A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics
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Kumbinarasaiah Srinivasa
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Mallanagoud Mulimani
Abstract
In this paper, we propose a numerical approach for solving the nonlinear fractional Klein–Gordon equation (FKGE), a model of significant importance in simulating nonlinear waves in quantum mechanics. Our method combines the Bernoulli wavelet collocation scheme with a functional integration matrix to obtain approximate solutions for the proposed model. Initially, we transform the main problem into a system of algebraic equations, which we solve using the Newton–Raphson method to extract the unknown coefficients and achieve the desired approximate solution. To theoretically validate our method, we conduct a comprehensive convergence analysis, demonstrating its uniform convergence. We perform numerical experiments on various examples with different parameters, presenting the results through tables and figures. Our findings indicate that employing more terms in the utilized techniques enhances accuracy. Furthermore, we compare our approach with existing methods from the literature, showcasing its performance in terms of computational cost, convergence rate, and solution accuracy. These examples illustrate how our techniques yield better approximate solutions for the nonlinear model at a low computational cost, as evidenced by the calculated CPU time and absolute error. Additionally, our method consistently provides better accuracy than other methods from the literature, suggesting its potential for solving more complex problems in physics and other scientific disciplines.
Acknowledgments
The authors would like to convey their thanks to the Editor and Reviewers for the helpful comments and suggestions which improved the work.
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Research ethics: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Competing interests: The authors state no conflict of interest.
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Research funding: The author expresses his affectionate thanks to the University Grants Commission (UGC), Govt. of India for the financial support under the UGC-BSR Research Start-Up Grant for 2021-2024:F.30-580/2021 (BSR) Dated: 23rd Nov. 2021.
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Data availability: Not applicable.
References
[1] K. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Netherlands, Elsevier, 1974.Suche in Google Scholar
[2] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore, World Scientific Publishing Company, 1999.10.1142/9789812817747Suche in Google Scholar
[3] R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” J. Phys. A: Math. Gen., vol. 37, no. 31, pp. R161–R208, 2004. https://doi.org/10.1088/0305-4470/37/31/r01.Suche in Google Scholar
[4] R. L. Bagley and P. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” J. Rheol., vol. 27, no. 3, pp. 201–210, 1983. https://doi.org/10.1122/1.549724.Suche in Google Scholar
[5] Y. Chatibi, E. H. El Kinani, and A. Ouhadan, “Variational calculus involving nonlocal fractional derivative with Mittag-Leffler kernel,” Chaos, Solitons Fractals, vol. 118, pp. 117–121, 2019. https://doi.org/10.1016/j.chaos.2018.11.017.Suche in Google Scholar
[6] B. A. Tayyan and A. H. Sakka, “Lie symmetry analysis of some conformable fractional partial differential equations,” Arab. J. Math., vol. 9, no. 1, pp. 201–212, 2020. https://doi.org/10.1007/s40065-018-0230-8.Suche in Google Scholar
[7] I. Podlubny, Fractional Differential Equations, USA, Academic Press, 1999.Suche in Google Scholar
[8] T. Q. Tang, Z. Shah, R. Jan, and E. Alzahrani, “Modeling the dynamics of tumor-immune cells interactions via fractional calculus,” Eur. Phys. J. Plus, vol. 137, no. 3, pp. 1–18, 2022. https://doi.org/10.1140/epjp/s13360-022-02591-0.Suche in Google Scholar
[9] A. Alshomrani, M. Z. Ullah, and D. Baleanu, “A new approach on the modelling, chaos control and synchronization of a fractional biological oscillator,” Adv. Differ. Equ., vol. 2021, no. 1, pp. 1–20, 2021. https://doi.org/10.1186/s13662-021-03224-z.Suche in Google Scholar
