International Scientific Journal


This paper extends the (2+1)-dimensional Eckhaus-type dispersive long wave equations in continuous medium to their fractional partner, which is a model of non-linear waves in fractal porous media. The derivation is shown briefly using He’s fractional derivative. Using the semi-inverse method, the variational principles are established for the fractional system, which up to now are not discovered. The obtained fractal variational principles are proved correct by minimizing the functional with the calculus of variations, and might find potential applications in numerical modeling.
PAPER REVISED: 2020-05-24
PAPER ACCEPTED: 2020-05-24
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THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Issue 2, PAGES [1277 - 1285]
  1. Ablowitz, M. J., Clarkson, P. A., Solitons, Non-Linear Evolution Equations and Inverse Scatting, Cambridge University Press, Cambridge, UK, 1991, 165-182
  2. Gu, C. H. B., et al., L. Guo, Soliton Theory and Its Application, Hangzhou: Zhejiang Science and Technology Publishing House, Zhejiang, China, 1990, pp. 76-95
  3. He, J. H., Some Asymptotic Methods for Strongly Non-Linear Equations, Int. J. Mod. Phys. B, 13 (2006), 20, pp. 1141-1199
  4. Wang, M. L., et al., Application of a Homogeneous Balance Method to Exact Solutions of Non-Linear Equations in Mathematical Physics, Phys Lett A, 216 (1996), 1-5, pp. 67-75
  5. Liu, S. S., Fu, Z. T., Expansion Method about the Jacobi Elliptic Function and Its Applications to Non-Linear Wave Equations, Acta Phys. Sin., 50 (2001), 11, pp. 2068-2073
  6. Ma, H. C., Exact Solutions of Non-Linear Fractional Partial Differential Equations by Fractional Sub-Equation Method, Thermal Science, 19 (2015), 4, pp. 1239-1244
  7. Li, Z. B., Exact Solutions of Time-fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16 (2012), 2, pp. 335-338
  8. 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
  9. Wang, L., Chen, X., Approximate Analytical Solutions of Time Fractional Whitham-Broer-Kaup Equations by a Residual Power Series Method, Entropy, 17 (2015), 9, pp. 6519-6533
  10. Wu, Y., Variational Approach to Higher-Order Water-Wave Equations, Chaos, Solitons and Fractals, 32 (2007), 1, pp. 195-203
  11. Baleanu, D., et al., A Modified Fractional Variational Iteration Method for Solving Non-Linear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operators, Thermal Science, 22 (2018), Suppl. 1, pp. S165-S175
  12. Durgun, D. D., Konuralp, A., Fractional Variational Iteration Method for Time-Fractional Non-Linear Functional Partial Differential Equation Having Proportional Delays, Thermal Science, 22 (2018), Suppl. 1, S33-S46
  13. He, J. H., Liu, F. J., Local Fractional Variational Iteration Method for Fractal Heat Transfer in Silk Cocoon Hierarchy, Non-Linear Science Letters A., 4 (2013), Jan., pp. 15-20
  14. Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
  15. Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
  16. Malomed, B. A., Weinstein, M. I., Soliton Dynamics in the Discrete Non-Linear Schrodinger Equation, Phys. Lett. A., 220 (1996), 1-3, pp. 91-96
  17. Malomed, B. A., Variational Methods in Non-Linear Fiber Optics and Related Fields, in: Progress in Optics (ed. E. Wolf), 2002, Vol. 43, Chapter 2, pp. 71-193
  18. Chong, C., et al., On the Validity of the Variational Approximation in Discrete Non-Linear Schrodinger Equations, Phys. D Non-linear Phenom., 241 (2011), 2, pp. 115-124
  19. He, J. H., Variational Principles for Some Non-Linear Partial Differential Equations with Variable Coefficients, Chaos Solitons and Fractals, 19 (2004), 4, pp. 847-851
  20. He, J. H., A Modified Li-He's Variational Principle for Plasma, International Journal of Numerical Methods for Heat and Fluid-Flow, (2019),
  21. He, J. H., An Alternative Approach to Establishment of a Variational Principle for the Torsional Problem of Piezoelastic Beams, Applied Mathematics Letters, 52 (2016), Feb., pp. 1-3
  22. He, J. H., Hamilton's Principle for Dynamical Elasticity, Applied Mathematics Letters, 72 (2017), Oct., pp. 65-69
  23. He, J. H., Generalized Equilibrium Equations for Shell Derived from a Generalized Variational Principle, Applied Mathematics Letters, 64 (2017), Feb., pp. 94-100
  24. He, J. H., Lagrange Crisis and Generalized Variational Principle for 3-D unsteady flow, International Journal of Numerical Methods for Heat and Fluid-Flow, 30 (2019), 3, pp. 1189-1196
  25. He, J. H., Sun, C., A Variational Principle for a Thin Film Equation, Journal of Mathematical Chemistry, 57 (2019), Aug., pp. 2075-2081
  26. Wang, Y., A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium, Fractals, 27 (2019), 4, 1950047
  27. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 08, 1950134
  28. He, J. H., Ain, Q. T., New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermo-Dynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681
  29. 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
  30. He, J. H., Variational Principle and Periodic Solution of the Kundu-Mukherjee-Naskar Equation, Results in Physics, 17 (2020), June, 103031
  31. El-Kalaawy, O. H., New Variational Principle-Exact Solutions and Conservation Laws for Modified Ion-Acoustic Shock Waves and Double Layers with Electron Degenerate in Plasma, Physics of Plasmas, 24 (2017), 3, 032308
  32. Ji, F. Y., A fractal Boussinesq Equation for Non-Linear Transverse Vibration of a Nanofiber-Reinforced Concrete Pillar, Applied Mathematical Modelling, 82 (2020), June, pp. 437-448
  33. Yan, Z. Y., The Investigation for (2+1)-Dimensional Eckhaus-Type Extension of the Dispersive Long Wave Equation, J. Phys. A:Math. Gen., 37 (2004), 3, pp. 841-850
  34. Zhang, J. F., et al., New Soliton Solutions to (2+1)-Dimensional Eckhaust-Type Dispersive Long Wave Equation, Journal of Zhejiang Normal University, (Nat. Sci.), (2005), 2, pp. 144-148
  35. Ain, Q. T., He, J. H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  36. He, F., Ji, Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
  37. He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  38. He, J. H., A Tutorial Review on Fractal Space-Time and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), June, pp. 3698-3718
  39. He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
  40. Liu, F. J., et al., He's Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silkworm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
  41. He, J. H., et al., A New Fractional Derivative and Its Application Explanation of Polar Bear Hairs, Journal of King Saud University-Science, 28 (2016), 2, pp. 190-192
  42. Wang, K. L., Liu, S. Y., He's Fractional Derivative for Non-Linear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
  43. Yue, Shen and Ji-Huan He, Variational Principle for a Generalized KdV Equation in a Fractal Space, Fractals, 20 (2020), 04, 2050069

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