Applications of the nonlinear Klein/Sinh-Gordon equations in modern physics: a numerical study

Ihteram Ali; Imtiaz Ahmad; Ihteram Ali; Imtiaz Ahmad

doi:10.3934/mmc.2024029

Mathematical Modelling and Control

2024, Volume 4, Issue 3: 361-373. doi: 10.3934/mmc.2024029

Previous Article Next Article

Research article

Applications of the nonlinear Klein/Sinh-Gordon equations in modern physics: a numerical study

Ihteram Ali ¹,
Imtiaz Ahmad ^{2
,
,}

1.
Department of Mathematics, Women University Swabi, Swabi 23430, Pakistan
2.
Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Kajang 43000, Selangor, Malaysia

Received: 27 August 2023 Revised: 18 December 2023 Accepted: 12 January 2024 Published: 12 September 2024

In this article, a hybrid numerical scheme based on Lucas and Fibonacci polynomials in combination with Störmer's method for the solution of Klein/Sinh-Gordon equations is proposed. Initially, the problem is transformed to a time-discrete form by using Störmer's technique. Then, with the help of Fibonacci polynomials, we approximate the derivatives of the function. The suggested technique is validated to both one and two-dimensional problems. The resultant findings are compared with existing numerical solutions and presented in a tabular form. The comparison reveals the superior accuracy of the scheme. The numerical convergence of the scheme is computed in each example.
- Klein/Sinh-Gordon equation,
- finite differences,
- Lucas polynomials,
- Fibonacci polynomials
Citation: Ihteram Ali, Imtiaz Ahmad. Applications of the nonlinear Klein/Sinh-Gordon equations in modern physics: a numerical study[J]. Mathematical Modelling and Control, 2024, 4(3): 361-373. doi: 10.3934/mmc.2024029

Related Papers:

Abstract

In this article, a hybrid numerical scheme based on Lucas and Fibonacci polynomials in combination with Störmer's method for the solution of Klein/Sinh-Gordon equations is proposed. Initially, the problem is transformed to a time-discrete form by using Störmer's technique. Then, with the help of Fibonacci polynomials, we approximate the derivatives of the function. The suggested technique is validated to both one and two-dimensional problems. The resultant findings are compared with existing numerical solutions and presented in a tabular form. The comparison reveals the superior accuracy of the scheme. The numerical convergence of the scheme is computed in each example.