[10] A. Tahmineh, “Study on the application of the fractional calculus in pharmacokinetic modeling,” Technol. Innov. Eng. Res., vol. 2, pp. 52–70, 2022. https://doi.org/10.9734/bpi/tier/v2/16062d.Suche in Google Scholar
[11] M. Di Paola and A. Pirrotta, Fractional Calculus in Visco-Elasticity, Cham, Springer, 2022, pp. 261–272.10.1007/978-3-030-94195-6_16Suche in Google Scholar
[12] S. Arora, T. Mathur, S. Agarwal, K. Tiwari, and P. Gupta, “Applications of fractional calculus in computer vision: a survey,” Neurocomputing, vol. 489, pp. 407–428, 2022. https://doi.org/10.1016/j.neucom.2021.10.122.Suche in Google Scholar
[13] H. A. Hejazi, M. I. Khan, A. Raza, K. Smida, S. U. Khan, and I. Tlili, “Inclined surface slip flow of nanoparticles with subject to mixed convection phenomenon: fractional calculus applications,” J. Indian Chem. Soc., vol. 99, no. 7, 2022, Art. no. 100564. https://doi.org/10.1016/j.jics.2022.100564.Suche in Google Scholar
[14] M. Hassouna, E. H. El Kinani, and A. Ouhadan, “Fractional calculus: applications in rheology,” Fract. Order Syst., vol. 1, pp. 513–549, 2022. https://doi.org/10.1016/b978-0-12-824293-3.00018-1.Suche in Google Scholar
[15] E. Wei, et al.., “Nonlinear viscoelastic-plastic creep model of rock based on fractional calculus,” Adv. Civ. Eng., vol. 2022, no. 6, pp. 1–7, 2022. https://doi.org/10.1155/2022/3063972.Suche in Google Scholar
[16] E. Viera-Martin, J. F. Gómez-Aguilar, J. E. Solís-Pérez, J. A. Hernández-Pérez, and R. F. Escobar-Jiménez, “Artificial neural networks: a practical review of applications involving fractional calculus,” Eur. Phys. J.: Spec. Top., vol. 231, no. 10, pp. 2059–2095, 2022. https://doi.org/10.1140/epjs/s11734-022-00455-3.Suche in Google Scholar PubMed PubMed Central
[17] Z. Khan, F. Jarad, A. Khan, and H. Khan, “Nonlinear discrete fractional sum inequalities related to the theory of discrete fractional calculus with applications,” Adv. Differ. Equ., vol. 2021, no. 1, p. 100, 2021. https://doi.org/10.1186/s13662-021-03257-4.Suche in Google Scholar
[18] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, vol. 4, United Kingdom, Butterworth-Heinemann, 1982.Suche in Google Scholar
[19] V. G. Nordström, “Uber die moglichkeit, das elektromagnetische feld und das gravitationsfeld zu vereinigen,” Phys. Z., vol. 15, pp. 504–506, 1914.Suche in Google Scholar
[20] D. Bambusi and S. Cuccagna, “On dispersion of small energy solutions of the nonlinear Klein-Gordon equation with a potential,” Am. J. Math., vol. 133, no. 5, pp. 1421–1468, 2011. https://doi.org/10.1353/ajm.2011.0034.Suche in Google Scholar
[21] W. Bao and X. Dong, “Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime,” Numer. Math., vol. 120, no. 2, pp. 189–229, 2012. https://doi.org/10.1007/s00211-011-0411-2.Suche in Google Scholar
[22] N. A. Khan, F. Riaz, and A. Ara, “A note on soliton solutions of Klein-Gordon-Zakharov equation by variational approach,” Nonlinear Sci., vol. 5, no. 3, pp. 135–139, 2016. https://doi.org/10.1515/nleng-2016-0001.Suche in Google Scholar
[23] M. Tamsir and V. K. Srivastava, “Analytical study of time-fractional order Klein-Gordon equation,” Alexandria Eng. J., vol. 55, no. 1, pp. 561–567, 2016. https://doi.org/10.1016/j.aej.2016.01.025.Suche in Google Scholar
[24] A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional Klein-Gordon equation,” Signal Process., vol. 91, no. 3, pp. 446–451, 2011. https://doi.org/10.1016/j.sigpro.2010.04.016.Suche in Google Scholar
[25] M. R. Ali, U. Ghosh, S. Sarkar, and S. Das, “Analytic solution of the fractional order nonlinear Schrödinger equation and the fractional order Klein Gordon equation,” Differ. Equ. Dyn. Syst., vol. 30, no. 5, pp. 499–512, 2022. https://doi.org/10.1007/s12591-022-00596-w.Suche in Google Scholar
[26] J. Liu, M. Nadeem, M. Habib, and A. Akgül, “Approximate solution of nonlinear time-fractional Klein-Gordon equations using Yang transform,” Symmetry, vol. 14, no. 5, p. 907, 2022. https://doi.org/10.3390/sym14050907.Suche in Google Scholar
[27] A. S. Hendy, T. R. Taha, D. Suragan, and M. A. Zaky, “An energy-preserving computational approach for the semilinear space fractional damped Klein-Gordon equation with a generalized scalar potential,” Appl. Math. Model., vol. 108, pp. 512–530, 2022. https://doi.org/10.1016/j.apm.2022.04.009.Suche in Google Scholar
