THERMAL SCIENCE

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Chu-Han Shang

,

Huai-An Yi

SOLUTIONS OF THE KDV-MKDV EQUATIONS ARISING IN NON-LINEAR ELASTIC RODS UNDER FRACTAL DIMENSION

ABSTRACT
A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately.
KEYWORDS
non-linear elastic rod, KdV-MKdV equation, He's fractal derivatives, He-Weierstrass function, soliton
PAPER SUBMITTED: 2022-11-09
PAPER REVISED: 2023-05-25
PAPER ACCEPTED: 2023-05-28
PUBLISHED ONLINE: 2024-05-18
DOI REFERENCE: https://doi.org/10.2298/TSCI2403125S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE Issue 3, PAGES [2125 - 2133]
REFERENCES
  1. Palanthandalam-Madapusi, H. J., Goyal, S., Robust Estimation of Non-linear Constitutive Law from Static Equilibrium Data for Modeling the Mechanics of DNA, Automatica, 47 (2011), 6, pp. 1175-1182
  2. Hearst, J. E., Shi, Y., New Solutions for the Stationary States of the Elastic Rod Model are Useful in the Representation of DNA Configuration in the Living Cell, Non-linear Science Today, 9 (1996), Aug., pp. 1-38
  3. Zhang, W., et al., Explicit Exact Solitary-Wave Solutions for Compound KdV-Type and Compound KdV-Burgers-Type Equations with Non-linear Terms of Any Order, Chaos Solitons & Fractals, 13 (2002), 2, pp. 311-319
  4. Ngo, D., et al., Highly Non-linear Solitary Wave Propagation in Y-Shaped Granular Crystals with Variable Branch Angles, Physical Review E, 85 (2012), 3-2, 036602
  5. Zhou, Y. Q., et al., The Complex Linear Solitary Wave Solutions of a Kind of Non-linear Wave Equations, Journal of Sichuan Normal University (Natural Science), 3 (2005), 1, pp. 73-75
  6. Zhuang, W., Yang, G. T., The Propagation of Solitary Waves in a Non-linear Elastic Rod, Applied Mathematics and Mechanics (English Edition), 7 (1986), July, pp. 615-626
  7. Han, Q., Zheng, X. F., Solitary Wave Solutions in Non-linear Elastic Rods and their Numerical Analysis (in Chinese), Journal of South China University of Technology (Natural Science Edition), 32 (2004), 4, pp. 92-96
  8. Hu, W. P., et al., Numerical Simulation on the Longitudinal Wave in Non-linear Elastic Rod (in Chinese), Chinese Journal of Computational Mechanics, 27 (2010), 1, pp. 8-13
  9. Guo, P., et al., Perturbation Analysis for Higher Order Equation of Longitudinal Oscillation of a Non-linear Elastic Rod (in Chinese), Journal of Northwest Normal University (Natural Science), 40 (2004), 1, pp. 38-41
  10. Kabir, M. M., Exact Traveling Wave Solutions for Non-linear Elastic Rod Equation, Journal of King Saud University: Science, 31 (2019), 3, pp. 390-397
  11. Celik, N., et al., A Model of Solitary Waves in a Non-linear Elastic Circular Rod: Abundant Different Type Exact Solutions and Conservation Laws, Chaos Solitons and Fractals, 143 (2021), 110486
  12. Guo, P., et al., Explicit Exact Solution of the Non-linear Elastic Bar Wave Equation, Applied Mathematics and Mechanics, 43 (2022), 8, pp. 869-876
  13. Ji, F. Y., et al., A fractal Boussinesq Equation for Non-linear Transverse Vibration of a Nanofiber-Reinforced Concrete Pillar, Applied Mathematical Modelling, 82 (2020), June, pp. 437-448
  14. Ma, H. J., Simplified Hamiltonian-based Frequency-Amplitude Formulation for Non-linear Vibration Systems, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 2, pp. 445-455
  15. He, J.-H., et al., Forced Non-linear Oscillator in a Fractal Space, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 1, pp. 1-20
  16. He, J.-H., et al.，2023， Pull-down Instability of the Quadratic Non-linear Oscillators, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 2, pp. 191-200
  17. He, C. H., et al., Hybrid Rayleigh -Van der Pol-Duffing Oscillator (HRVD): Stability Analysis and Controller, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 1, pp. 244-268
  18. He, C. H., et al., Controlling the Kinematics of a Spring-Pendulum System Using an Energy Harvesting Device, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 3, pp. 1234-1257
  19. He, C. H., El-Dib, Y. O., A Heuristic Review on the Homotopy Perturbation Method for Non-conservative Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 41 (2022), 2, pp. 572-603
  20. He, J.-H., et al., Homotopy Perturbation Method for Strongly Non-linear Oscillators, Mathematics and Computers in Simulation, 204 (2023), Feb., pp. 243-258
  21. He, J.-H., et al., A Good Initial Guess for Approximating Non-linear Oscillators by the Homotopy Perturbation Method, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 1, pp. 21-29
