Numerical Solution of Fractional Sine-Gordon Equation Using Spectral Method and Homogenization

References

[1] A. G. Popov and E. V. Maevskii, Analytical approaches to the study of the sine-Gordon equation and pseudospherical surfaces. J. Math. Sci. 142(5) (2007), 2377–2418.10.1007/s10958-007-0183-5Suche in Google Scholar

[2] R. Rajaraman, Solitons and instantons: an introduction to solitons and instantons in quantum field theory, Elsevier, Amsterdam, 1987.Suche in Google Scholar

[3] T. A. Kurura, O. Okoya and O. Ongati, Finite difference solution of (2+ 1)-dimensional sine-Gordon equation: a model for investigating the effects of varying surface damping parameter on Josephson current flowing through the long Josephson junction, Int. J. Sci. Res. 5(12) (2016), 1249–1255.Suche in Google Scholar

[4] M. Remoissenet, Waves called solitons. Concepts and Experiments, Springer, Berlin, 2003.Suche in Google Scholar

[5] A. C. Scott, Propagation of magnetic flux on a long Josephson tunnel junction. Il Nuovo Cimento B 69(2) (1970), 241–261.10.1007/BF02710988Suche in Google Scholar

[6] R. Hirota, Exact three-soliton solution of the two-dimensional sine-Gordon equation. J. Phys. Soc. Jpn. 35(5) (1973), 1566–1566.10.1143/JPSJ.35.1566Suche in Google Scholar

[7] A. M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. Appl. Math. Comput. 167(2) (2005), 1196–1210.10.1016/j.amc.2004.08.005Suche in Google Scholar

[8] S. Liu, Z. Fu and S. Liu, Exact solutions to sine-Gordon-type equations. Phys. Lett. A 351 (2006), 59–63.10.1016/j.physleta.2005.10.054Suche in Google Scholar

[9] G. L. Lamb Jr., Elements of soliton theory, Wiley-Interscience, New York, 1980.Suche in Google Scholar

[10] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer, New York, 2009.10.1007/978-3-642-00251-9Suche in Google Scholar

[11] C. Zheng, Numerical solution to the sine-Gordon equation defined on the whole real axis. SIAM J. Sci. Comput. 29(6) (2007), 2494–2506.10.1137/050640643Suche in Google Scholar

[12] D. Kaya, A numerical solution of the sine-Gordon equation using the modified decomposition method. Appl. Math. Comput. 143(2) (2003), 309–317.10.1016/S0096-3003(02)00363-6Suche in Google Scholar

[13] S. Abbasbandy, Numerical solution of nonlinear Klein Gordon equations by variational iteration method. Int. J. Numer. Method Eng. 70(7) (2007), 876–881.10.1002/nme.1924Suche in Google Scholar

[14] B. Batiha, M. S. M. Noorani and I. Hashim, Numerical solution of sine-Gordon equation by variational iteration method. Phys. Lett. A 370(5) (2007), 437–440.10.1016/j.physleta.2007.05.087Suche in Google Scholar

[15] J. Biazar and F. Mohammadi, Application of differential transform method to the sine-Gordon equation. Int. J. Nonlinear Sci. 10(2) (2010), 190–195.Suche in Google Scholar

[16] J. Rashidinia and R. Mohammadi, Tension spline solution of nonlinear sine-Gordon equation. Numer. Algorithms 56(1) (2011), 129–142.10.1007/s11075-010-9377-xSuche in Google Scholar

[17] H. S. Shukla and M. Tamsir, Numerical solution of nonlinear sine-Gordon equation by using the modified cubic B-spline differential quadrature method. Beni-Suef Univ. J. Basic Appl. Sci., 2016 (accepted for publication).10.1063/1.4906256Suche in Google Scholar

[18] M. T. Darvishi, F. Khani, S. Hamedi-Nezhad and S. W. Ryu, 2010. New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations. Int. J. Comput. Math. 87(4), 908–919.10.1080/00207160802247596Suche in Google Scholar

[19] T. R. Rao, Numerical solution of sine-gordon equations through reduced differential transform method. Global J. Pure Appl. Math. 13(7) (2017), 3879–3888.Suche in Google Scholar

[20] M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simul. 79(3) (2008), 700–715.10.1016/j.matcom.2008.04.018Suche in Google Scholar

[21] K. Djidjeli, W. G. Price and E. H. Twizell, Numerical solutions of a damped sine-Gordon equation in two space variables. J. Eng. Math. 29(4) (1995), 347–369.10.1007/BF00042761Suche in Google Scholar

[22] F. Yin, T. Tian, J. Song and M. Zhu, Spectral methods using Legendre wavelets for nonlinear Klein Sine-Gordon equations. J. Comput. Appl. Math. 275 (2015), 321–334.10.1016/j.cam.2014.07.014Suche in Google Scholar

[23] E. Keshavarz, Y. Ordokhani and M. Razzaghi, A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control 22(18) (2016), 3889—3903.10.1177/1077546314567181Suche in Google Scholar

[24] Z. Liu, S. Lü and F. Liu, Fully discrete spectral methods for solving time fractional nonlinear Sine-Gordon equation with smooth and non-smooth solutions. Appl. Math. Comput. 333 (2018), 213—224.10.1016/j.amc.2018.03.069Suche in Google Scholar

[25] M. Dehghan, M. Abbaszadeh and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein Gordon equations. Eng. Anal. Boundary Elem. 50 (2015), 412–434.10.1016/j.enganabound.2014.09.008Suche in Google Scholar

[26] K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol. 111, Elsevier, Dover Publications, New York, 1974.Suche in Google Scholar

[27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.Suche in Google Scholar

[28] E. H. Doha, A. H. Bhrawy and S. S. Ezz-Eldien, A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36(10) (2012), 4931–4943.10.1016/j.apm.2011.12.031Suche in Google Scholar

[29] E. Kreyszig, Introductory functional analysis with applications, Wiley, New York, 1989.Suche in Google Scholar

[30] M. Behroozifar and A. Sazmand, An approximate solution based on Jacobi polynomials for time-fractional convection diffusion equation. Appl. Math. Comput. 296 (2017), 1–17.10.1016/j.amc.2016.09.028Suche in Google Scholar

[31] E. H. Doha, A. H. Bhrawy and R. M. Hafez, On shifted Jacobi spectral method for high-order multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 17(10) (2012), 3802–3810.10.1016/j.cnsns.2012.02.027Suche in Google Scholar