A Framework for the Numerical Solution of Klein–Gordon Eequations Using Three New Collocation Methods
محورهای موضوعی : فصلنامه ریاضی
1 - Department of Mathematics, Islamic Azad University, Isfahan(Khorasgan) branch, Isfahan, Iran.
کلید واژه: Klein−Gordon equations, Legendre interpolation, Legendre−Gauss−Radau collocation, Domain decomposition, Quasilinearization.,
چکیده مقاله :
The Klein−Gordon equation is a relativistic version of the Schr¨odinger equation and has a large range of applications in contemporary physics. In this paper, we present a unified framework for the numerical solution of nonlinear Klein−Gordon equations using three new collocation schemes. The solutions are approximated by the two−dimensional Legendre−Gauss−Radau interpolation directly. Moreover, using the properties of Jacobi polynomials, the partial derivatives of the solutions are expressed in terms of Jacobi polynomials that makes the approximations numerically more stable. We first derive a single−domain collocation method (SDC) that is suited for problems in small and moderate domains. Next, a multi−domain collocation method (MDC) is presented for large domains. The proposed MDC method solves the problem step by step in subdomains whilst is of the BN−stability. We then construct a single−domain iterative collocation method (SDIC). The proposed SDIC method is based on the quasilinearization (QL) technique and is suited for highly nonlinear problems. The key properties of the SDC, MDC and SDIC schemes are explained and the convergence of the QL approach applied to PDEs with the differential operator of the wave equation is assessed. Numerical examples are included to assess the accuracy and features of each collocation scheme.
The Klein−Gordon equation is a relativistic version of the Schr¨odinger equation and has a large range of applications in contemporary physics. In this paper, we present a unified framework for the numerical solution of nonlinear Klein−Gordon equations using three new collocation schemes. The solutions are approximated by the two−dimensional Legendre−Gauss−Radau interpolation directly. Moreover, using the properties of Jacobi polynomials, the partial derivatives of the solutions are expressed in terms of Jacobi polynomials that makes the approximations numerically more stable. We first derive a single−domain collocation method (SDC) that is suited for problems in small and moderate domains. Next, a multi−domain collocation method (MDC) is presented for large domains. The proposed MDC method solves the problem step by step in subdomains whilst is of the BN−stability. We then construct a single−domain iterative collocation method (SDIC). The proposed SDIC method is based on the quasilinearization (QL) technique and is suited for highly nonlinear problems. The key properties of the SDC, MDC and SDIC schemes are explained and the convergence of the QL approach applied to PDEs with the differential operator of the wave equation is assessed. Numerical examples are included to assess the accuracy and features of each collocation scheme.
[1] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics,
Springer, Verlag Berlin Heidelberg (1988).
[2] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in
Single Domains, Springer, Berlin (2006).
[3] Z.h. Wang, B. Guo, Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput. 52 (2012) 226–255.
[4] R.E. Bellman, R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems,
Elsevier Publishing, New York, NY, USA, 1965.
[5] A.M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein–Gordon equations, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 889–901.
[6] S. Abbasbandy, Numerical solution of non-linear Klein–Gordon equations by variational
iteration method, Int. J. Numer. Meth. Engng. 70 (2007) 876–881.
15
[7] J. Li, T. Wang, Analysis of a conservative fourth-order compact finite difference scheme for
the Klein–Gordon–Dirac equation, Comput. Appl. Math. 40 (114) (2021).
[8] S.G. Venkatesh, S. Raja Balachandar, S.K. Ayyaswamy, K. Krishnaveni, An efficient approach for solving Klein–Gordon equation arising in quantum field theory using wavelets,
Comput. Appl. Math. 37 (2018) 81–98.
[9] N. Hayashi, P.I. Naumkin, The initial value problem for the quadratic nonlinear KleinGordon equation, Adv. Math. Phys, vol. 2010, Article ID 504324, 35 pages, 2010.
https://doi.org/10.1155/2010/504324.
[10] I.L. Elviyanti, B.N. Pratiwi, A. Suparmi, C. Cari, The application of minimal length in KleinGordon equation with hulthen potential using asymptotic iteration method, Adv. Math.
Phys, vol. 2018, Article ID 9658679, 8 pages, 2018. https://doi.org/10.1155/2018/9658679.
[11] A.T. Deresse, Y.O. Mussa, A.K. Gizaw, Analytical solution of two-dimensional SineGordon equation, Adv. Math. Phys, vol. 2021, Article ID 6610021, 15 pages, 2021.
https://doi.org/10.1155/2021/6610021.
