[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.
Send comment about this article