, J.H. He, A fractal modification of the surface coverage

model for an electrochemical arsenicsensor, Electrochim. Acta. 296 (2019) 491–493.

24. Q.L. Wang, X.Y. Shi, J.H. He, Z.B. Li, Fractal calculus and its application to

explanation of biomechanism of polar bearhairs, Fractals. 26 (2018) 1850086.

25. Y. Wang, Q. Deng, Fractal derivative model for tsunami travelling, Fractals (2019)

http://dx.doi.org/10.1142/S0218348X19500178.

26. Y.Wang, J. Ye An, Amplitude-frequency relationship to a fractional Duffing

oscillator arising in microphysics and tsunamimotion, J. Low Freq. Noise V. A.

https://doi.org/10.1177/1461348418795813.

27. Y. Wu, J.H. He, Homotopy perturbation method for nonlinear oscillators with

coordinate dependent mass, Results Phys. 10 (2018) 270–271.

28. B.I. Lev, V.B. Tymchyshyn, A.G. Zagorodny, On certain properties of nonlinear

oscillator with coordinate-dependent mass, Phys. Lett. A. 381 (2017) 3417–3423.

29. Z.F. Ren, G.F. Hu, He’s frequency-amplitude formulation with average residuals for

nonlinear oscillators, J. Low Freq.

30. Noise V.A. http://dx.doi.org/10.1177/1461348418812327.

Downloads

Published

30.06.2021

How to Cite

Laplace Transform: Developing the Variational Iteration Method. (2021). International Journal of Psychosocial Rehabilitation, 25(3), 758-763. https://doi.org/10.61841/8mvjbj31