THERMAL SCIENCE
International Scientific Journal
Find this paper on
AN ANALYTICAL APPROACH TO FRACTIONAL BOUSINESQ-BURGES EQUATIONS
ABSTRACT
This paper proposes an analytical approach to fractional calculus by the fractional complex transform and the modified variational iteration method. The fractional Bousinesq-Burges equations are used as an example to reveal the main merits of the present technology.
KEYWORDS
fractional Bousinesq-Burges equation, fractional complex transform, modified variational iteration method
PAPER SUBMITTED: 2019-04-27
PAPER REVISED: 2019-11-01
PAPER ACCEPTED: 2019-11-01
PUBLISHED ONLINE: 2020-06-21
DOI REFERENCE: https://doi.org/10.2298/TSCI2004581L
CITATION EXPORT: view in browser or download as text file
REFERENCES
[1] Kumar, S., et al., Two Analytical Methods for Time-fractional Nonlinear Coupled Boussinesq-Burger's Equations Arise in Propagation of Shallow Water Waves, Nonlinear Dynamics, 85 (2016), 2, pp. 699-715, 10.1007/s11071-016-2716-2
[2] He, J. H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147, 10.2298/tsci11s1145h
[3] He, J. H., et al., A New Fractional Derivative and its Application to Explanation of Polar Bear Hairs, Journal of King Saud University-Science, 28 (2016), 2, pp. 190-192, 10.1016/j.jksus.2015.03.004
[4] He J. H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276, 10.1016/j.rinp.2018.06.011
[5] He, J. H., A Simple Approach to One-Dimensional Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), 113565, 10.1016/j.jelechem.2019.113565
[6] Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 6, 1850086, 10.1142/s0218348x1850086x
[7] Wang, Y., Deng, Q., Fractal Derivative Model for Tsunami Travelling, Fractals, 27 (2019), 1, 1950017, 10.1142/s0218348x19500178
[8] He, J. H., A Tutorial Review on Fractal Space Time and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-718, 10.1007/s10773-014-2123-8
[9] Geng, X. G., Wu, Y. T., Finite-Band Solutions of the Classical Boussinesq-Burgers Equations, Journal of Mathematical Physics, 40 (1999), 6, pp. 2971-2982, 10.1063/1.532739
[10] Zhang, J., et al., Quasi-Periodic Solution of the (2+1)-Dimensional Boussinesq-Burgers Soliton Equation, Physica A: Statistical Mechanics and its Applications, 319 (2003), Mar., pp. 213-232, 10.1016/s0378-4371(02)01526-1
[11] Yu, D. N., et al., Homotopy Perturbation Method with an Auxiliary Parameter for Nonlinear Oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 38 (2019), 3-4, pp. 1540-1554, 10.1177/1461348418811028
[12] Wu,Y., He, J.H., Homotopy Perturbation Method for Nonlinear Oscillators with Coordinate-Dependent Mass, Results in Physics, 10 (2018), Sept., pp. 270-271, 10.1016/j.rinp.2018.06.015
[13] He, J. H., Homotopy Perturbation Method with Two Expanding Parameters, Indian Journal of Physics, 88 (2014), 2, pp. 193-196, 10.1007/s12648-013-0378-1
[14] He, J. H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012 (2012), ID 857612, 10.1155/2012/857612
[15] He, J. H., Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Me-dia, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 1-2, pp. 57-68, 10.1016/s0045-7825(98)00108-x
[16] He, J. H., Variational Iteration Method-a Kind of Non-linear Analytical Technique: Some Examples, International Journal of Non-Linear Mechanics, 34 (1999), 4, pp. 699-708, 10.1016/s0020-7462(98)00048-1
[17] Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138, 10.1016/j.aml.2019.01.016
[18] Lu, J. F., Ma, L., The VIM-Padé Technique for Strongly Nonlinear Oscillators with Cubic and Harmonic Restoring Force, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 1272-1278, 10.1177/1461348418813612
[19] Ren, Z. F., et al., He's Multiple Scales Method for Nonlinear Vibrations, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 1708-1712, 10.1177/1461348419861450
[20] He, J. H., The Simpler, the Better: Analytical Methods for Nonlinear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 1252-1260, 10.1177/1461348419844145
[21] Li, F. Q., Nadeem, M., He-Laplace Method for Nonlinear Vibration in Shallow Water Waves, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 1305-1313, 10.1177/1461348418817869
[22] He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334, 10.2298/tsci110503068h
[23] He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133, 10.2298/tsci1904131h
[24] Li, Z. B., et al., Exact Solutions of Time-Fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16 (2012), 2, pp. 335-338, 10.2298/tsci110503069l
[25] He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
[26] Ain, Q. T., He, J. H., On Two-Scale Dimension and its Applications, Thermal Science 23 (2019), 3B, pp. 1701-1712, 10.2298/tsci190408138a
[27] Abassy, T. A., et al., Toward a Modified Variational Iteration Method, Journal of Computational and Applied Mathematics, 207 (2007), 1, pp. 137-147, 10.1016/j.cam.2006.07.019
[28] Lu, J. F., Numerical Analysis of the (2+1)-Dimensional Boiti-Leon-Pempinelli Equation, Thermal Science, 21 (2017), 4, pp. 1657-1663, 10.2298/tsci160715050l
[29] Ren, Z. F., et al., He's Multiple Scales Method for Nonlinear Vibrations, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1708-1712
[30] He, J. H., et al., A New Fractional Derivative And Its Application To Explanation Of Polar Bear Hairs, Journal of King Saud University - Science, 28 (2016), 2, pp.190-192
[31] Wang, Y., et al., A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium, Fractals, 27 (2019), 4, ID 1950047, 10.1142/s0218348x19500476
[32] Wang, K. L., He, C. H. A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, ID 1950134, 10.1142/s0218348x19501342
[33] Wang, Y. H., CTE Method to the Interaction Solutions of Boussinesq-Burgers Equations, Applied Mathematics Letters, 38 (2014), Dec., pp. 100-105, 10.1016/j.aml.2014.07.014
[34] He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934, 10.1007/s10910-019-01048-7
PDF VERSION [DOWNLOAD]
© 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


