THERMAL SCIENCE

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Caixia Liu

PERIODIC SOLUTION OF FRACTAL PHI-4 EQUATION

ABSTRACT
This paper focuses on a fractal Phi-4 equation with time-space fractal derivatives, though its solitary solutions have been deeply studied, its periodic solution was rarely revealed due to its strong non-linearity. Now the condition is completely changed, He’s frequency formulation provides with a universal tool to having a deep insight into the periodic property of the fractal Phi-4 equation. The two-scale transform is used to convert approximately the fractal Phi-4 equation a differential model, and a criterion is suggested for the existence of a periodic solution of the equation, the effect of fractal orders on the periodic property is also elucidated.
KEYWORDS
fractal calculus, periodic solution, solitary wave, two-scale mathematics, Duffing oscillator
PAPER SUBMITTED: 2020-05-02
PAPER REVISED: 2020-05-31
PAPER ACCEPTED: 2020-05-31
PUBLISHED ONLINE: 2021-01-31
DOI REFERENCE: https://doi.org/10.2298/TSCI200502032L
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Issue 2, PAGES [1345 - 1350]
REFERENCES
  1. Korpinar, Z., Some Analytical Solutions by Mapping Methods for Non-Linear Conformable Time-Fractional Phi-4 Equation, Thermal Science, 23 (2019), S6, pp. S1815-S1822
  2. Tariq, H., Akram, G., New Approach for Exact Solutions of Time Fractional Cahn-Allen Equation and Time Fractional Phi-4 Equation, Physics A, 473 (2017), May, pp. 352-362
  3. Alzahrani, A. K., et al., The Unsteady Liquid Film Flow of the Carbon Nanotubes Engine Oil Nanofluid over a Non-Linear Radially Extending Surface, Thermal Science, 24 (2020), 2A, pp. 951-963
  4. Sirisubtawee, S., et al., Exact Traveling Wave Solutions of the Space-Time Fractional Complex Ginzburg-Landau Equation and the Space-Time Fractional Phi-4 Equation Using Reliable Methods, Advances in Difference Equations, 2019 (2019), May, 219
  5. He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  6. He, J. H., A Short Review on Analytical Methods for to a Fully Fourth Order Non-Linear Integral Boundary Value Problem with Fractal Derivatives, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2020), 11, pp. 4933-4943
  7. Wang, K. L., et al., Physical Insight of Local Fractional Calculus and Its Application Fractional Kdv-Burgers-Kuramoto Equation, Fractals, 27 (2019), 7, 1950122
  8. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 29 (2019), 8, 1950134
  9. Wang, Y., et al., A Fractal Derivative Model for Snow's Thermal Insulation Property, Thermal Science, 23 (2019), 4, pp. 2351-2354
  10. Liu, H. Y., et al., A Fractal Rate Model for Adsorption Kinetics at Solid/Solution Interface, Thermal Science, 23 (2019), 4, pp. 2477-2480
  11. Shen, Y., El-Dib, Y. O., A Periodic Solution of the Fractional Sine-Gordon Equation Arising in Architectural Engineering, Journal of Low Frequency Noise Vibration and Active Control, On-line first, doi.org/10.1177/1461348420917565, 2020
  12. Yang, Z. P., et al., A Fractal Model for Pressure Drop through a Cigarette Filter, Thermal Science, 24 (2020), 4, pp. 2653-2659
  13. He, J. H., Thermal Science for the Real World: Reality and Challenge, Thermal Science, 24 (2020), 4, pp. 2289-2294
  14. He, J. H., A Fractal Variational Theory for 1-D Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, 2050024
  15. He, C. H., et al., Taylor Series Solution for Fractal Bratu-Type Equation Arising in Electrospinning Process, Fractals, 28 (2020), 1, 2050011
  16. He, J. H., A Simple Approach to 1-D Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), 113565
  17. Zhang, J. J., et al., Some Analytical Methods for Singular Boundary Value Problem in a Fractal Space, Appl. Comput. Math., 18 (2019), 3, pp. 225-235
  18. 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
  19. Shen, Y., and He, J. H., Variational Principle for a Generalized KdV Equation in a Fractal Space, Fractals, 28 (2020), 04, 2050069
  20. Li, X. J., et al., A Fractal Two-Phase Flow Model for the Fiber Motion in a Polymer Filling Process, Fractals, 28 (2020), 05, 2050093
  21. Ain, Q. T., He, J. H., On Two-Scale Dimension and Its Application, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  22. He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
  23. He, J.H., Ain, Q.T. New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermodynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681
  24. He, J. H., The Simplest Approach to Non-Linear Ocillators, Results Phys, 15 (2019), 102546
  25. He, J. H., Some Asymptotic Methods for Strongly Non-Linear Equations, Int. J. Mod. Phys. B, 13 (2006), 20, pp. 1141-1199
  26. He, J. H., The Simpler, The Better: Analytical Methods for Non-Linear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1252-1260
  27. Yao, X., He, J. H., On Fabrication of Nanoscale Non-Smooth Fibers with High Geometric Potential and Nanoparticle's Non-Linear Vibration, Thermal Science, 24 (2020), 4, pp. 2491-2497
  28. He, J. H., Variational Principle and Periodic Solution of the Kund-Mukherjee-Naskar equation, Results in Physics, 17 (2020), June, 103031
  29. He, C.H., et al., Fangzhu: An Ancient Chinese Nanotechnology for Water Collection from Air: History,Mathematical Insight, Promises and Challenges, Mathematical Methods in the Applied Sciences, On-line first, doi.org/10.1002/mma.6384, 2020
  30. Liu, C. X., A Short Remark on He's Frequency Formulation, Journal of Low Frequency Noise, Vibration and Active Control, On-line first, doi.org/10.1177/1461348420926331, 2020
  31. Li, X. X., et al., 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
  32. Ren, Z. F., and Wu, J. B., He's Frequency-Amplitude Formulation for Non-Linear Oscillator with Damping, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1045-1049
  33. Ren, Z. F., and Hu, G. F., He's Frequency-Amplitude Formulation with Average Residuals for Non-Linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1050-1059
  34. Wang, Q. L., et al., A Short Remark on Ren-Hu's Modification of He's Frequency-Amplitude Formulation and the Temperature Oscillation in a Polar Bear Hair, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1374-1377
  35. Ren, Z. F., and Hu, G. F., Discussion on the Accuracies of He's Frequency-Amplitude Formulation and Its Modification with Average Residuals, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1713-1715
  36. He, C. H., et al., A Complement to Period/Frequency Estimation of a Non-Linear Oscillator, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 992-995
  37. Tao, Z. L., et al., Approximate Frequency-Amplitude Relationship for a Singular Oscillator, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1036-1040
  38. Yu, D. N., et al., Homotopy Perturbation Method with an Auxiliary Parameter for Non-Linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1540-1554
  39. He, J. H., and Xiao, J., A Short Review on Analytical Methods for the Capillary Oscillator in a Nanoscale Deformable Tube, Mathematical Methods in the Applied Sciences, On-line first, doi.org/10.1002/ mma.6321, 2020
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Periodic solution of fractal Phi-4 equation

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