[28] Q. A. Huang, G. Zhang, and B. Wu, “Fully-discrete energy-preserving scheme for the space-fractional Klein–Gordon equation via Lagrange multiplier type scalar auxiliary variable approach,” Math. Comput. Simul., vol. 192, pp. 265–277, 2022. https://doi.org/10.1016/j.matcom.2021.09.002.Suche in Google Scholar
[29] Y. H. Youssri, “Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein-Gordon equation,” Int. J. Mod. Phys. C, vol. 33, no. 4, 2022, Art. no. 2250049. https://doi.org/10.1142/s0129183122500498.Suche in Google Scholar
[30] A. Rayal and S. R. Verma, “Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation,” Appl. Numer. Math., vol. 174, pp. 191–220, 2022. https://doi.org/10.1016/j.apnum.2022.01.015.Suche in Google Scholar
[31] A. M. A. El-Sayed, A. Elsaid, and D. Hammad, “A reliable treatment of homotopy perturbation method for solving the nonlinear Klein-Gordon equation of arbitrary (fractional) orders,” J. Appl. Math., vol. 2012, 2012, Art. no. 581481. https://doi.org/10.1155/2012/581481.Suche in Google Scholar
[32] C. K. Chui, Wavelets: A Mathematical Tool for Signal Analysis, United States, Society for Industrial and Applied Mathematics, 1997.10.1137/1.9780898719727Suche in Google Scholar
[33] L. I. Yuanlu, “Solving a nonlinear fractional differential equation using Chebyshev wavelets,” Commun. Nonlinear Sci. Numer. Simul., vol. 15, no. 9, pp. 2284–2292, 2010. https://doi.org/10.1016/j.cnsns.2009.09.020.Suche in Google Scholar
[34] S. Kumbinarasaiah and R. Yeshwanth, “A study on Chlamydia transmission in United States through the Haar wavelet technique,” Results Control Optim., vol. 12, 2024, Art. no. 100396. https://doi.org/10.1016/j.rico.2024.100396.Suche in Google Scholar
[35] M. P. Preetham and S. Kumbinarasaiah, “Analysis of hybrid nanofluid MHD flow and heat transfer between two surfaces in a rotating system in the presence of Joule heating and thermal radiation by Fibonacci wavelet,” J. Nanofluids, vol. 13, no. 1, pp. 1–4, 2024. https://doi.org/10.1166/jon.2024.2121.Suche in Google Scholar
[36] S. Kumbinarasaiah and M. Mulimani, “Fibonacci wavelets approach for the fractional Rosenau-Hyman equations,” Results Control Optim., vol. 11, 2023, Art. no. 100221. https://doi.org/10.1016/j.rico.2023.100221.Suche in Google Scholar
[37] J. Xie, “Numerical computation of fractional partial differential equations with variable coefficients utilizing the modified fractional Legendre wavelets and error analysis,” Math. Methods Appl. Sci., vol. 44, no. 8, pp. 7150–7164, 2021. https://doi.org/10.1002/mma.7252.Suche in Google Scholar
[38] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Amsterdam, Elsevier, 2006.Suche in Google Scholar
[39] E. Keshavarz, Y. Ordokhani, and M. Razzaghi, “The Bernoulli wavelets operational matrix of integration and its applications for the solution of linear and nonlinear problems in calculus of variations,” Appl. Math. Comput., vol. 351, pp. 83–98, 2019. https://doi.org/10.1016/j.amc.2018.12.032.Suche in Google Scholar
[40] S. Mohammadi, S. R. Hejazi, and H. Seifi, “Bernoulli wavelet method for numerical solution of Fokker-Planck-Kolmogorov time fractional differential equations,” Comput. Sci. Eng., vol. 2, no. 1, pp. 143–163, 2022.Suche in Google Scholar
[41] S. Kumbinarasaiah and M. Mulimani, “A study on the non-linear Murray equation through the Bernoulli wavelet approach,” Int. J. Appl. Comput. Math., vol. 9, no. 3, p. 40, 2023. https://doi.org/10.1007/s40819-023-01500-y.Suche in Google Scholar
[42] A. Khan, M. Faheem, and A. Raza, “Solution of third-order Emden-Fowler type equations using wavelet methods,” Eng. Comput., vol. 38, no. 6, pp. 2850–2881, 2021. https://doi.org/10.1108/ec-04-2020-0218.Suche in Google Scholar
[43] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Fractional-order Bernoulli wavelets and their applications,” Appl. Math. Model., vol. 40, nos. 17–18, pp. 8087–8107, 2016. https://doi.org/10.1016/j.apm.2016.04.026.Suche in Google Scholar
[44] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet,” J. Comput. Appl. Math., vol. 309, pp. 493–510, 2017. https://doi.org/10.1016/j.cam.2016.06.005.Suche in Google Scholar
[45] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations,” Numer. Algorithm, vol. 74, pp. 223–245, 2017. https://doi.org/10.1007/s11075-016-0146-3.Suche in Google Scholar