References

[1]	N. H. Sweilam, M. M. Khader, M. Adel, On the stability analysis of weighted average finite difference methods for fractional wave equation, Fract. Differ. Calc., 2 (2012), 17–29. https://doi.org/10.7153/fdc-02-02 doi: 10.7153/fdc-02-02
[2]	M. Adel, M. Khader, H. Ahmad, T. Assiri, Approximate analytical solutions for the blood ethanol concentration system and predator-prey equations by using variational iteration method, AIMS Math., 8 (2023), 19083–19096. https://doi.org/10.3934/math.2023974 doi: 10.3934/math.2023974
[3]	Siraj-ul-Islam, I. Ahmad, Local meshless method for PDEs arising from models of wound healing, Appl. Math. Model., 48 (2017), 688–710. https://doi.org/10.1016/j.apm.2017.04.015 doi: 10.1016/j.apm.2017.04.015
[4]	M. Adel, M. M. Khader, S. Algelany, High-dimensional chaotic lorenz system: numerical treatment using changhee polynomials of the appell type, Fractal Fract., 7 (2023), 398. https://doi.org/10.3390/fractalfract7050398 doi: 10.3390/fractalfract7050398
[5]	M. Adel, M. E. Ramadan, H. Ahmad, T. Botmart, Sobolev-type nonlinear Hilfer fractional stochastic differential equations with noninstantaneous impulsive, AIMS Math., 7 (2022), 20105–20125. https://doi.org/10.3934/math.20221100 doi: 10.3934/math.20221100
[6]	X. Liu, M. Ahsan, M. Ahmad, M. Nisar, X. Liu, I. Ahmad, et al., Applications of Haar wavelet-finite difference hybrid method and its convergence for hyperbolic nonlinear Schrödinger equation with energy and mass conversion, Energies, 14 (2021), 7831. https://doi.org/10.3390/en14237831 doi: 10.3390/en14237831
[7]	M. Ahsan, I. Ahmad, M. Ahmad, I. Hussian, A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation, Math. Comput. Simul., 165 (2019), 13–25. https://doi.org/10.1016/j.matcom.2019.02.011 doi: 10.1016/j.matcom.2019.02.011
[8]	I. Ahmad, M. Ahsan, I. Hussain, P. Kumam, W. Kumam, Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11 (2019), 394. https://doi.org/10.3390/sym11030394 doi: 10.3390/sym11030394
[9]	F. Wang, J. Zhang, I. Ahmad, A. Farooq, H. Ahmad, A novel meshfree strategy for a viscous wave equation with variable coefficients, Front. Phys., 9 (2021), 701512. https://doi.org/10.3389/fphy.2021.701512 doi: 10.3389/fphy.2021.701512
[10]	M. Ahsan, S. Lin, M. Ahmad, M. Nisar, I. Ahmad, H. Ahmed, et al., A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation, Open Phys., 19 (2021), 722–734. https://doi.org/10.1515/phys-2021-0080 doi: 10.1515/phys-2021-0080
[11]	P. Thounthong, M. N. Khan, I. Hussain, I. Ahmad, P. Kumam, Symmetric radial basis function method for simulation of elliptic partial differential equations, Mathematics, 6 (2018), 327. https://doi.org/10.3390/math6120327 doi: 10.3390/math6120327
[12]	J. F. Li, I. Ahmad, H. Ahmad, D. Shah, Y. M. Chu, P. Thounthong, et al., Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method, Open Phys., 18 (2020), 1063–1072. https://doi.org/10.1515/phys-2020-0222 doi: 10.1515/phys-2020-0222
[13]	I. Ahmad, M. Riaz, M. Ayaz, M. Arif, S. Islam, P. Kumam, Numerical simulation of partial differential equations via local meshless method, Symmetry, 11 (2019), 257. https://doi.org/10.3390/sym11020257 doi: 10.3390/sym11020257
[14]	I. Ahmad, I. Ali, R. Jan, S. A. Idris, M. Mousa, Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater, Plos One, 18 (2023), e0294348. https://doi.org/10.1371/journal.pone.0294348 doi: 10.1371/journal.pone.0294348
[15]	I. Ahmad, Siraj-ul-Islam, Mehnaz, S. Zaman, Local meshless differential quadrature collocation method for time-fractional PDEs, Discrete Cont. Dyn. Syst., 13 (2020), 2641–2654. https://doi.org/10.3934/dcdss.2020223 doi: 10.3934/dcdss.2020223
[16]	Ö. Oruç, A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin-Bona-Mahony-Burgers equation, Comput. Math. Appl., 74 (2017), 3042–3057. https://doi.org/10.1016/j.camwa.2017.07.046 doi: 10.1016/j.camwa.2017.07.046
[17]	I. Ahmad, A. A. Bakar, R. Jan, S. Yussof, Dynamic behaviors of a modified computer virus model: insights into parameters and network attributes, Alex. Eng. J., 103 (2024), 266–277. https://doi.org/10.1016/j.aej.2024.06.009 doi: 10.1016/j.aej.2024.06.009
