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A REMARK ON A STRONG MINIMUM CONDITION OF A FRACTAL VARIATIONAL PRINCIPLE

ABSTRACT
The fractal variational principle gives a good physical understanding of a discontinuous problem in an energy way, and it is a good tool to revealing the physical phenomenon which cannot be done by the traditional variational theory. A minimum variational principle is very important in ensuring the convergence of artificial intelligence algorithms for numerical simulation and image processing. The strong minimum condition of a fractal variational principle in a fractal space is discussed, and two examples are given to illustrate its simplicity and feasibility.
KEYWORDS
PAPER SUBMITTED: 2023-04-08
PAPER REVISED: 2023-08-08
PAPER ACCEPTED: 2023-08-10
PUBLISHED ONLINE: 2024-05-18
DOI REFERENCE: https://doi.org/10.2298/TSCI2403371N
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE Issue 3, PAGES [2371 - 2377]
REFERENCES
  1. Wang, S. Q., A Variational Approach to Non-linear Two-Point Boundary Value Problems, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2452-2455
  2. Wang, S. Q., et al., Variational Iteration Method for Solving Integro-Differential Equations, Physics Letters A, 367 (2007), 3, pp. 188-191
  3. 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
  4. Ma, H. J., Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 2, pp. 445-455
  5. He, J.-H., et al., Forced Nonlinear Oscillator in a Fractal Space, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 1, pp. 1-20
  6. He, J.-H., et al., Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Oscillators, Facta Universitatis Series: Mechanical Engineering, 19 (2021), 2, pp. 199-208
  7. He, J.-H., Hamilton's Principle for Dynamical Elasticity, Applied Mathematics Letters, 72 (2017), Oct., pp. 65-69
  8. Wang, S. Q., et al., Diabetic Retinopathy Diagnosis Using Multichannel Generative Adversarial Network WITH Semisupervision, IEEE Transactions on Automation Science and Engineering, 18 (2020), 2, pp. 574-585
  9. Yu, W., et al., Tensorizing GAN with High-Order Pooling for Alzheimer's Disease Assessment, IEEE Transactions on Neural Networks and Learning Systems, 33 (2022), 9, pp. 4945-4959
  10. Hu, S. Y., et al., Brain MR to PET Synthesis via Bidirectional Generative Adversarial Network, Proceedings, 23rd International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2020, LNCS, Lima, Peru, 2020, pp. 698-707
  11. You, S. R., et al., Fine Perceptive Gans for Brain MR Image Super-Resolution in Wavelet Domain, IEEE Transactions on Neural Networks and Learning Systems, 34 (2023), 11, pp. 8802-8814
  12. Hu, S. Y., et al., Bidirectional Mapping Generative Adversarial Networks for Brain MR to PET Synthesis, IEEE Transactions on Medical Imaging, 41 (2021), 1, pp. 145-157
  13. Yu, W., et al. Morphological Feature Visualization of Alzheimer's Disease via Multidirectional Perception Gan, IEEE Transactions on Neural Networks and Learning Systems, 34 (2023), 8, pp. 4401-4415
  14. Hu, S. Y., et al., Medical Image Reconstruction Using Generative Adversarial Network for Alzheimer Disease Assessment with Class-Imbalance Problem, Proceedings, IEEE 6th International Conference on Computer and Communications (ICCC), Chengdu, China, 2020, pp. 1323-1327
  15. Kuo, P. H., et al., Machine Tool Chattering Monitoring by Chen-Lee Chaotic System-Based Deep Convolutional Generative Adversarial Nets, Structural Health Monitoring, 22 (2023), 6
  16. Lin, C. Y., et al., Application of Chaotic Encryption and Decryption in Wireless Transmission from Sensory Toolholders on Machine Tools, IEEE Sensors Journal, 23 (2023), 11, pp. 11453-11468
  17. Lu, J. F., Ma, L., Numerical Analysis of Space-Time Fractional Benjamin-Bona-Mahony Equation, Thermal Science, 27 (2023), 3A, pp. 1755-1762
