## THERMAL SCIENCE

International Scientific Journal

### VARIATIONAL PRINCIPLES FOR TWO KINDS OF NON-LINEAR GEOPHYSICAL KDV EQUATION WITH FRACTAL DERIVATIVES

**ABSTRACT**

It is an important and difficult inverse problem to construct variational principles from complex models directly, because their variational formulations are theoretical bases for many methods to solve or analyze the non-linear problems. At first, this paper extends two kinds of non-linear geophysical KdV equations in continuum mechanics to their fractional partners in fractal porous media or with irregular boundaries. Then, by designing skillfully, the trial-Lagrange functional, variational principles are successfully established for the non-linear geophysical KdV equation with Coriolis term, and the high-order extended KdV equation with fractal derivatives, respectively. Furthermore, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations.

**KEYWORDS**

PAPER SUBMITTED: 2020-10-04

PAPER REVISED: 2021-10-01

PAPER ACCEPTED: 2021-10-01

PUBLISHED ONLINE: 2022-07-16

**THERMAL SCIENCE** YEAR

**2022**, VOLUME

**26**, ISSUE

**Issue 3**, PAGES [2505 - 2515]

- Ablowitz, M. J., Clarkson, P. A., Solitons, Non-linear Evolution Equations and Inverse Scatting, Cambridge University Press, Cambridge, UK, 1991
- Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Higher education press, Beijing, China, 2009
- Wazwaz, A. M., Linear and Non-linear Integral Equations: Methods and Applications, Higher educa-tion press, Beijing, China, 2011
- Wang, K. L., et al., Physical Insight of Local Fractional Calculus and its Application to Fractional Kdv-Burgers Equation, Fractals, 27 (2019), 7, 1950122
- He, J.‐H., et al., Homotopy Perturbation Method for the Fractal Toda Oscillator. Fractal Fract., 5 (2021), Sept., 93
- Wu, Y., Variational Approach to Higher-Order Water-Wave Equations, Chaos Solitons Fractals, 32 (2007), 1, pp. 195-203
- Gazzola, F., et al., Variational Formulation of the Melan Equation, Math. Methods Appl. Sci., 41 (2018), 3, pp. 943-951
- Liu, Y. P., et al., A Fractal Langmuir Kinetic Equation and Its Solution Structure, Thermal Science, 25 (2021), 2, pp. 1351-1354
- Liu, X. Y., et al., Computer Simulation of Pantograph Delay Differential Equation, Thermal Science, 25 (2021), 2, pp. 1381-1385
- Baleanu, D., A Modified Fractional Variational Iteration Method for Solving Non-Linear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator, Thermal Science, 22 (2018), Suppl. 1, pp. S165-S175
- Durgun, D. D., Fractional Variational Iteration Method for Time-Fractional Non-Linear Functional Par-tial Differential Equation Having Proportional Delays, Thermal Science, 22 (2018), Suppl. 1, pp. S33-S46
- Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Itera-tion Method, Thermal Science, 17 (2013), 2, pp. 625-628
- He, J. H., Maximal Thermo-Geometric Parameter in a Non-linear Heat Conduction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), 2, pp. 605-608
- Malomed, B. A., Variational Methods in Non-Linear fiber Optics and Related fields, Prog. Opt., 43 (2002), pp. 71-193
- Chong, C., Pelinovsky, D. E., Variational Approximations of Bifurcations of Asymmetric Solitons in Cubic-Quintic Non-Linear Schrödinger Lattices, Discret. Contin. Dyn. Syst., 4 (2011), 5, pp. 1019-1031
- Kaup, D. J., Variational Solutions for the Discrete Non-Linear Schrödinger Equation, Math. Comput. Simul., 69 (2005), 3-4, pp. 322-333
- Putri, N. Z., et al., Variational Approximations for Intersite Soliton in a Cubic-Quintic Discrete Non-Linear Schrödinger Equation, J. Phys. Conf. Ser., 1317 (2019), 012015
- Yao, S. W., Variational Principle for Non-Linear Fractional Wave Equation in a Fractal Space, Thermal Science, 25 (2021), 2, pp. 1243-1247
- Tian, Y., Wang, K. L., Conservation Laws for Partial Differential Equations Based on the Polynomial Characteristic Method, Thermal Science, 24 (2020), 4, pp. 2529-2534
- He, J. H., et al., Dynamic Pull-In for Micro-Electromechanical Device with a Current-Carrying Conduc-tor, Journal of Low Frequency Noise Vibration and Active Control., 40 (2021), 2, pp.1059-1066
- He, J. H., Variational Principles for Some Non-Linear Partial Differential Equations with Variable Coef-ficients, Chaos Solitons Fractals, 19 (2004), 4, pp. 847-851
- He, J. H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathemat-ics Letters, 119 (2021), Sept., 107199