[18]	H. Ahmad, T. A. Khan, I. Ahmad, P. S. Stanimirović, Y. M. Chu, A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations, Results Phys., 19 (2020), 103462. https://doi.org/10.1016/j.rinp.2020.103462 doi: 10.1016/j.rinp.2020.103462
[19]	A. M. Wazwaz, The tanh and the Sine-Cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation, Appl. Math. Comput., 167 (2005), 1179–1195. https://doi.org/10.1016/j.amc.2004.08.006 doi: 10.1016/j.amc.2004.08.006
[20]	M. Dehghan, A. Shokri, Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, Comput. Appl. Math., 230 (2009), 400–410. https://doi.org/10.1016/j.cam.2008.12.011 doi: 10.1016/j.cam.2008.12.011
[21]	E. Yomba, A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations, Phys. Lett. A, 372 (2008), 1048–1060. https://doi.org/10.1016/j.physleta.2007.09.003 doi: 10.1016/j.physleta.2007.09.003
[22]	X. Antoine, X. Zhao, Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes, J. Comput. Phys., 448 (2022), 110728. https://doi.org/10.1016/j.jcp.2021.110728 doi: 10.1016/j.jcp.2021.110728
[23]	N. J. Mauser, Y. Zhang, X. Zhao, On the rotating nonlinear Klein-Gordon equation: nonrelativistic limit and numerical methods, Multiscale Model. Simul., 18 (2020), 999–1024. https://doi.org/10.1137/18M1233509 doi: 10.1137/18M1233509
[24]	W. Bao, X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886. https://doi.org/10.1016/j.jcp.2019.108886 doi: 10.1016/j.jcp.2019.108886
[25]	S. M. El-Sayed, The decomposition method for studying the Klein-Gordon equation, Chaos Solitons Fract., 18 (2003), 1025–1030. https://doi.org/10.1016/S0960-0779(02)00647-1 doi: 10.1016/S0960-0779(02)00647-1
[26]	M. Dehghan, A. Mohebbi, Z. Asgari, Fourth-order compact solution of the nonlinear Klein-Gordon equation, Numer. Algorithms, 52 (2009), 523–540. https://doi.org/10.1007/s11075-009-9296-x doi: 10.1007/s11075-009-9296-x
[27]	N. Wang, M. Li, C. Huang, Unconditional energy dissipation and error estimates of the SAV Fourier spectral method for nonlinear fractional generalized wave equation, J. Sci. Comput., 88 (2021), 19. https://doi.org/10.1007/s10915-021-01534-8 doi: 10.1007/s10915-021-01534-8
[28]	S. A. Khuri, A. Sayfy, A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation, Appl. Math. Comput., 216 (2010), 1047–1056. https://doi.org/10.1016/j.amc.2010.01.122 doi: 10.1016/j.amc.2010.01.122
[29]	A. Hussain, S. Haq, M. Uddin, Numerical solution of Klein-Gordon and sine-Gordon equations by meshless method of lines, Eng. Anal. Bound. Elem., 37 (2013), 1351–1366. https://doi.org/10.1016/j.enganabound.2013.07.001 doi: 10.1016/j.enganabound.2013.07.001
[30]	I. Ahmad, M. Ahsan, I. Hussain, P. Kumam, W. Kumam, Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11 (2019), 394. https://doi.org/10.3390/sym11030394 doi: 10.3390/sym11030394
[31]	H. Kheiri, A. Jabbari, Exact solutions for the double sinh-Gordon and generalized form of the double sinh-Gordon equations by using (G'/G)-expansion method, Turk. J. Phys., 34 (2011), 73–82. https://doi.org/10.3906/fiz-0909-7 doi: 10.3906/fiz-0909-7
[32]	A. M. Wazwaz, Exact solutions for the generalized sinh-Gordon and the generalized sinh-Gordon equations, Chaos Solitons Fract., 28 (2006), 127–135. https://doi.org/10.1016/j.chaos.2005.05.017 doi: 10.1016/j.chaos.2005.05.017
[33]	X. Li, S. Zhang, Y. Wang, H. Chen, Analysis and application of the element-free galerkin method for nonlinear sinh-Gordon and generalized sinh-Gordon equations, Comput. Math. Appl., 71 (2016), 1655–1678. https://doi.org/10.1016/j.camwa.2016.03.007 doi: 10.1016/j.camwa.2016.03.007
[34]	M. Dehghan, A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simul., 79 (2008), 700–715. https://doi.org/10.1016/j.matcom.2008.04.018 doi: 10.1016/j.matcom.2008.04.018
[35]	C. W. Chang, C. S. Liu, An implicit Lie-group iterative scheme for solving the nonlinear Klein-Gordon and sine-Gordon equations, Appl. Math. Modell., 40 (2016), 1157–1167. https://doi.org/10.1016/j.apm.2015.06.028 doi: 10.1016/j.apm.2015.06.028
[36]	M. Dehghan, M. Abbaszadeh, A. Mohebbi, The numerical solution of the two-dimensional sinh-Gordon equation via three meshless methods, Eng. Anal. Bound. Elem., 51 (2015), 220–235. https://doi.org/10.1016/j.enganabound.2014.10.015 doi: 10.1016/j.enganabound.2014.10.015