  18. Lu, J., Chen, L., Numerical Analysis of a Fractal Modification of Yao-Cheng Oscillator, Results in Physics, 38 (2022), 105602
  19. Lu, J., Ma, L., Numerical Analysis of a Fractional Nonlinear Oscillator with Coordinate-Dependent Mass, Results in Physics, 43 (2022), 106108
  20. 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
  21. He, J.-H., Variational Principles for some Nonlinear Partial Differential Equations with Variable Coefficients, Chaos, Solitons & Fractals, 19 (2004), 4, pp. 847-851
  22. Li, X. J., et al., Multi-Scale Numerical Approach to the Polymer Filling Process in the Weld Line Region, Facta Universitatis-Series Mechanical Engineering, 20 (2022), 2, pp. 363-380
  23. Batista, M., On a Strong Minimum of Stable Forms of Elastica, Mechanics Research Communications, 107 (2020), 103522
  24. He, J.-H., Hamilton's Principle and Generalized Variational Principles of Linear Thermopiezoelectricity, ASME J. App. Mech., 68 (2001), 4, pp. 666-667
  25. He, J.-H., Variational Principle and Periodic Solution of the Kundu-Mukherjee-Naskar Equation, Results in Physics, 17 (2020), 103031
  26. Ling, W. W., Wu, P. X., Variational Theory for a Kind of Non-Linear Model for Water Waves, Thermal Science, 25 (2021), 2B, pp. 1249-1254
  27. He, J.-H., A Fractal Variational Theory for One-Dimensional Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, pp. 20500243
  28. He, J.-H., On the Fractal Variational Principle for the Telegraph Equation, Fractals, 29 (2021), 1, 2150022
  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., Liu, C., Variational Principle for Singular Waves, Chaos, Solitons & Fractals, 172 (2023), pp. 113566
  31. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, pp. 1950134
  32. He, J.-H., et al., Solitary Waves Travelling along an Unsmooth Boundary, Results in Physics, 24 (2021), pp. 104104
  33. 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, pp. 2150214
  34. Wang, Y., et al., A Variational Formulation for Anisotropic Wave Travelling in a Porous Medium, Fractals, 27 (2019), 4, 1950047
  35. He, J.-H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathematics Letters, 119 (2021), 107199
  36. Zhao, L., et al., Promises and Challenges of Fractal Thermodynamics, Thermal Science, 27 (2023), 3A, pp. 1735-1740
  37. 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
  38. 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
  39. Ain, Q. T., He, J.-H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  40. He, J.-H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  41. 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
  42. Feng, G. Q., Niu, J. Y., An Analytical Solution of the Fractal Toda Oscillator, Results in Physics, 44 (2023), 106208
  43. Feng, G. Q., Dynamic Pull-Down Theory for the Toda Oscillator, International Journal of Modern Physics B, On-line first, doi.org/10.1142/30217979224502928
  44. He, J.-H., Yang, Q., et al., Pull-Down Instability of the Quadratic Nonlinear Oscillators, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 2, pp. 191-200
  45. Shen, Y., et al., Nonlinear Vibration with Discontinuities in a Fractal Space: Its Variational Formulation and Periodic Property, Fractals, 31 (2023), 7, 2350070
  46. He, J.-H., A Tutorial Review Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  47. He, J.-H., et al., A Variational Principle for a Nonlinear Oscillator Arising in the Microelectromechanical System, Journal of Applied and Computational Mechanics, 7 (2021), 1, pp. 78-83
  48. Tian, D., et al., Fractal N/MEMS: from Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2150030
  49. 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
  50. Skrzypacz, P., et al., Dynamic Pull-in and Oscillations of Current-Carrying Filaments in Magnetic Micro-Electro-Mechanical System, Communications in Nonlinear Science and Numerical Simulation, 109 (2022), 106350

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