  22. He, J.-H., et al., Stability of Three Degrees-of-Freedom Auto-Parametric System, Alexandria Engineering Journal, 61 (2022), 11, pp. 8393-8415
  23. Lv, G. J., Dynamic Behaviors for the Graphene Nano/Microelectromechanical System in a Fractal Space, Journal of Low Frequency Noise, Vibration & Active Control, 42 (2023), 3,
  24. Kou, P. H., et al., Novel Fractional-Order Convolutional Neural Network Based Chatter Diagnosis Approach in Turning Process with Chaos Error Mapping, Non-linear Dynamics, 111 (2023), 8, pp. 7547-7564
  25. Kuo, P. H., et al., A Thermal Displacement Prediction System with an Automatic LRGTVAC-PSO Optimized Branch Structured Bidirectional GRU Neural Network, IEEE Sensors Journal, 23 (2023), 12, pp. 12574-12586
  26. He, J.-H., et al., Pull-in Stability of a Fractal System and Its Pull-in Plateau, Fractals, 30 (2022), 9, 2250185
  27. Tian, D., et al., Fractal N/MEMS: from Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2, 2150030
  28. 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
  29. He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, International Journal of Numerical Methods for Heat & Fluid Flow, 33 (2023), 1, pp. 351-359
  30. He, C. H., et al., Low Frequency Property of a Fractal Vibration Model for a Concrete Beam, Fractals, 29 (2021), 5, 2150117
  31. He, C. H., Liu, C., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046
  32. He, J.-H., et al., Homotopy Perturbation Method for Fractal Duffing Oscillator with Arbitrary Conditions, Fractals, 30 (2022), 9, 22501651
  33. He, J.-H., et al., Fractal Oscillation and Its Frequency-Amplitude Property, Fractals, 29 (2021), 4, 2150105
  34. 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
  35. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
  36. He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), 104104
  37. He, J.-H., On the Fractal Variational Principle for the Telegraph Equation, Fractals, 29 (2021), 1, 2150022
  38. Wu, P. X., et al., Solitary Waves of the Variant Boussinesq-Burgers Equation in a Fractal-Dimensional Space, Fractal, 30 (2022), 3, 2250056
  39. He, C. H., Liu, C., Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete's Strength, Facta Universitatis Series: Mechanical Engineering, 21 (2023), 1, pp. 137-150
  40. He, C. H., et al., A Fractal Model for the Internal Temperature Response of a Porous Concrete, Applied and Computational Mathematics, 21 (2022), 1, pp. 71-77
  41. He J.-H., et al., A Tutorial Introduction to the Two-Scale Fractal Calculus and Its Application to the Fractal Zhiber-Shabat Oscillator, Fractals, 29 (2021), 8, 2150268
  42. He, J.-H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
  43. Qian, M. Y., He, J.-H., Two-Scale Thermal Science for Modern Life - Making the Impossible Possible, Thermal Science, 26 (2022), 3B, pp. 2409-2412
  44. He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), May, 104104
  45. Sun, J. S., Variational Principle and Solitary Wave of the Fractal Fourth-Order Non-linear Ablowitz-Kaup-Newell-Segur Water Wave Model, Fractals, 31 (2023), 5, 2350036
  46. Sun, J. S., Approximate Analytic Solution of the Fractal Klein-Gordon Equation, Thermal Science, 25 (2021), 2, pp. 1489-1494
  47. He, J.-H., Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics with Emphasis on Turbomachinery Aerodynamics, International Journal of Turbo & Jet-Engines, 14 (1997), 1, pp. 23-28
  48. He, J.-H., Variational Approach to Impulsive Differential Equations Using the Semi-Inverse Method, Zeitschrift fur Naturforschung Section A, 66 (2011), 10-11, pp. 632-634
  49. 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
  50. He, C. H., Liu, C., Variational Principle for Singular Waves, Chaos, Solitons & Fractals, 172 (2023), 113566
  51. Wang, S. Q., A Variational Approach to Non-linear Two-Point Boundary Value Problems, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2452-2455
  52. Shen, Y. Y., et al., Subcarrier-Pairing-Based Resource Optimization for OFDM Wireless Powered Relay Transmissions with Time Switching Scheme, IEEE Transactions on Signal Processing, 65 (2016), 5, pp. 1130-1145
  53. He, J.-H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Appl. Math. Lett., 119 (2021), 107199
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Solutions of the KdV-MKdV equations arising in non-linear elastic rods under fractal dimension

© 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