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Jianshe Sun

FRACTAL MODIFICATION OF SCHRöDINGER EQUATION AND ITS FRACTAL VARIATIONAL PRINCIPLE

ABSTRACT
With the help of a new fractal derivative, a fractal model for variable coefficients and highly non-linear Schrödinger equations on a non-smooth boundary are acquired. The variational principles of the fractal variable coefficients and highly non-linear Schrödinger equations are built successfully by coupling fractal semi-inverse and He’s two-scale transformation methods, which are helpful to reveal the symmetry, to discover the conserved quantity, and the obtained variational principles have widespread applications in numerical simulation.
KEYWORDS
variational principle, fractal semi-inverse method, fractal derivative, Schrödinger equations, He’s two-scale transform method
PAPER SUBMITTED: 2022-05-04
PAPER REVISED: 2022-07-14
PAPER ACCEPTED: 2022-07-15
PUBLISHED ONLINE: 2023-06-11
DOI REFERENCE: https://doi.org/10.2298/TSCI2303029S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Issue 3, PAGES [2029 - 2037]
REFERENCES
  1. Zhou, X. W., Wang, L., A Variational Principle for Coupled Non-linear Schrödinger Equations with Variable Coefficients and High Non-linearity, Comput. Math. Appl., 61 (2011), 8, pp. 2035 -2038
  2. Ain, Q. T., et al., The Fractional Complex Transform: A Novel Approach to the Time-Fractional Schrödinger Equation, Fractals, 28 (2020), 7, 2050141
  3. He, J.-H., El-Dib, Y. O., Periodic Property of the Time-Fractional Kundu-Mukherjee-Naskar Equation, Results in Physics, 19 (2020), Dec., 103345
  4. 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
  5. Yao, L., Chang, J. R., Variational Principles for Non-linear Schrödinger Equation with High Non-linearity, J. Non-linear Sci. Appl., 1 (2008), 1, pp. 1-4
  6. Ozis, T., Yildirim, A., Application of He's Semi-Inverse Method to the Non-linear Schrödinger Equation, Comput. Math. Appl., 54 (2007), 7-8, pp. 1039-1042
  7. He, J.-H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
  8. He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), May, 104104
  9. Wu, P. X., et al., Solitary Waves of the Variant Boussinesq-Nurgers Equation in a Fractal Dimensional Space, Fractals, 30 (2022), 3, pp. 1-10
  10. 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
  11. He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, International Journal of Numerical Methods for Heat & Fluid Flow, 33 (2022), 1, pp. 351-359
  12. He, C. H., et al., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046
  13. Wang, K. L., Wei, C. F., A Powerful and Simple Frequency Formula to Non-linear Fractal Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1373-1379
  14. Feng, G. Q., He's Frequency Formula to Fractal Undamped Duffing Equation, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 4, pp. 1671-1676
  15. Zuo, Y.-T., Liu, H.-J., Fractal Approach to Mechanical and Electrical Properties of Graphene/Sic Composites, Facta Universitatis-Series Mechanical Engineering, 19 (2021), 2, pp. 271-284
  16. Tian, D., et al., 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
  17. Tian, D., et al., Fractal Pull-in Stability Theory for Microelectromechanical Systems, Frontiers in Phys-ics, 9 (2021), Mar., 606011
  18. Anjum, N., et al., Two-scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 2150182
  19. He, J.-H., et al., Evans Model for Dynamic Economics Revised, AIMS Mathematics, 6 (2021), 9, pp. 9194-9206
  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., El-Dib, Y. O., A Tutorial Introduction to the Two-Scale Fractal Calculus and Its Application to the Fractal Zhiber-Shabat Oscillator, Fractals, 29 (2021), 8, 2150268
  22. Ain, Q. T., He, J. H., On Two-scale Dimension and its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  23. He, J.-H., Seeing with a Single Scale is Always Unbelieving: From Magic to Two-Scale Fractal, Thermal Science, 25 (2021), 2B, pp. 1217-1219
  24. He, J.-H., When Mathematics Meets Thermal Science, The Simpler is the Better, Thermal Science, 25 (2021), 3B, pp. 2039-2042
  25. He, J.-H., Fractal Calculus and Its Geometrical Explanation, Results. Phys., 10 (2018), Sept., pp. 272 -276
  26. He, J.-H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), June, pp. 3698-3718
  27. 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 & Active Control, 40 (2021), 2, pp. 683-691
  28. 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, 2150214
  29. Liu, Y. P., et al., A Fractal Langmuir Kinetic Equation and Its Solution Structure, Thermal Science, 25 (2021), 2B, pp. 1351-1354
  30. Liu, X. Y., et al., Optimization of a Fractal Electrode-Level Charge Transport Model, Thermal Science, 25 (2021), 3B, pp. 2213-2220
  31. Yang X. J., An Insight on the Fractal Power Law Flow: From a Hausdorff Vector Calculus Perspective, Fractals, 30 (2022), 22500542
  32. Sun, J. S., Analytical Approximate Solutions Of (N+1)-Dimensional Fractal Harry Dym Equations, Fractals, 26 (2018), 6, 1850094
  33. Sun, J. S., Approximate Analytic Solution of the Fractal Klein-Gordon Equation, Thermal Science, 25 (2021), 2B, pp. 1489-1494
  34. Sun, J. S., Traveling Wave Solution of fractal KDV-Burgers-Kuramoto Equation Within Local Fractional Differential Operator, Fractals, 29 (2021), 7, 2150231
  35. Liu, J.G., et al., A New Perspective to Study the Third Order Modified KdV Equation on Fractal Set, Fractals, 28 (2020), 6, 2050110
  36. Feng, Y. Y., et al., New Perspective Aimed at Local Fractional Order Memristor Model on Cantor Sets, Fractals, 29 (2021), 1, 2150011
  37. Yang, X. J., Machado, J. A. T., A New Fractal Non-linear Burgers' Equation Arising in the Acoustic Signals Propagation, Math. Meth. Appl. Sci., 42 (2019), 18, pp. 7539-7544
  38. Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science: New York, USA, 2012
  39. Yang, X. J., The Zero-Mass Renormalization Group Differential Equations and Limit Cycles in Non-Smooth Initial Value Problems, Japan Agri. Res. Quart., 3 (2012), 9, pp. 229-235
  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. Ma, H. J., Simplified Hamiltonian-Based Frequency-Amplitude Formulation For Non-linear Vibration System, Facta Universitatis. Series: Mechanical Engineering, 20 (2022), 2, pp. 445-455
  42. He, J.-H. A Fractal Variational Theory for One-Dimensional Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, 2050024
  43. Li, Z. B., He, J. H., Fractional Complex Transform for Fractional Differential Equations, Math. Comput. Appl., 15 (2010), 5, pp. 970-973
  44. He, J.-H, et al., Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
  45. Wang, Y., et al., A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium, Fractals, 27 (2019), 4, 1950047
  46. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
  47. He, J.-H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, J. Appl. Comput. Mech., 6 (2020), 4, pp. 735-740
  48. Wang, K. L., Wang, H., Fractal Variational Principles for Two Different Types of Fractal Plasma Models with Variable Coefficients, Fractals, 30 (2022), 3, 22500438
  49. He, J.-H., On the Fractal Variational Principle for the Telegraph Equation, Fractals, 29 (2021), 1, 2150022
  50. He, J.-H., Qian, M. Y., A Fractal Approach to the Diffusion Process of Red Ink in a Saline Water, Thermal Science, 26 (2022), 3B, pp. 2447-2451
  51. 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
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Fractal modification of Schrödinger equation and its fractal variational principle

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