[12] S.A. Khuri, A. Sayfy, A spline collocation approach for the numerical solution of a generalized
nonlinear Klein–Gordon equation, J. Appl. Math. Comput. 216 (4) (2010) 1047–1056.
[13] M. Lakestani, M. Dehghan, Collocation and finite difference-collocation methods for the
solution of nonlinear Klein–Gordon equation, J. Computer Phys. Commun. 181 (8) (2010)
1392–1401.
[14] M. Dehghan , A. Mohebbi, Z. Asgari, Fourth-order compact solution of the nonlinear Klein–
Gordon equation, J. Numer. Algor. 52 (2009) 523–540.
[15] M. Dehghan , A. Ghesmati, Application of the dual reciprocity boundary integral equation
technique to solve the nonlinear Klein–Gordon equation, J. Computer Phys. Commun. 181
(2010) 1410–1418.
[16] J. Rashidinia, M. Ghasemia, R. Jalilian, Numerical solution of the nonlinear Klein–Gordon
equation, J. Comput. Appl. Math. 233 (2010) 1866–1878.
[17] S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some solutions of the linear and nonlinear
Klein–Gordon equations using the optimal homotopy asymptotic method, J. Appl. Math.
Comput. 216 (10) (2010) 2898–2909.
[18] J. Chen, J. Yang, H. Yang, Single soliton solutions of the coupled nonlinear Klein–Gordon
equations with power law nonlinearity, J. Appl. Math. Comput. 246 (2014) 184–191.
[19] F. Yin, T. Tian, J. Song, M. Zhu, Spectral methods using Legendre wavelets for nonlinear
Klein, Sine–Gordon equations, J. Comput and Appl. Math. 275 (2015) 321–334.
[20] M.S. Semary, H.N. Hassan, A new approach for a class of nonlinear boundary value problems
with multiple solutions, J. Association of Arab Universities for Basic and Applied Sciences.
17 (2015) 27–35.
[21] B.N. Saray, M. Lakestani, C. Cattani, Evaluation of mixed Crank–Nicolson scheme and
Tau method for the solution of Klein–Gordon equation, J. Appl. Math. Comput. 331 (2018)
169–181.
16
[22] E.Y. Deeba, S.A. Khuri, A decomposition method for solving the nonlinear Klein–Gordon
equation, J. Comput. Phys. 124 (4) (1996) 42–48.
[23] A.S.V. Ravi Kanth, K. Aruna, Differential transform method for solving the linear and
nonlinear Klein–Gordon equation, Comput. Phys. Commun. 180 (5) (2009) 708–711.
[24] E.S. Suharto, N.I.Z. Othman, M.A.S. Nasir, Nonstandard finite difference scheme associated with harmonic mean averaging for the nonlinear Klein–Gordon equation, J. Advanced
Research in Applied Sciences and Engineering Technology. 6 (1) (2017) 9–19.
[25] M. Dehghan, A. Shokri, Numerical solution of the nonlinear Klein–Gordon equation using
radial basis functions, J. Comput. Appl. Math. 230 (2) (2009) 400–410.
[26] A.Y. Majeed, A.M. Bewar, Approximate solutions for solving the Klein–Gordon and sine–
Gordon equations, J. Arab Universities for Basic and Applied Sciences. 22 (2017) 83–90.
[27] B. Batiha, M.S.M. Noorani, I. Hashim, Numerical solution of sine-Gordon equation by variational iteration method, Physics Letters A. 370 (5-6) (2007) 437–440.
[28] D. Kaya, A numerical solution of the sine–Gordon equation using the modified decomposition
method, J. Appl. Math. Comput. 143 (23) (2003) 309-317.
[29] M. Dehghan, A. Shokri, A numerical method for one-dimensional nonlinear Sine–Gordon
equation using collocation and radial basis functions, Numer. Methods Partial Diff. Eq. 24
(2)(2008) 687–698.
[30] M.O. Korpusov, A.N. Levashov, D.V. Lukyanenko, Analytical-Numerical Study of FiniteTime Blow-up of the Solution to the Initial-Boundary Value Problem for the Nonlinear
Klein-Gordon Equation, Comput. Math. Math. Phys. 90 (2020) 1452–1460.
[31] E.G. Ekomasov, R.K. Salimov, Pseudo-spectral fourier method as applied to finding localized
spherical soliton solutions of (3+1)-dimensional Klein-Gordon equations, Comput. Math.
Math. Phys. 56 (2016) 1604–1610.