[46] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations,” Appl. Numer. Math., vol. 122, pp. 66–81, 2017. https://doi.org/10.1016/j.apnum.2017.08.002.Suche in Google Scholar
[47] F. Li, H. M. Baskonus, S. Kumbinarasaiah, G. Manohara, W. Gao, and E. Ilahan, “An efficient numerical scheme for biological models in the frame of Bernoulli wavelets,” Comput. Model. Eng. Sci., vol. 137, no. 3, pp. 2381–2408, 2023. https://doi.org/10.32604/cmes.2023.028069.Suche in Google Scholar
[48] S. Kumbinarasaiah and M. Mulimani, “Bernoulli wavelets numerical approach for the nonlinear Klein-Gordon and Benjamin-Bona-Mahony equation,” Int. J. Appl. Comput. Math., vol. 9, no. 5, p. 108, 2023. https://doi.org/10.1007/s40819-023-01555-x.Suche in Google Scholar
[49] A. H. Bhrawy, T. M. Taha, and J. A. T. Machado, “A review of operational matrices and spectral techniques for fractional calculus,” Nonlinear Dyn., vol. 81, no. 3, pp. 1023–1052, 2015. https://doi.org/10.1007/s11071-015-2087-0.Suche in Google Scholar
[50] M. Izadi, S. K. Yadav, and G. Methi, “Two efficient numerical techniques for solutions of fractional shallow water equation,” Partial Differ. Equ. Appl. Math., vol. 9, 2024, Art. no. 100619. https://doi.org/10.1016/j.padiff.2024.100619.Suche in Google Scholar
[51] M. Izadi and H. M. Srivastava, “Numerical treatments of nonlinear Burgers-Fisher equation via a combined approximation technique,” Kuwait J. Sci., vol. 51, no. 2, 2024, Art. no. 100163. https://doi.org/10.1016/j.kjs.2023.12.003.Suche in Google Scholar
[52] M. Izadi, A. El-Mesady, and W. Adel, “A novel Touchard polynomial-based spectral matrix collocation method for solving the Lotka-Volterra competition system with diffusion,” Math. Model. Numer. Simul. Appl., vol. 4, no. 1, pp. 37–65, 2024. https://doi.org/10.53391/mmnsa.1408997.Suche in Google Scholar
[53] R. M. Ganji, H. Jafari, M. Kgarose, and A. Mohammadi, “Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials,” Alexandria Eng. J., vol. 60, no. 5, pp. 4563–4571, 2021. https://doi.org/10.1016/j.aej.2021.03.026.Suche in Google Scholar
[54] A. M. Nagy, “Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method,” Appl. Math. Comput., vol. 310, pp. 139–148, 2017. https://doi.org/10.1016/j.amc.2017.04.021.Suche in Google Scholar
[55] R. T. Alqahtani, “Approximate solution of nonlinear fractional Klein-Gordon equation using spectral collocation method,” Appl. Math., vol. 6, no. 13, pp. 2175–2181, 2015. https://doi.org/10.4236/am.2015.613190.Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Analytical and numerical studies of a cancer invasion model with nonlocal diffusion
- Exploration of different multi-peak solitons and vibrant breather type waves’ solutions of nonlinear Schrödinger equations with advanced dispersion and cubic–quintic nonlinearity, unveiling their applications
- Leader-following consensus tracking control for fractional-order multi-motor systems via disturbance-observer
- A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics
- On the construction of various soliton solutions of two space-time fractional nonlinear models
- RANS study of surface roughness effects on ship resistance
- Complexiton and interaction solutions to a specific form of extended Calogero–Bogoyavlenskii–Schiff equation via its bilinear form
- An investigation on explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation
- The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Analytical and numerical studies of a cancer invasion model with nonlocal diffusion
- Exploration of different multi-peak solitons and vibrant breather type waves’ solutions of nonlinear Schrödinger equations with advanced dispersion and cubic–quintic nonlinearity, unveiling their applications
- Leader-following consensus tracking control for fractional-order multi-motor systems via disturbance-observer
- A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics
- On the construction of various soliton solutions of two space-time fractional nonlinear models
- RANS study of surface roughness effects on ship resistance
- Complexiton and interaction solutions to a specific form of extended Calogero–Bogoyavlenskii–Schiff equation via its bilinear form
- An investigation on explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation
- The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations
