1 Introduction
The variational iteration method [1] is considered the most powerful tool after Newton’s iteration method for solving a wide range of physical problems [2–6]. It has been employed in solving seepage flows with fractional derivatives and nonlinear oscillators, making it a widely used primary mathematical tool for various nonlinear equations. Given its significance, many scholars, including J.H. He [7], S. Momani [8], and Z.M. Odibat [9], have extensively researched this method. A key advantage of VIM over other analytical methods is that it does not require linearization or manipulation of nonlinear terms. By using a suitable initial guess and incorporating a Lagrange multiplier, one can obtain exact or highly accurate solutions for various physical problems. However, the identification of the multiplier can be difficult without a solid understanding of the intricate theory of variational calculus [10–12], which can be challenging for some practitioners. In recent years, the integral transform has been extensively used in numerical simulation due to its rapid convergence and ease of use. It has significant practical implications in addressing various real-world engineering challenges, including electrical, industrial, mechanical, and civil engineering. In many instances, the choice of an appropriate integration transform can simplify the analysis. The choice of transformation becomes very important when we investigate different problems. This short opinion proposes a more accessible and comprehensive method for easily and effectively identifying the multiplier: the introduction of a generalized integral transform. This transform generalizes Fourier series, Laplace transforms, and other transformations, such as the Sumudu transform [13] and the Aboodh transform [14, 15]. This approach is highly appealing and promising and it does not require specialized knowledge of variational calculus. Furthermore, the procedure can be used in all mathematics textbooks.
2 The determination of the lagrange multiplier by the He-transform
Considering a general nonlinear oscillator equation in the following form:with initial conditions
We represent Eq. 1 aswhere is unknown frequency,
According to the variational iteration method (VIM) [1], the correction functional which is essentially a convolution for Eq. 3 can be expressed aswhere is a Lagrange multiplier, and its value can be selectively determined by stationary conditions of Eq. 4 with respect to using variation theory [10–12]. is the th approximate solution and is a restricted variation, i.e., .
Next, we present an alternative method for determining the multiplier. Based on the seminal contributions of Abassy [16], Mokhtari [17], and Hesameddini [18], the Laplace transform was initially incorporated into the variational iteration method [19]. It is worth considering whether or not there is a more representative integral transform than the Laplace transform in the context of VIM. J.H. He proposed in 2023 [20] a new generalized integral transform, which not only includes various integral transforms falling under the category of the Laplace transform, but also retains the properties of the Fourier transform as a special case, such as existence and linearity. This new transform offers a new perspective for the identification of Lagrange multipliers with extreme ease [21–23]. In the following, we will use this new generalized integral transform to identify the Lagrange multiplier.
He’s integral transform [20] of an integrable function has the following definition
Here is the image of , is the integral transformation operator, and denotes the transformation variable. The superscript is from the integer range.
The Lagrange multiplier can usually be expressed as [1].
The correction function given in Eq. 4 is essentially the convolution, so we can easily use the He-transform. By substituting Eq. 5 into Eq. 4 and applying the He-transform to both sides of the resulting equation, we obtain the final transformation of the correction function by employing the linearity theorem and the differentiation theorem [20], as follows:
The optimal value of can be determined by taking Eq. 6 to be stationary with respect to , assuming that. ,
Eq. 7 leads to the following result
Applying the inverse He-transform to Eq. 8 yields the following result
This is the same as that in Ref. [24], showing that the He-transform works more easily and more effectively.
3 Conclusion remark
In this opinion, we elucidate that the He-transform facilitates the identification of the Lagrange multiplier, making the variational iteration method more promising for solving physical problems. We hope that this short opinion can attract a wide audience from various fields, such as mathematics, physics, mechanics, and engineering. As current studies primarily concentrate on solving nonlinear oscillators with an initial value of zero, we will apply the method to solve nonlinear oscillators with generalized initial values in the future.
Statements
Author contributions
Q-RS: Writing–original draft, Writing–review and editing. J-GZ: Funding acquisition, Investigation, Project administration, Writing–review and editing.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The paper was supported by the national natural science foundation of China (61863022) and the key project of Gansu natural science foundation (23JRRA882).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
HeJH. Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech (1999) 34:699–708. 10.1016/S0020-7462(98)00048-1
2.
TomarSSinghMVajraveluKRamosH. Simplifying the variational iteration method: a new approach to obtain the Lagrange multiplier. Math Comput Simul (2023) 204:640–4. 10.1016/j.matcom.2022.09.003
3.
KhuriSAssadiR. An extended variational iteration method for fractional BVPs encountered in engineering applications. Int J Numer Methods (2023) 33:2671–81. 10.1108/HFF-02-2023-0073
4.
AnjumNHeJHHeCHGepreelKA. Variational iteration method for prediction of the pull-in instability condition of micro/nanoelectromechanical systems. Phys Mesomech (2023) 26:5–14. 10.55652/1683-805X_2023_26_1_5
5.
TianYFengGQ. A short review on approximate analytical methods for nonlinear problems. Therm Sci (2022) 26:2607–18. 10.2298/TSCI2203607T
6.
HeJH. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons and Fractals (2004) 19:847–51. 10.1016/s0960-0779(03)00265-0
7.
HeJH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B (2006) 20:1141–99. 10.1142/S0217979206033796
8.
MomaniSAbuasadS. Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fractals (2006) 27:1119–23. 10.1016/j.chaos.2005.04.113
9.
OdibatZMMomaniS. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonl Sci Num Simulation (2006) 7:27–34. 10.1515/IJNSNS.2006.7.1.27
10.
ZuoYT. Variational principle for a fractal lubrication problem. Fractals (2024). 10.1142/S0218348X24500804
11.
HeJH. Generalized equilibrium equations for shell derived from a generalized variational principle. Appl Math Lett (2017) 64:94–100. 10.1016/j.aml.2016.08.008
12.
HeCH. A variational principle for a fractal nano/microelectromechanical (N/MEMS) system. Int J Numer Methods (2022) 33:351–9. 10.1108/HFF-03-2022-0191
13.
KılıçmanAGadainHE. On the applications of Laplace and Sumudu transforms. J Franklin Inst (2010) 347:848–62. 10.1016/j.jfranklin.2010.03.008
14.
ManimegalaiKZephaniaCFSBeraPKBeraPDasSKSilT. Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method. The Eur Phys J Plus (2019) 134:1–10. 10.1140/epjp/i2019-12824-6
15.
TaoHAnjumNYangYJ. The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus. Front Phys (2023) 11:1168795. 10.3389/fphy.2023.1168795
16.
AbassyTAEl-TawilMAEi-ZoheiryH. Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with laplace transforms and the pade technique. Comput Math Appl (2007) 54:940–54. 10.1013/j.camwa.2006.12.067
17.
MokhtariRMohammadiM. Some remarks on the variational iteration method. Int J Nonlinear Sci Numer Simul (2009) 10:67–74. 10.1515/IJNSNS.2009.10.1.67
18.
HesameddiniELatifizadehH. Reconstruction of variational iteration algorithms using the laplace transform. Int J Nonlinear Sci Numer Simul (2009) 10:1377–82. 10.1515/IJNSNS.2009.10.11-12.1377
19.
RehmanSHussainARahmanJUNaceedAMunirT. Modified Laplace based variational iteration method for the mechanical vibrations and its applications. acta mechanica et automatica (2022) 16:98–102. 10.2478/ama-2022-0012
20.
HeJ-HAnjumNHeC-HAlsolamiAA. Beyond Laplace and Fourier transforms challenges and future prospects. Therm Sci (2023) 27:5075–89. 10.2298/TSCI230804224H
21.
AnjumNHeJH. A dual Lagrange multiplier approach for the dynamics of the mechanical systems. J Appl Comput Mech (2024). 10.22055/jacm.2024.45944.4439
22.
TangWAnjumNHeJH. Variational iteration method for the nanobeams-based N/MEMS system. MethodsX (2023) 11:102465. 10.1016/j.mex.2023.102465
23.
AnjumNSulemanMLuDHeJHRamzanM. Numerical iteration for nonlinear oscillators by Elzaki transform. J Low Frequency Nois, Vibration Active Control (2020) 39:879–84. 10.1177/1461348419873470
24.
AnjumNHeJH. Laplace transform: making the variational iteration method easier. Appl Math Lett (2019) 92:134–8. 10.1016/j.aml.2019.01.016
Summary
Keywords
He-transform, nonlinear oscillator, variational iteration method, variational principle, lagrange multiplier
Citation
Song Q-R and Zhang J-G (2024) He-transform: breakthrough advancement for the variational iteration method. Front. Phys. 12:1411691. doi: 10.3389/fphy.2024.1411691
Received
03 April 2024
Accepted
29 April 2024
Published
20 June 2024
Volume
12 - 2024
Edited by
Chun-Hui He, Xi’an University of Architecture and Technology, China
Reviewed by
Dan Tian, Xi’an University of Architecture and Technology, China
Naveed Anjum, Government College University, Pakistan
Updates
Copyright
© 2024 Song and Zhang.
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*Correspondence: Jian-Gang Zhang, Zhangjg7715776@126.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.