Numerical study of sine-Gordon equations using Bessel collocation method

Iranian Journal of Numerical Analysis and Optimization

مقاله 8، دوره 13، Issue 4 - شماره پیاپی 27، اسفند 2023، صفحه 728-746   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2023.81484.1229

نویسندگان

S. Arora؛ I. Bala^*

Department of Mathematics, Punjabi University, Patiala, 147002, Punjab, India.

چکیده

The nonlinear space time dynamics have been discussed in terms of a hyper-bolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel poly-nomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously dif-ferentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the present tech-nique has been discussed through absolute error, L2-norm, and L∞-norm.
The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form.

کلیدواژه‌ها

Sine-Gordon equation؛ Bessel polynomials؛ Wave equation؛ Or-thogonal Collocation

مراجع

[1] Ablowitz, M.J., Herbst, B.M. and Schober, C.M. Numerical simula-tion of quasi-periodic solutions of the sine-Gordon equation, Phys. D 87 (1995), 37–47.
[2] Ablowitz, M.J., Herbst, B.M. and Schober, C.M. On the numerical solu-tion of the sine-Gordon equation, J. Comput. Phys. 131 (1997), 354–367.
[3] Ablowitz, M.J., Kruskal, M.D. and Ladik, J.F. Solitary wave collisions, SIAM J. Appl. Math. 36 (1979), 428–443.
[4] Abramowitz, M. and Stegun, I.A. Handbook of mathematical functions with formulas and mathematical tables, with corrections Superintendent of Documents. National Bureau of Standards Applied Mathematics Se-ries, No. 55 U. S. Government Printing Office, Washington, D.C., 1965,
[5] Argyris, J. and Haase, M. An engineer’s guide to soliton phenomena: application of the finite element method, Comput. Meth. Appl. Mech. Eng. 61(1) (1987), 71–122.
[6] Arora, S., Dhaliwal, S.S. and Kukreja, V.K. Solution of two point bound-ary value problems using orthogonal collocation on finite elements, Appl. Math. Comput. 171 (2005), 358–370.
[7] Bailey, W.N. Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32 Stechert-Hafner, Inc., New York, 1964,
[8] Barone, A., Esposito, F., Magee, C.J., and Scott, A.C. Theory and ap-plications of the sine-Gordon equation, Riv. Nuovo Cimento. 1 (1971), 227–267.
[9] Dehghan, M. and Shokri, A. A numerical method for one-dimensional nonlinear Sine-Gordon equation using collocation and radial basis func-tions, Numer. Methods. Partial. Differ. Eq. 24(2) (2008), 687–698.
[10] Evans, W.D., Everitt, W.N., Kwon, K.H. and Littlejohn, L.L. Real or-thogonalizing weights for Bessel polynomials, J. Comput. Appl. Math. 49 (1993), 51–57.
[11] Everitt, W.N. and Markett, C. On a generalization of Bessel functions satisfying higher-order differential equations, J. Comput. Appl. Math. 54 (1994), 325–349.
[12] Ferguson, N.B. and Finlayson, B.A. Transient chemical reaction analysis by orthogonal collocation, Chem. Eng. J. 1(4) (1970), 327–336.
[13] Finlayson, B.A. Packed bed reactor analysis by orthogonal collocation, Chem. Eng. Sci. 26 (1971), 1081–1091.
[14] Gautschi, W. Numerical Analysis, Second Edition, Springer-Verlag, New York, 2012.

[15] Ilati M. and Dehghan, M. The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations, Eng. Anal. Bound. Elem. 52 (2015), 99–109.
[16] Jianga, C., Sun, J., Li, H. and Wang, Y. A fourth-order AVF method for the numerical integration of sine-Gordon equation, Appl. Math. Comput. 313 (2017), 144–158.
[17] Koornwinder, T.H. Orthogonal polynomial with weight function (1 −x)α(1 − x)β + M δ(x − 1) + N δ(x − 1), Can. Math. Bull. 27(2) (1984), 205–214.
[18] Kumar, D., Singh, J., Kumar and S., Sushila Numerical computation of Klein–Gordon equations arising in quantum field theory by using homo-topy analysis transform method, Alex. Eng. J. 53 (2014), 469–474.
[19] Martin-Vergara, F., Rus, F. and Villatoro, F.R. Padé numerical schemes for the sine-Gordon equation, Appl. Math. Comput. 358 (2019), 232–243.
[20] McLachlan, N.W. Bessel functions for engineers, University of Illinois, Oxford University Press, London, England. 1961.
[21] Miller, J.J.H., O’Riordan, E. and Shishkin, G.I. Fitted Numerical meth-ods for singular perturbation problems, error estimate in the maximum norm for linear problems in one and two dimensions, World Scientific, 1996.
[22] Mishra, P., Sharma, K.K., Pani, A.K. and Fairweather, G. Orthogonal spline collocation for singularly perturbed reaction diffusion problems in one dimension, Int. J. Numer. Anal. Model. 16(4) (2019), 647–667.
[23] Mittal, R.C., and Bhatia, R. Numerical solution of nonlinear Sine-Gordon equation by modified cubic B-Spline collocation method, Int. J. Partial Differ. Equ. 2014 (2014), 1–8.
[24] Perring, J.K. and Skyrme, T.H.R. A model unified field equation, Nucl. Phys. 31 (1962), 550–555.
[25] Prenter, P.M. Splines and variational methods, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975.
[26] Rainville, E.D. Special functions, Reprint of 1960 first edition. Chelsea Publishing Co., Bronx, N.Y., 1971.
[27] Saray, B.N., Lakestani, M. and Cattani, C. Evaluation of mixed Crank-Nicolson scheme and tau method for the solution of Klein-Gordon equa-tion, App. Math. Comput. 331 (2018), 169–181.
[28] Scott, A.C., Chu, F.Y.F. and McLaughlin, D.W. The soliton: A new concept in applied science, Proc. IEEE 61 (1973), 1443–1483.

