THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Bin Chen

,

Jun-Feng Lu

TWO ANALYTICAL METHODS FOR TIME FRACTIONAL CAUDREY-DODD-GIBBON-SAWADA-KOTERA EQUATION

ABSTRACT
This paper focuses on solving the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation (FCDGSKE). We propose two analytical methods based on the fractional complex transform, the variational iteration method and the homotopy perturbation method. The approximated solutions to the initial value problems associated with FCDGSKE are provided without linearization and complicated calculation. Numerical results show the main merits of the analytical approaches.
KEYWORDS
time FCDGSKE, fractional complex transform, homotopy perturbation method, variational iteration method
PAPER SUBMITTED: 2020-10-04
PAPER REVISED: 2021-10-01
PAPER ACCEPTED: 2021-10-01
PUBLISHED ONLINE: 2022-07-16
DOI REFERENCE: https://doi.org/10.2298/TSCI2203535C
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 3, PAGES [2535 - 2543]
REFERENCES
  1. Baleanu, D., et al., Lie Symmetry Analysis, Exact Solutions and Conservation Laws for the Time Frac-tional Caudrey-Dodd-Gibbon-Sawada-Kotera Equation, Communications in Non-linear Science and Numerical Simulation, 59 (2018), June, pp. 222-234
  2. He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  3. He, J. H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  4. Aiyer, R. N., et al., Solitons and Discrete Eigenfunctions of the Recursion Operator of Non-Linear Evo-lution Equations, I. The Caudrey-Dodd-Gibbon-Sawada-Kotera Equation, Journal of Physics A, 19 (1986), 18, pp. 3755-3770
  5. Jiang B. O., Bi. Q. S., A Study on the Bilinear Caudrey-Dodd-Gibbon Equation, Non-linear Analysis-Theory Methods& Applications, 72 (2010), 12, pp. 4530-4533
  6. Yang, L., Trial Equation Method for Exact Travelling Solutions of Fifth Order Caudrey-Dodd-Gibbon Equation, Scholars Journal of Engineering and Technology, 2 (2014), 3, pp. 472-476
  7. Parker, A., A Reformulation of the Dressing Method for the Sawada-Kotera Equation, Inverse Problems, 17 (2001), 4, pp. 885-895
  8. Satsuma, J., Kaup, D. J., A Bäcklund Transformation for a Higher Order Korteweg-de Vries Equation, Journal of the Physical Society of Japan, 43 (1977), 2, pp. 692-697
  9. Salas, A. H., et al., Computing Multi-Soliton Solutions to Caudrey-Dodd-Gibbon Equation by Hirota's Method, International Journal of Physics Sciences, 34 (2011), 6, pp. 7729-7737
  10. Xu, Y. G., et al., Solving the Fififth Order Caudrey-Dodd-Gibbon (CDG) Equation Using the Exp-function Method, Applied Mathematics and Computation, 206 (2008), 1, pp. 70-73
  11. Gu, Y. Y., Analytical Solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation via Symbol Cal-culation Approach, Journal of Function Spaces, 2020 (2020), ID 5042724
  12. Geng, X., et al., Riemann Theta Function Solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera Hier-archy, Journal of Geometry and Physics, 140 (2019), June, pp. 85-103
  13. 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
  14. He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
  15. 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
  16. Dan, D. D., et al., Using Piecewise Reproducing Kernel Method and Legendre Polynomial for Solving a Class of the Time Variable Fractional Order Advection-Reaction-Diffusion Equation, Thermal Science, 25 (2021), 2B, pp. 1261-1268
  17. Manafian, J., Lakestani, M., N-Lump and Interaction Solutions of Localized Waves to the (2+1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation, Journal of Geome-try and Physics, 150 (2020), Feb., 103598
  18. Li, Z. B., et al., Exact Solutions of Time-Fractional Heat Conduction Equation by the Fractional Com-plex Transform, Thermal Science, 16 (2012), 2, pp. 335-338
  19. He, J. H., Li, Z. B., Converting Fractional Differential Equations Into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  20. Li, Z. B., He, J. H., Fractional Complex Transform for Fractional Differential Equations, Mathematical and Computational Applications, 15 (2010), 5, pp. 970-973
  21. Tian, Y., Liu, J., Direct Algebraic Method for Solving Fractional Fokas Equation, Thermal Science, 25 (2021), 3, pp. 2235-2244
  22. 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
  23. Habib, S., et al., Study of Non-Linear Hirota-Satsuma Coupled KdV and Coupled mKdV System with Time Fractional Derivative, Fractals, 29 (2021), 5, 2150108
  24. He, J. H., et al., Approximate Periodic Solutions to Microelectromechanical System Oscillator Subject to Magnetostatic Excitation, Mathematical Methods in the Applied Sciences, On-line first, doi.org/10.1002/mma.7018
  25. Lu, J. F., Variational Iteration Method for Solving Two-Point Boundary Value Problems, Journal of Computational and Applied Mathematics, 207 (2007), 1, pp. 92-95
  26. 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
  27. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
  28. 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
  29. 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
  30. He, J.‐H., et al., Homotopy Perturbation Method for the Fractal Toda Oscillator, Fractal Fract., 5 (2021), 93, 5030093
  31. He, J. H., Homotopy Perturbation Method for Bifurcation of Non-Linear Problems, International Jour-nal of Non-Linear Sciences and Numerical Simulation, 6 (2005), 2, pp. 207-208
  32. 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
  33. Liu, X. Y., et al., Fractal Calculus for Modeling Electrochemical Capacitors Under Dynamical Cycling, Thermal Science, 25 (2021), 2, pp. 1317-1320
  34. He, J.-H., El-Dib, Y. O., Homotopy Perturbation Method for Fangzhu Oscillator, Journal of Mathemati-cal Chemistry, 58 (2020), 10, pp. 2245-2253
  35. Lu, J. F., An Analytical Approach to Fractional Bousinesq-Burges Equations, Thermal Science, 24 (2020), 4, pp. 2581-2588
  36. 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
  37. 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
  38. Anjum, N., et al., Two-Scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 21501826
  39. 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
  40. Tian, D., et al., Fractal N/MEMS: From Pull-In Instability to Pull-In Stability, Fractals, 29 (2021), 2, 2150030
  41. 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, pp. 1380-1386
  42. 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
  43. He, J. H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathemat-ics Letters, 119 (2021), Sept., 107199
  44. He, C. H., et al, Low Frequency Property of a Fractal Vibration Model for a Concrete Beam, Fractals, 29 (2021), 5, 150117
PDF VERSION [DOWNLOAD]
Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

© 2024 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