- Li, Y., He, C. H., A Short Remark on Kalaawy's Variational Principle for Plasma, International Journal of Numerical Methods for Heat & Fluid Flow, 27 (2017), 10, pp. 2203-2206
- He, J. H., et al., A Fractal Modification of Chen-Lee-Liu Equation and its Fractal Variational Principle, International Journal of Modern Physics B, 35 (2021), 21, 2150214
- Tian, Y., Wang, K. L., Conservation Laws for Partial Differential Equations Based on the Polynomial Characteristic Method, Thermal Science, 24 (2020), 4, pp. 2529-2534
- Khan, Y., Fractal Higher-order Dispersions Model and Its Fractal Variational Principle Arising in the Field of Physcial Process, Fluctuation and Noise Letters, 20 (2021), 4, 2150034
- He, J. H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), May, 104104
- He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
- Wang, K. J., Generalized Variational Principle and Periodic Wave Solution to the Modified Equal Width-Burgers Equation in Non-Linear Dispersion Media, Physics Letters A, 419 (2021), Dec., 127723
- Cao, X. Q., et al., Variational Theory for (2+1)-Dimensional Fractional Dispersive Long Wave Equa-tions, Thermal Science, 25 (2021), 2, pp. 1277-1285
- Cao, X. Q., Variational Principles for Two Kinds of Extended Korteweg-de Vries Equations, Chin. Phys. B, 20 (2011), 9, pp. 94-102
- Cao, X. Q., Generalized Variational Principles for Boussinesq Equation Systems, Acta Phys. Sin., 60 (2011), 8, pp. 105-113
- Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950132
- Wang, K. L., Variational Principle for Non-Linear Oscillator Arising in a Fractal Nano/Microelectromechanical System, Mathematical Methods in the Applied Sciences, On-line first, doi.org/10.1002/mma.6726, 2020
- El-Kalaawy, O. H., Variational Principle, Conservation Laws and Exact Solutions for Dust Ion Acoustic Shock Waves Modeling Modified Burger Equation, Comput. Math. Appl., 72 (2016), 4, pp. 1013-1041
- Anjum, N., et al., Two-Scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 21501826-744
- He, J. H., Seeing with a Single Scale is Always Unbelieving: From Magic to Two-Scale Fractal, Ther-mal Science, 25 (2021), 2, pp. 1217-1219
- He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
- He, J. H. A Tutorial Review on Fractal Space-time and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Liu, F. J., et al., He's Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silk-worm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
- Wang, K. L., Liu, S. Y., He's Fractional Derivative for Non-linear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
- Karunakar, P., Chakraverty, S., Effect of Coriolis Constant on Geophysical Korteweg-de Vries Equation, Journal of Ocean Engineering and Science, 4 (2019), 2, pp. 113-121
- Kaya, D., Explicit and Numerical Solutions of Some Fifth-order KdV Equation by Decomposition Method, Appl. Math. Comput., 144 (2003), 2-3, pp. 353-363
- Kaya, D., EI-Sayed, S. M., On a Generalized Fifth-Order KdV Equations, Phys. Lett. A, 310 (2003), 1, pp. 44-51
- Zhang, Y., Chen, D. Y., The Novel Multi Solitary Wave Solution to the Fifth-Order KdV Equation, Chin. Phys. B, 10 (2004), 10, pp. 1606-1610
- Wazwaz, A. M., A Study on Compacton-Like Solutions for the Modified KdV and Fifth Order KdV-Like Equations, Appl. Math. Comput., 147 (2004), 2, pp. 439-447
- Wazwaz, A. M., Helal, M. A., Variants of the Generalized Fifth-Order KdV Equation with Compact and Noncompact Structures, Chaos Solitons and Fract., 21 (2004), 3, pp. 579-589
- Li, J., et al., Simulation Investigation on the Internal Wave via the Analytical Solution of Korteweg-de Vries Equation (in Chinese), Marine Science Bulletin, 30 (2011), 1, p. 23-28
- Jiang, Z. H., et al., Ocean Internal Waves Interpreted as Oscillation Travelling Waves in Consideration of Ocean Dissipation, Chin. Phys. B, 23 (2014), 5, 050302
- Wang, Z., Zhu Y. K., Theory, Modelling and Computation of Non-Linear Ocean Internal Waves, Chi-nese Journal of Theoretical and Applied Mechanics, 51 (2019), 6, pp. 1589-1604
- Lee, C. Y, Beardsley R. C., The Generation of Long Non-Linear Internal Waves in a Weakly Stratified Shear flow, Journal of Geophysical Research, 79 (1974), 3, pp. 453-462
- Helfrich, K. R., Melville, W. K., Long Non-Linear Internal Waves, Annual Review of Fluid Mechanics, 38 (2006), Jan., pp. 395-425