[37]	P. L. Christiansen, P. S. Lomdahl, Numerical study of 2+1 dimensional sine-Gordon solitons, Phys. D, 2 (1981), 482–494. https://doi.org/10.1016/0167-2789(81)90023-3 doi: 10.1016/0167-2789(81)90023-3
[38]	R. Jiwari, S. Pandit, R. Mittal, Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Comput. Phys. Commun., 183 (2012), 600–616. https://doi.org/10.1016/j.cpc.2011.12.004 doi: 10.1016/j.cpc.2011.12.004
[39]	W. Abd-Elhameed, Y. Youssri, Connection formulae between generalized Lucas polynomials and some Jacobi polynomials: application to certain types of fourth-order BVPs, Int. J. Appl. Comput. Math., 6 (2020), 45. https://doi.org/10.1007/s40819-020-0799-4 doi: 10.1007/s40819-020-0799-4
[40]	M. Çetin, M. Sezer, C. Güler, Lucas polynomial approach for system of high-order linear differential equations and residual error estimation, Math. Probl. Eng., 2015 (2015), 625984. https://doi.org/10.1155/2015/625984 doi: 10.1155/2015/625984
[41]	F. Mirzaee, S. F. Hoseini, Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations, Appl. Math. Comput., 273 (2016), 637–644. https://doi.org/10.1016/j.amc.2015.10.035 doi: 10.1016/j.amc.2015.10.035
[42]	N. Bayku, M. Sezer, Hybrid Taylor-Lucas collocation method for numerical solution of high-order Pantograph type delay differential equations with variables delays, Appl. Math. Inf. Sci., 11 (2017), 1795–1801. https://doi.org/10.18576/amis/110627 doi: 10.18576/amis/110627
[43]	N. A. Nayied, F. A. Shah, M. A. Khanday, Fibonacci wavelet method for the numerical solution of nonlinear reaction-diffusion equations of Fisher-type, J. Math., 2023 (2023), 1705607. https://doi.org/10.1155/2023/1705607 doi: 10.1155/2023/1705607
[44]	F. A. Shah, M. Irfan, K. S. Nisar, R. T. Matoog, E. E. Mahmoud, Fibonacci wavelet method for solving time-fractional telegraph equations with dirichlet boundary conditions, Results Phys., 24 (2021), 104123. https://doi.org/10.1016/j.rinp.2021.104123 doi: 10.1016/j.rinp.2021.104123
[45]	H. M. Srivastava, F. A. Shah, R. Abass, An application of the gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation, Russ. J. Math. Phys., 26 (2019), 77–93. https://doi.org/10.1134/S1061920819010096 doi: 10.1134/S1061920819010096
[46]	K. S. Nisar, F. A. Shah, A numerical scheme based on gegenbauer wavelets for solving a class of relaxation-oscillation equations of fractional order, Math. Sci., 17 (2023), 233–245. https://doi.org/10.1007/s40096-022-00465-1 doi: 10.1007/s40096-022-00465-1
[47]	Ö. Oruç, A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 14–25. https://doi.org/10.1016/j.cnsns.2017.09.006 doi: 10.1016/j.cnsns.2017.09.006
[48]	I. Ali, S. Haq, K. S. Nisar, D. Baleanu, An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations, Adv. Differ. Equations, 2021 (2021), 43. https://doi.org/10.1186/s13662-020-03160-4 doi: 10.1186/s13662-020-03160-4
[49]	S. Haq, I. Ali, Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials, Eng. Comput., 38 (2021), 2059–2068. https://doi.org/10.1007/s00366-021-01327-5 doi: 10.1007/s00366-021-01327-5
[50]	S. Haq, I. Ali, K. S. Nisar, A computational study of two-dimensional reaction-diffusion Brusselator system with applications in chemical processes, Alex. Eng. J., 60 (2021), 4381–4392. https://doi.org/10.1016/j.aej.2021.02.064 doi: 10.1016/j.aej.2021.02.064
[51]	I. Ahmad, A. A. Bakar, I. Ali, S. Haq, S. Yussof, A. H. Ali, Computational analysis of time-fractional models in energy infrastructure applications, Alex. Eng. J., 82 (2023), 426–436. https://doi.org/10.1016/j.aej.2023.09.057 doi: 10.1016/j.aej.2023.09.057
[52]	I. Ali, S. Haq, S. F. Aldosary, K. S. Nisar, F. Ahmad, Numerical solution of one-and two-dimensional time-fractional Burgers equation via lucas polynomials coupled with finite difference method, Alex. Eng. J., 61 (2022), 6077–6087. https://doi.org/10.1016/j.aej.2021.11.032 doi: 10.1016/j.aej.2021.11.032
[53]	J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral methods for time-dependent problems, Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511618352

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)