THERMAL SCIENCE

International Scientific Journal

Shuxian DengORCID

,

Xinxin Ge

THE VARIATIONAL ITERATION METHOD FOR WHITHAM-BROER-KAUP SYSTEM WITH LOCAL FRACTIONAL DERIVATIVES

ABSTRACT
The Whitham-Broer-Kaup equations are modified using local fractional derivatives, and the equations are then solved by the variational iteration method. Yang-Laplace transform method is adopted to make the solution process simpler.
KEYWORDS
Whitham-Broer-Kaup system, fractal media, Yang-Laplace transform, variational iteration method
PAPER SUBMITTED: 2020-10-05
PAPER REVISED: 2021-10-01
PAPER ACCEPTED: 2021-10-01
PUBLISHED ONLINE: 2022-07-16
DOI REFERENCE: https://doi.org/10.2298/TSCI2203419D
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE No. 3, PAGES [2419 - 2426]
REFERENCES
[1] Deng, S. X., Ge, X. X., Fractional Fokker-Planck Equation in a Fractal Medium, Thermal Science, 24 (2020), 4, pp. 2589-2595, 10.2298/tsci2004589d
[2] He, C. H., et al., Low Frequency Property of a Fractal Vibration Model for a Concrete Beam, Fractals, 29 (2021), 5, 150117, 10.1142/s0218348x21501176
[3] Liu, F. J., et al., Thermal Oscillation Arising in a Heat Shock of a Porous Hierarchy and Its Application, Facta Universitatis Series: Mechanical Engineering, On-line first, 210317054, 2021, 10.22190/fume210317054l
[4] Li, X. X., He, J. H., Along the Evolution Process: Kleiber's 3/4 Law Makes Way for Rubner's Surface Law: A Fractal Approach, Fractals, 27 (2019), 2, 1950015, 10.1142/s0218348x19500154
[5] Tian, D., et al., Hall-Petch Effect and Inverse Hall-Petch Effect: A Fractal Unification, Fractals, 26 (2018), 6, 1850083, 10.1142/s0218348x18500834
[6] He, J. H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), May, 104104, 10.1016/j.rinp.2021.104104
[7] Li, X. J., Wang, D., Effects of a Cavity's Fractal Boundary on the Free Front Interface of the Polymer Filling Stage, Fractals, 29 (2021), 7, 2150225-784, 10.1142/s0218348x2150225x
[8] Yang, X. J., Local Fractional Integral Transforms, Progress Non-linear Science, 4 (2011), pp. 1-25
[9] He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199, 10.1142/s0218348x21501991
[10] Han, C., et al., Numerical Solutions of Space Fractional Variable-Coefficient KdV-Modified KdV Equa-tion by Fourier Spectral Method, Fractals, 29 (2021), 8, 21502467, 10.1142/s0218348x21502467
[11] Wang, K. J., On New Abundant Exact Traveling Wave Solutions to the Local Fractional Gardner Equa-tion Defined on Cantor Sets, Mathematical Methods in the Applied Sciences, 45 (2022), 4, pp. 1904-1915, 10.1002/mma.7897
[12] Yan, Z., Zhang, H., New Explicit Solitary Wave Solutions and Periodic Wave Solutions for Whitham-Broer-Kaup Equation in Shallow Water, Physics Letters A , 285 (2001), 5-6, pp. 355-362, 10.1016/s0375-9601(01)00376-0
[13] Rafei, M., Daniali, H., Application of the Variational Iteration Method to the Whitham-Broer-Kaup Equations, Computers and Mathematics with Applications, 54(2007), 7-8, pp. 1079-1085, 10.1016/j.camwa.2006.12.054
[14] Wang, M., Li, X., Simplified Homogeneous Balance Method and Its Applications to the Whitham-Broer-Kaup Model Equations, Journal of Applied Mathematics & Physics, 8 (2014), 2, pp. 823-827, 10.4236/jamp.2014.28091
[15] Wang, L., Chen, X., Approximate Analytical Solutions of Time Fractional Whitham-Broer-Kaup Equa-tions by a Residual Power Series Method, Entropy, 9 (2015), 17, pp. 6519-6533, 10.3390/e17096519
[16] He, C. H., et al., Hybrid Rayleigh-Van Der Pol-Duffing Oscillator: Stability Analysis and Controller, Journal of Low Frequency Noise Vibration and Active Control, 41 (2021), 1, pp. 244-268, 10.1177/14613484211026407
[17] Tian, D., et al., Fractal N/MEMS: From Pull-In Instability to Pull-In Stability, Fractals, 29 (2021), 2, 2150030, 10.1142/s0218348x21500304
[18] Tian, D., He, C. H., A Fractal Micro-Electromechanical System and Its Pull-In Stability, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, 1380-1386, 10.1177/1461348420984041
[19] 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, 10.1177/1461348419847298
[20] Wu, Y., Liu, Y. P., Residual Calculation in He's Frequency-Amplitude Formulation, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 2, pp. 1040-1047
