THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Shuxian Deng

,

Xinxin Ge

APPROXIMATE ANALYTICAL SOLUTION FOR PHI-FOUR EQUATION WITH HE’S FRACTAL DERIVATIVE

ABSTRACT
This paper, for the first time ever, proposes a Laplace-like integral transform, which is called as He-Laplace transform, its basic properties are elucidated. The homotopy perturbation method coupled with this new transform becomes much effective in solving fractal differential equations. Phi-four equation with He’s derivative is used as an example to reveal the main merits of the present technology.
KEYWORDS
Phi-four equation, He’s fractal derivative, He-Laplace transform, homotopy perturbation method
PAPER SUBMITTED: 2019-12-31
PAPER REVISED: 2020-06-28
PAPER ACCEPTED: 2020-06-28
PUBLISHED ONLINE: 2021-03-27
DOI REFERENCE: https://doi.org/10.2298/TSCI191231127D
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Issue 3, PAGES [2369 - 2375]
REFERENCES
  1. He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  2. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 29 (2019), 8, ID 1950134
  3. Wang, Y., et al., A Fractal Derivative Model for Snow's Thermal Insulation Property, Thermal Science, 23 (2019), 4, pp. 2351-2354
  4. Liu, H. Y., et al., A Fractal Rate Model for Adsorption Kinetics at Solid/Solution Interface, Thermal Science, 23 (2019), 4, pp. 2477-2480
  5. Wang, Q. L., et al., Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Hairs (vol. 26, 1850086, 2018), Fractals, 27 (2019), 5, ID 1992001
  6. Wang, Q. L., et al., Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Hairs (vol. 26, 1850086, 2018), Fractals, 26 (2018), 6, ID 1850086
  7. He, J. H., A Fractal Variational Theory for One-Dimensional Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, ID 2050024
  8. 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), Dec., ID 113565
  9. 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
  10. 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
  11. Shen, Y., He, J. H., Variational Principle for a Generalized KdV Equation in a Fractal Space, Fractals, 28 (2020), 4, ID 20500693
  12. Li, X. J., et al., A Fractal Two-Phase Flow Model for the Fiber Motion in a Polymer Filling Process, Fractals, 28 (2020), 5, ID 20500930
  13. Fan, J., et al., Fractal Calculus for Analysis of Wool Fiber: Mathematical Insight of Its Biomechanism, Journal of Engineered Fibers and Fabrics, 14 (2019), Aug., 1558925019872200
  14. Lin, L., Yao, S. W., Release Oscillation in a Hollow Fiber - Part 1: Mathematical Model and Fast Estimation of Its Frequency, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1707-1703
  15. Deng, X., et al., Travelling Wave Solutions for a Non-linear Variant of the PHI-Four Equation, Mathematical and Computer Modelling, 49 (2009), 3-4, pp. 617-622
  16. He, J. H., Exp-Function Method for Fractional Differential Equations, International Journal of Non-linear Sciences and Numerical Simulation, 14 (2013), 6, pp. 363-366
  17. Tariq, H., et al. New Approach for Exact Solutions of Time Fractional Cahn-Allen Equation and Time Fractional Phi-4 Equation, Physica A, 473 (2017), 1, pp. 352-362
  18. 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
  19. Ain, Q. T., He, J. H. On Two-Scale Dimension and Its Application, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  20. He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Therm. Sci., 23 (2019), 4, pp. 2131-2133
  21. He, J. H., Thermal Science for the Real World: Reality and Challenge, Thermal Science, 24 (2020), 4, pp. 2289-2294
  22. He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  23. 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
  24. 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
  25. Kuang, W. X., et al., Homotopy Perturbation Method With an Auxiliary Term for the Optimal Design of a Tangent Non-linear Packaging System, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1075-1080
  26. Yao, S. W., Cheng, Z. B., The Homotopy Perturbation Method for a Non-linear Oscillator With a Damping, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1110-1112
  27. He, J. H., Jin, X., 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
  28. Gondal, M. A., et al., Homotopy Perturbation Method for Non-linear Exponential Boundary Layer Equation using Laplace Transformation, He's Polynomials and Pade Technology He's Polynomials and Pade Technology, International Journal of Non-linear Sciences and Numerical Simulation, 11 (2010), 12, pp. 1145-1153
  29. Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
  30. Nadeem, M., Li, F., He-Laplace Method for Non-linear Vibration Systems and Non-linear Wave Equations, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1060-1074
  31. Suleman, M., et al., He-Laplace Method for General Non-linear Periodic Solitary Solution of Vibration Equations, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. -12971304
  32. Li, F., 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. 1313-1305
  33. Wang, Y., et al., Using Reproducing Kernel for Solving a Class of Fractional Partial Differential Equation with Non-Classical Conditions, Applied Mathematics and Computation, 219 (2013), 11, pp. 5918-5925
  34. Wang, Y., et al., New Algorithm for Second-Order Boundary Value Problems of Integro-Differential Equation, Journal of Computational and Applied Mathematics, 229 (2009), 1, pp. 1-6
PDF VERSION [DOWNLOAD]
Approximate analytical solution for Phi-four equation with He’s fractal derivative

© 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