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APPROXIMATE ANALYTIC SOLUTION OF THE FRACTAL KLEIN-GORDON EQUATION

ABSTRACT
The linear and non-linear Klein-Gordon equations are considered. The fractional complex transform is used to convert the equations on a continuous space/time to fractals ones on Cantor sets, the resultant equations are solved by local fractional reduced differential transform method. Three examples are given to show the effectiveness of the technology.
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PAPER SUBMITTED: 2020-03-01
PAPER REVISED: 2020-05-28
PAPER ACCEPTED: 2020-06-20
PUBLISHED ONLINE: 2021-01-31
DOI REFERENCE: https://doi.org/10.2298/TSCI200301051S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Issue 2, PAGES [1489 - 1494]
REFERENCES
  1. Wazwaz, A.-M., Compactons, Solitons and Periodic Solutions for Some Forms of Non-Linear Klein-Gordon Equations, Chaos, Solitons and Fractals, 28 (2006), 4, pp. 1005-1013
  2. El-Sayed, S. M., The Decomposition Method for Studying the Klein-Gordon Equation, Chaos, Solitons and Fractals, 18 (2003), 5, pp. 1025-1030
  3. Kanth, A. S. V. R., Aruna, K., Differential Transform Method for Solving the Linear and Non-Linear Klein-Gordon Equation, Computer Physics Communications, 180 (2009), 5, pp. 708-711
  4. Golmankhaneh, A. K., et al., On Non-Linear Fractional Klein-Gordon Equation, Signal Processing, 91 (2011), 3, pp. 446-451
  5. Gepreel, K. A., Mohamed, M. S., Analytical Approximate Solution for Non-Linear Space-Time Fractional Klein-Gordon Equation, Chinese Physics B, 22 (2013), 1, 010201
  6. Veeresha, P., et al., An Efficient Technique for Non-Linear Time-Fractional Klein-Fock-Gordon Equation, Applied Mathematics and Computation, 364 (2020), 124637
  7. Tamsir, M., Srivastava, V. K., Analytical Study of Time-Fractional Order Klein-Gordon Equation, Alexandria Engineering Journal, 55 (2016), 1, pp. 561-567
  8. Yang, X. J., Machado J. A. T., A New Fractal Non-Linear Burgers' Equation Arising in the Acoustic Signals Propagation, Mathematical Methods in the Applied Sciences, 42 (2019), 18, pp. 1-6
  9. Sun, J. S., Analytical Approximate Solutions of (N+1)-Dimensional Fractal Harry Dym Equations, Fractals, 26 (2018), 6, 1850094
  10. He, J. H., A Simple Approach to 1-D Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemistry, 854 (2019), 113565
  11. Wang, K. L., He, C. H., Physical Insight of Local Fractional Calculus and Its Application Fractional KdV-Burgers-Kuramoto, Fractals, 27 (2019), 7, 1950122
  12. He, J. H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  13. Li, Z. B., et al., Exact Solutions of Time Fractional Heat Conduction Equation by the Fractional Complex Transform, Thermal Science, 16 (2012), 2, pp. 335-338
  14. 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
  15. Yang, X. J., et al., Transport Equations in Fractal Porous Media within Fractional Complex Transform Method, Proceedings of the Romanian Academy A, 14 (2013), 4, pp. 287-292
  16. Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
  17. Zhou, J. K., Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986
  18. Keskin, Y., Oturanc, G., Reduced Differential Transform Method for Partial Differential Equations, International Journal of Non-Linear Sciences and Numerical Simulation, 10 (2009), 6, pp. 741-750
  19. Yang, A. M., et al., Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets, Abstract and Applied Analysis, 2014 (2014), ID372741
  20. Kumar D., et al., A Hybrid Computational Approach for Klein-Gordon Equations on Cantor Sets, Non-Linear Dyn, 87 (2017), 1, pp. 511-517
  21. 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
  22. 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

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