[21] Feng, G. Q., He's Frequency Formula to Fractal Undamped Duffing Equation, Journal of Low Frequen-cy Noise Vibration and Active Control, 40 (2021), 4, pp. 1671-1676, 10.1177/1461348421992608
[22] Li, X. X., He, C. H., Homotopy Perturbation Method Coupled with the Enhanced Perturbation Method, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1399-1403, 10.1177/1461348418800554
[23] Anjum, N., et al., Li-He's Modified Homotopy Perturbation Method for Doubly-Clamped Electrically Actuated Microbeams-Based Microelectromechanical System, Facta Universitatis: Mechanical Engi-neering, 19 (2021), 4, pp. 601-612, 10.22190/fume210112025a
[24] Ji, Q. P., et al., Li-He's Modified Homotopy Perturbation Method Coupled with the Energy Method for the Dropping Shock Response of a Tangent Non-Linear Packaging System, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 2, pp. 675-682, 10.1177/1461348420914457
[25] 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, 10.1142/s0217979221502143
[26] He, J. H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathemat-ics Letters, 119 (2021), Sep., 107199, 10.1016/j.aml.2021.107199
[27] Wang, K. J., On New Abundant Exact Traveling Wave Solutions to the Local Fractional Gardner Equa-tion Defined on Cantor Sets, Mathematical Methods in the Applied Sciences, 45 (2021), 4, pp. 1904-1915
[28] 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, 10.1016/j.physleta.2021.127723
[29] Wang, K. J., Zhang, P. L., Investigation of the Periodic Solution of the Time-Space Fractional Sasa-Satsuma Equation Arising in the Monomode Optical Fibers, EPL, On-line first, [doi.org/10.1209/]( 02955075/ac2a62, 2021, 10.1209/0295-5075/ac5c78
[30] Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
[31] He, J. H., Variational Iteration Method - Some Recent Results and New Interpretations, Journal of Com-putational and Applied Mathematics, 207 (2007), 1, pp. 3-7, 10.1016/j.cam.2006.07.009
[32] He, J. H., Wu, X. H., Variational Iteration Method: New Development and Applications, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 881-894, 10.1016/j.camwa.2006.12.083
[33] Nadeem, M., He, J. H., He-Laplace Variational Iteration Method for Solving the Non-Linear Equations Arising in Chemical Kinetics and Population Dynamics, Journal of Mathematical Chemistry, 59(2021), 5, pp. 1234-1245, 10.1007/s10910-021-01236-4
[34] Li, F. Q., Nadeem, M., He-Laplace Method for Non-Linear Vibration in Shallow Water Waves, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4 , pp. 1305-1313, 10.1177/1461348418817869
[35] Anjum, N., He, J. H., Analysis of Non-Linear Vibration of Nano/Microelectromechanical System Switch Induced by Electromagnetic Force Under Zero Initial Conditions, Alexandria Engineering Journal, 59 (2020), 6, pp. 4343-4352, 10.1016/j.aej.2020.07.039
[36] Ling, W. W., Wu, P. X., Variational Principle of the Whitham-Broer-Kaup Equation in Shallow Water Wave with Fractal Derivatives, Thermal Science, 25 (2021) 2B, pp. 1249-1254, 10.2298/tsci180510087l
[37] Wang, K. L., et al., Physical Insight of Local Fractional Calculus and Its Application to Fractional KdV-Burgers-Kuramoto Equation, Fractals, 27 (2019), 7, 1950122, 10.1142/s0218348x19501226
[38] Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134, 10.1142/s0218348x19501342
[39] Tian, Y., Wan, J. X., Exact Solutions of Space-Time Fractional 2+1 Dimensional Breaking Soliton Equation, Thermal Science, 25 (2021), 2, pp. 1229-1235, 10.2298/tsci200421016t
[40] Tian, Y., Liu, J., A Modified Exp-Function Method for Fractional Partial Differential Equations, Ther-mal Science, 25 (2021), 2, pp. 1237-1241, 10.2298/tsci200428017t
[41] Tian, Y., Liu, J., Direct Algebraic Method for Solving Fractional Fokas Equation, Thermal Science, 25 (2021), 3, pp. 2235-2244, 10.2298/tsci200306111t
[42] Anjum, N., et al., Two-scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 21501826, 10.1142/s0218348x21501826
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The variational iteration method for Whitham-Broer-Kaup system with local fractional derivatives

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence