Abstract
Thin film arises in various applications from electrochemistry to nano devices, many mathematical tools were adopted to study the problem, e.g. Lie symmetries and conservation laws, however, the variational approach is rare. This paper shows that the semi-inverse method is an effective approach to establishment of a variational formulation for the thin film equation. A detailed derivation process is given, a special skill for construction of a heuristic trial-functional is elucidated.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Variational principle plays a key role in both numerical and analytical analyses of a practical problem, it suggests an energy conservation for the whole solution domain, and a variational-based numerical algorithm guarantees the energy conservation at each point, while a variational-based analytical solution is an optimal one for a given trial-solution and valid for the whole solution domain.
Recently Recio et al. [1] studied the following equation
where f, g and h are functions of u.
Equation (1) can describe a thin film problem, which can be found widely applications in electrochemistry [2], cell culture [3], fiber fabrication [4], nanoscale adhesion [5], coating [6], wetting [7] and micro/nano devices [8]. Many analytical methods and numerical methods were applied to study such problems [9,10,11,12,13,14,15]. This paper aims at establishing a variational formulation for Eq. (1) by the semi-inverse method [16,17,18,19,20,21,22,23,24,25,26,27].
2 The semi-inverse method and the variational formulation
The semi-inverse method [16,17,18,19] is to establish a variational principle directly from governing equations. In order to effectively use the method, we write Eq. (1) in a conservation form
where F, G and H satisfy the following relationships
According to Eq. (2), we can introduce a special function \( \varphi \), satisfying the following relations
The defined special function (\( \varphi \)) is potential-like function. By the semi-inverse method [16,17,18,19], we can establish a trial functional in the form
where \( \sigma \) is an unknown function of u and its derivatives. The advantage of the above trial-functional is the stationary condition with respect to \( \varphi \) is Eq. (2), but we can not identify \( \sigma \), so we modify Eq. (8) in the form
where a and b are constants to be further determined.
The Euler–Lagrange equations of Eq. (9) are
where \( \delta \sigma /\delta u \) is the variational derivative. In this paper it can be written in the form
In view of Eq. (6), we can write Eq. (10) in the form
which should be Eq. (2), so we have
In view of Eqs. (6) and (7), we can write Eq. (11) in the form
or
In order to identify \( \sigma \) in Eq. (16), we set
Equation (16) becomes
From Eq. (18) we can determine \( \sigma \) easily, which satisfies the following relation
We, therefore, obtain the following variational formulation
where \( \sigma \) is defined in Eq. (19).
Remark
if a = 1 as suggested in Eq. (8), we have difficulty in identifying \( \sigma \) from Eq. (16).
3 An example
We consider a special case of Eq. (1)
Hereby \( f(u) = 1 \), \( g(u) = u \),\( h(u) = u^{2} \), \( F(u) = u \),\( G(u) = \frac{1}{2}u^{2} \), \( H(u) = \frac{1}{3}u^{3} \), and
From Eq. (24) \( \sigma \) can be identified as
We, therefore, obtain the following variational principle for Eq. (21):
which is subject to the constraint of Eq. (22).
Proof
The stationary conditions of Eq. (26) with respect to \( \varphi \) and u are
In view of Eq. (22), we find Eqs. (27) and (28) are equivalent to Eq. (21) and Eq. (23), respectively.
4 Discussion and conclusion
A suitable construction of a trial functional is of great importance for the establishment of a variational principle. Equation (8) does not work, because if we set a = 1, Eq. (16) becomes
or
In Eq. (30), we have difficulty in identification of \( \sigma \) due to the terms involving \( u_{xxx} \) and \( u_{x} \).
We can easily determine \( \sigma \) for all even-order derivatives from the following equation
where \( \alpha \), \( \beta \), and \( \delta \) are constants, m is a function of u. From Eq. (31) \( \sigma \) can be determined as
where M is defined as
In this paper, we find that the semi-inverse method provides an effective tool to finding a needed variational principle for a practical problem, the derivation process is explained step by step, so that it can be easily followed. Recently Wang et al. [27] successfully applied the semi-inverse method to fractal calculus [28, 29], and obtained a variational principle for wave traveling in a fractal space.
The variational principle is a foundation of the variational iteration method [30, 31], which is now widely applied in fractional calculus, and the present paper might give a hint for an effective identification of Lagrange multiplier in the fractional variational iteration method [30,31,32,33,34,35,36].
References
E. Recio, T.M. Garrido, R. de la Rosa et al., Conservation laws and Lie symmetries a (2 + 1)-dimensional thin film equation. J. Math. Chem. 57(5), 1243–1251 (2019)
X.-X. Li, D. Tian, C.-H. He, J.-H. He, A fractal modification of the surface coverage model for an electrochemical arsenic sensor. Electrochim. Acta 296, 491–493 (2019)
J. Fan, Y.R. Zhang, Y. Liu et al., Explanation of the cell orientation in a nanofiber membrane by the geometric potential theory. Results Phys. 15, 102537 (2019)
Z.P. Yang, F. Dou, T. Yu et al., On the cross-section of shaped fibers in the dry spinning process: physical explanation by the geometric potential theory. Results Phys. 14, 102347 (2019)
X.X. Li, J.H. He, Nanoscale adhesion and attachment oscillation under the geometric potential. Part 1: the formation mechanism of nanofiber membrane in the electrospinning. Results Phys. 12, 1405–1410 (2019)
A. Saeed, Z. Shah, S. Islam et al., Three-dimensional casson nanofluid thin film flow over an inclined rotating disk with the impact of heat generation/consumption and thermal radiation. Coatings 9(4), 248 (2019)
C.-J. Zhou, D. Tian, J.-H. He, What factors affect lotus effect? Therm. Sci. 22, 1737–1743 (2018)
J.H. He, From micro to nano and from science to technology: nano age makes the impossible possible. Micro Nanosyst. 12(1), 1–2 (2010)
J. Manafian, C.T. Sindi, An optimal homotopy asymptotic method applied to the nonlinear thin film flow problems. Int. J. Numer. Methods Heat Fluid Flow 28(12), 2816–2841 (2018)
N. Faraz, Y. Khan, Thin film flow of an unsteady Maxwell fluid over a shrinking/stretching sheet with variable fluid properties. Int. J. Numer. Methods Heat Fluid Flow 28(7), 1596–1612 (2018)
F. Ghani, T. Gul, S. Islam et al., Unsteady magnetohydrodynamics thin film flow of a third grade fluid over an oscillating inclined belt embedded in a porous medium. Therm. Sci. 21(2), 875–887 (2017)
Q.T. Ain, J.H. He, On two-scale dimension and its applications. Therm. Sci. 23(3B), 1707–1712 (2019)
J.H. He, F.Y. Ji, Two-scale mathematics and fractional calculus for thermodynamics. Therm. Sci. 57(8), 1932–1934 (2019)
J.H. He, F.Y. Ji, Taylor series solution for Lane–Emden equation. J. Math. Chem. (2019). https://doi.org/10.1007/s10910-019-01048-7
J.H. He, The simplest approach to nonlinear oscillators. Results Phys. 15, 102546 (2019)
J.H. He, Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19(4), 847–851 (2004)
J.H. He, J. Zhang, Semi-inverse method for establishment of variational theory for incremental thermoelasticity with voids, in Variational and Extremum Principles in Macroscopic Systems, ed. by S. Sieniutycz, H. Farkas (Elsevier, Amsterdam, 2005), pp. 75–95
J.H. He, A modified Li–He’s variational principle for plasma. Int. J. Numer. Methods Heat Fluid Flow (2019). https://doi.org/10.1108/HFF-06-2019-0523
J.H. He, Lagrange crisis and generalized variational principle for 3D unsteady flow. Int. J. Numer. Methods Heat Fluid Flow (2019). https://doi.org/10.1108/HFF-07-2019-0577
Y. Wu, J.H. He, A remark on Samuelson’s variational principle in economics. Appl. Math. Lett. 84, 143–147 (2018)
J.H. He, Hamilton’s principle for dynamical elasticity. Appl. Math. Lett. 72, 65–69 (2017)
K. Libarir, A. Zerarka, A semi-inversevariational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation. J. Mod. Opt. 65(8), 987–993 (2018)
J. Manafian, P. Bolghar, A. Mohammadalian, Abundant soliton solutions of the resonant nonlinear Schrodinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method. Opt. Quant. Electron. 49(10), 322 (2017)
O.H. El-Kalaawy, New variational principle-exact solutions and conservation laws for modified ion-acoustic shock waves and double layers with electron degenerate in plasma. Phys. Plasmas 24(3), 032308 (2017)
A. Biswas, Q. Zhou, S.P. Moshokoa et al., Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations. Optik 145, 14–17 (2017)
Y. Li, C.H. He, A short remark on Kalaawy’s variational principle for plasma. Int. J. Numer. Methods Heat Fluid Flow 27(10), 2203–2206 (2017)
Y. Wang, J.Y. An, X.Q. Wang, A variational formulation for anisotropic wave traveling in a porous medium. Fractals 27(4), 1950047 (2019)
J.H. He, A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53(11), 3698–3718 (2014)
J.H. He, Fractal calculus and its geometrical explanation. Results Phys. 10, 272–276 (2018)
N. Anjum, J.H. He, Laplace transform: making the variational iteration method easier. Appl. Math. Lett. 92, 134–138 (2019)
J.H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)
D. Baleanu, H.K. Jassim, H. Khan, A modified fractional variational iteration method for solving nonlinear gas dynamic and coupled KdV equations involving local fractional operator. Therm. Sci. 22, S165–S175 (2018)
D. Dogan Durgun, A. Konuralp, Fractional variational iteration method for time-fractional nonlinear functional partial differential equation having proportional delays. Therm. Sci. 22, S33–S46 (2018)
M. Inc, H. Khan, D. Baleanu et al., Modified variational iteration method for straight fins with temperature dependent thermal conductivity. Therm. Sci. 22, S229–S236 (2018)
H. Jafari, H.K. Jassim, J. Vahidi, Reduced differential transform and variational iteration methods for 3-D diffusion model in fractal heat transfer within local fractional operators. Therm. Sci. 22, S301–S307 (2018)
Y. Wang, Y.F. Zhang, Z.J. Liu, An explanation of local fractional variational iteration method and its application to local fractional modified Kortewed-de Vries equation. Therm. Sci. 22, 23–27 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
He, JH., Sun, C. A variational principle for a thin film equation. J Math Chem 57, 2075–2081 (2019). https://doi.org/10.1007/s10910-019-01063-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-019-01063-8