Abstract
The effectiveness of the local fractional reduced differential transformation method (LFRDTM) for the approximation of the solution related to the extended n-term local fractional Klein-Gordon equation is the main aim of this paper in which fractional complex transform and local fractional derivative have been employed to analyze the n-term Klein-Gordon equations, and Cantor sets. The proposed method, along with the existence of the solutions demonstrated through some examples, provides a powerful mathematical means in solving fractional linear differential equations. Considering these points, the paper also provides an accurate and effective method to solve complex physical systems that display fractal or self-similar behavior across various scales. In conclusion, the fractional complex transform with the local fractional differential transform method has been proven to be a robust and flexible approach towards obtaining effective approximate solutions of local fractional partial differential equations.
Keywords
Local fractional calculus local fractional differential equations; reduced differential transform method fractional Klein-Gordon equation.
References
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- Keskin, Y. and G. Oturanc, 2009 Reduced differential transform method for partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation 10: 741–750.
- Kolwankar, K. M. and A. D. Gangal, 1996 Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdisciplinary Journal of Nonlinear Science 6: 505–513.
- Kumar, D., J. Singh, and D. Baleanu, 2017 A hybrid computational approach for klein–gordon equations on cantor sets. Nonlinear Dynamics 87: 511–517.
- Sun, J., 2018 Analytical approximate solutions of (n+ 1)- dimensional fractal harry dym equations. Fractals 26: 1850094.
- Wang, K.-L., K.-J. Wang, and C.-H. He, 2019 Physical insight of local fractional calculus and its application to fractional kdv– burgers–kuramoto equation. Fractals 27: 1950122.
- Yang, A.-M., Y.-Z. Zhang, C. Cattani, G.-N. Xie, M. M. Rashidi, et al., 2014 Application of local fractional series expansion method to solve klein-gordon equations on cantor sets. In Abstract and Applied Analysis, volume 2014, Hindawi.
- Yang, X.-J., 2012 Advanced local fractional calculus and its applications. Yang, X.-J. and J. Tenreiro Machado, 2019 A new fractal nonlinear burgers’ equation arising in the acoustic signals propagation. Mathematical Methods in the Applied Sciences 42: 7539–7544.
- Zhang, Y., C. Cattani, and X.-J. Yang, 2015 Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains. Entropy 17: 6753–6764.
- Zhang, Y. and X.-J. Yang, 2016 An efficient analytical method for solving local fractional nonlinear pdes arising in mathematical physics. Applied Mathematical Modelling 40: 1793–1799.
Abstract
References
- Acan, O., M. M. Al Qurashi, and D. Baleanu, 2017 Reduced differential transform method for solving time and space local fractional partial differential equations. Journal of Nonlinear Sciences & Applications (JNSA) 10.
- Chu, Y.-M., M. Jneid, A. Chaouk, M. Inc, H. Rezazadeh, et al., 2023 Local time fractional reduced differential transform method for solving local time fractional telegraph equations. Fractals 0: null.
- Dubey, V. P., D. Kumar, J. Singh, A. M. Alshehri, and S. Dubey, 2022 Analysis of local fractional klein-gordon equations arising in relativistic fractal quantum mechanics.Waves in Random and Complex Media 0: 1–21.
- Jafari, H., H. K. Jassim, S. P. Moshokoa, V. M. Ariyan, and F. Tchier, 2016 Reduced differential transform method for partial differential equations within local fractional derivative operators. Advances in Mechanical Engineering 8: 1687814016633013.
- Kanth, A. R. and K. Aruna, 2009 Differential transform method for solving the linear and nonlinear klein–gordon equation. Computer Physics Communications 180: 708–711.
- Keskin, Y. and G. Oturanc, 2009 Reduced differential transform method for partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation 10: 741–750.
- Kolwankar, K. M. and A. D. Gangal, 1996 Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdisciplinary Journal of Nonlinear Science 6: 505–513.
- Kumar, D., J. Singh, and D. Baleanu, 2017 A hybrid computational approach for klein–gordon equations on cantor sets. Nonlinear Dynamics 87: 511–517.
- Sun, J., 2018 Analytical approximate solutions of (n+ 1)- dimensional fractal harry dym equations. Fractals 26: 1850094.
- Wang, K.-L., K.-J. Wang, and C.-H. He, 2019 Physical insight of local fractional calculus and its application to fractional kdv– burgers–kuramoto equation. Fractals 27: 1950122.
- Yang, A.-M., Y.-Z. Zhang, C. Cattani, G.-N. Xie, M. M. Rashidi, et al., 2014 Application of local fractional series expansion method to solve klein-gordon equations on cantor sets. In Abstract and Applied Analysis, volume 2014, Hindawi.
- Yang, X.-J., 2012 Advanced local fractional calculus and its applications. Yang, X.-J. and J. Tenreiro Machado, 2019 A new fractal nonlinear burgers’ equation arising in the acoustic signals propagation. Mathematical Methods in the Applied Sciences 42: 7539–7544.
- Zhang, Y., C. Cattani, and X.-J. Yang, 2015 Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains. Entropy 17: 6753–6764.
- Zhang, Y. and X.-J. Yang, 2016 An efficient analytical method for solving local fractional nonlinear pdes arising in mathematical physics. Applied Mathematical Modelling 40: 1793–1799.
Details
| Primary Language | English |
|---|---|
| Subjects | Applied Mathematics |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | December 31, 2023 |
| Published in Issue | Year 2023 Volume: 5 Issue: 4 |
Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science
The published articles in CHTA are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License 