[29] Shan, Y., Liu, W. and Wu, B. Space-time Legendre-Gauss-Lobatto col- location method for two-dimensional generalized sine-Gordon equation, Appl. Numer. Math. 122 (2017), 92–107.
[30] Shukla, H.S. and Tamsir M. Numerical solution of nonlinear Sine–Gordon equation by using the modified cubic B-spline differential quadra-ture method, Beni-Seuf Univ. J. Basic Appl. Sci. 7 (2018), 359–366.
[31] Shukla, H.S., Tamsir, M. and Srivastava, V. K. Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method, AIP Adv. 5(1) (2015), 017121.
[32] Sirendaoreji A new auxiliary equation and exact travelling wave solutions of nonlinear equations, Phys. Lett. A 356(2) (2006), 124–130.
[33] Sirendaoreji Auxiliary equation method and new solutions of Klein–Gordon equations, Chaos Solitons Fractals 31(4) (2007), 943–950.
[34] Sneddon, I. N. Special functions of mathematical physics and chemistry, Oliver and Boyd, Edinburgh-London; Interscience Publishers, Inc., New York, 1956.
[35] Stempak, K. A weighted uniform Lp-estimate of Bessel functions: a note on a paper of K. Guo: “A uniform Lp estimate of Bessel functions and distributions supported on Sn−1” [Proc. Amer. Math. Soc. 125 (1997),
no. 5, 1329–1340; MR1363462 (97g:46047)], Proc. Amer. Math. Soc. 128(10) (2000), 2943–2945.
[36] Strauss, W. and Vazquez, L. Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys. 28 (1978), 271–278.
[37] Tasbozan, O., Yagmurlu, N.M.,Ucar, Y. and Esen, A. Numerical solu-tions of the Sine-Gordon equation by collocation method, Sohag J. Math. 3(1) (2016), 1–6.
[38] Villadsen, J.V. and Stewart, W.E. Solution of boundary value problem by orthogonal collocation, Chem. Eng. Sci. 22 (1967), 1483–1501.
[39] Watson, G.N. A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.
[40] Wazwaz, A.M. New travelling wave solutions to the Boussinesq and the Klein–Gordon equations, Commun. Nonlinear Sci. Numer. Simul. 13(5) (2008), 889–901.
[41] Wingate, C. Numerical search for a ϕ4 breather mode, SIAM J. Appl. Math. 43 (1983), 120–140.

[42] Yüzbaşı, Ş. A numerical approach for solving a class of the nonlin-ear Lane–Emden type equations arising in astrophysics, Math. Methods Appl. Sci. 34(8) (2011), 2218–2230.
[43] Yüzbaşı, Ş. A numerical approach for solving the high-order linear singu-lar differential-difference equations, Comput. Math. Appl. 62(5) (2011), 2289–2303.
[44] Yüzbaşı, Ş. A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential-difference equations, Comput. Math. Appl. 64(6) (2012), 1691–1705.
[45] Yüzbaşı, Ş. Bessel collocation approach for solving continuous population models for single and interacting species, Appl. Math. Model. 36 (2012), 3787–3802.
[46] Yüzbaşı, Ş., Şahin, N. and Sezer, M. A Bessel polynomial approach for solving linear neutral delay differential equations with variable coef-ficients, J. Adv. Res. Appl. Math. 3(1) (2011), 81–101.
[47] Yüzbaşı, Ş., Şahin, N. and Sezer, M. Bessel matrix method for solving high-order linear Fredholm integro-differential equations, J. Adv. Res. Appl. Math. 3(2) (2011), 23–47.
[48] Yüzbaşı, Ş., Şahin, N. and Sezer, M. Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases, Comput. Math. Appl. 61(10) (2011), 3079–3096.