THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

ITERATIVE SOLUTION FOR A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS IN FRACTAL SPACES

ABSTRACT
A class of fractal PDE is successfully established by He's fractal derivative in a fractal space, and their variational principles are obtained by the semi-inverse method. The Fourier-Rabbani-He method and the Ritz-like method are used to solve the given fractal equations with initial value conditions. The example is a great demonstration of how the Fourier-Rabbani-He method is a powerful and simple tool that can be used in different ways.
KEYWORDS
PAPER SUBMITTED: 2024-03-05
PAPER REVISED: 2024-06-01
PAPER ACCEPTED: 2024-06-02
PUBLISHED ONLINE: 2025-07-06
DOI REFERENCE: https://doi.org/10.2298/TSCI2503821T
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2025, VOLUME 29, ISSUE Issue 3, PAGES [1821 - 1828]
REFERENCES
  1. Nadeem, M., He, J.-H., The Homotopy Perturbation Method for Fractional Differential Equations: Part 2, Two-Scale Transform, International Journal of Numerical Methods for Heat & Fluid Flow, 32 (2022), 2, pp. 559-567
  2. He, J.-H., et al., Good Initial Guess for Approximating Non-Linear Oscillators by the Homotopy Perturbation Method, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 1, pp. 21-29
  3. 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
  4. 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
  5. Anjum, N., He, J.-H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
  6. Tang, W., et al., Variational iteration Method for the Nanobeams-Based N/MEMS System, MethodsX, 11 (2023), 102465
  7. Golmankhaneh, A. K.,Tunc, C., Sumudu Transform in Fractal Calculus, Applied Mathematics and Computation, 350 (2019), June, pp. 386-401
  8. Nadeem, M., et al., The Homotopy Perturbation Method for Fractional Differential Equations: Part 1 Mohand Transform, International Journal of Numerical Methods for Heat & Fluid flow, 31 (2021), 11, pp. 3490-3504
  9. Mohammed, O. H., Salim, H.A., Computational Methods Based Laplace Decomposition for Solving Non-Linear System of Fractional Order Differential Equations, Alexandria Engineering Journal, 57 (2018), 4, pp. 3549-3557
  10. Shah, R., et al., Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay, Mathematics-Basel, 7 (2019), 6, 532
  11. He, J.-H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934
  12. He, J.-H., Taylor Series Solution for a Third Order Boundary Value Problem Arising in Architectural Engineering, Ain Shams Engineering Journal, 11 (2020), 4, pp. 1411-1414
  13. He, C. H., et al., Taylor Series Solution for Fractal Bratu-Type Equation Arising in Electrospinning Process, Fractals, 28 (2020), 1, 20500115
  14. He, J.-H., Wu, X. H., Exp-Function Method for Non-Linear Wave Equations, Chaos Soliton &Fractals, 30 (2006), 3, pp. 700-708
  15. 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
  16. Lv, G. J., et al., Shock-like Waves with Finite Amplitudes, Journal of Computational Applied Mechanics, 55 (2024), 1, pp. 1-7
  17. Rabbani, M., et al., Some Computational Convergent Iterative Algorithms to Solve Non-Linear Problems, Mathematical Sciences, 17 (2023), 2, pp. 145-156
  18. Golmankhaneh, A. K., Tunc, C., Sumudu Transform in Fractal Calculus, Applied Mathematics and Computation, 350 (2019), June, pp. 386-401
  19. Nadeem, M., et al., The Homotopy Perturbation Method for Fractional Differential Equations: Part 1 Mohand Transform, International Journal of Numerical Methods for Heat & Fluid Flow, 31 (2021), 11, pp. 3490-3504
  20. Mohammed, O. H., Salim, H. A., Computational Methods Based Laplace Decomposition for Solving Non-Linear System of Fractional Order Differential Equations, Alexandria Engineering Journal, 57 (2018), 4, pp. 3549-3557
  21. Shah, R., et al., Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay, Mathematics, 7 (2019), 6, 532
  22. Song, Q.-R., Zhang, J.-G., He-transform: Breakthrough Advancement for the Variational Iteration Method, Front. Phys., 12 (2024), 1411691
  23. He, J.-H., et al., Beyond Laplace and Fourier Transforms: Challenges and Future Prospects, Thermal Science, 27 (2023), 6B, pp. 5075-5089
  24. Zwillinger, D., Handbook of Differential Equations, 3rd ed., Academic Press, New York, USA, 1998
  25. Anjum, N., et al., Li-He's Modified Homotopy Perturbation Method for Doubly-Clamped Electrically Actuated Microbeams-Based Microelectromechanical System, Facta Universitatis, Series: Mechanical Engineering, 19 (2021), 4, pp. 601-612
  26. He, J.-H., El-Dib, Y. O., The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings, Facta Universitatis, Series: Mechanical Engineering, 19 (2021), 4, pp. 735-750
  27. He, J.-H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  28. He, J.-H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical physics, 53 (2014), June, pp. 3698-3718
  29. Ain, Q. T., He, J.-H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  30. He, J.-H., Ji, F.Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics. Thermal Science, 23 (2019), 4, pp. 2131-2133
  31. 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
  32. He, J.-H., Variational Principle and Periodic Solution of the Kundu-Mukherjee-Naskar Equation, Results in Physics, 17 (2020), 103031
  33. Zuo, Y. T., Variational Principle for a Fractal Lubrication Problem, Fractals, 32 (2024), 5, pp. 1-6
  34. Jiao, M.-L., et al. Variational Principle for Schrodinger-KdV System with the M-fractional Derivatives, Journal of Computational Applied Mechanics, 55 (2024), 2, pp. 235-241
  35. 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
  36. He, C. H., Liu, C., Variational Principle for Singular Waves, Chaos, Solitons & Fractals, 172 (2023), 113566
  37. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
  38. Wu, Y., He, J.-H., Variational Principle for the Kaup-Newell System, Journal of Computational Applied Mechanics, 54 (2023), 3, pp. 405-409
  39. He, J. H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathematics Letters, 119 (2021), 107199
  40. He, J.-H., Variational Principles for Some Non-Linear Partial Differential Equations with Variable Coefficients, Chaos, Solitons & Fractals, 19 (2004), 4, pp. 847-851
  41. He, J.-H., Variational Approach for Non-Linear Oscillators, Chaos Solitons & Fractals, 34 (2007), 5, pp. 1430-1439
  42. He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), 104104
  43. Anjum, N., et al., Free Vibration of a Tapered Beam by the Aboodh Transform-based Variational Iteration Method, Journal of Computational Applied Mechanics, 55 (2024), 3, pp. 440-450
  44. Almousa, M., et al., Sumudu and Elzaki Integral Transforms for Solving Systems of Integral and Ordinary Differential Equations, Advances in Differential Equations and Control Processes, 31 (2024) , 11, pp. 43-60

2025 Society of Thermal Engineers of Serbia. Published by the VinĨa 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