Numerical solution of a coupled modified Korteweg–de Vries equation by the Galerkin method using quadratic B-splines

REFERENCES

• N.K.R. Amein and M.A. Ramadan, A small time solutions for the KdV equation using Bubnov–Galerkin finite element method, J. Egypt. Math. Soc. 19 (2011), pp. 118–125. doi: 10.1016/j.joems.2011.10.005
   Google Scholar
• A.H.A. Ali, The modified extended tanh-function method for solving coupled MKdV and coupled Hirota–Satsuma coupled KdV equations, Phys. Lett. A 363 (2007), pp. 420–425. doi: 10.1016/j.physleta.2006.11.076
   Web of Science ®Google Scholar
• D. Bhatta and M.I. Bhatti, Numerical solution of KdV equation using modified Bernstein polynomials, Appl. Math. Comput. 174 (2006), pp. 1255–1268. doi: 10.1016/j.amc.2005.05.049
   Web of Science ®Google Scholar
• D. Bo Cao, J. Ren Yan, and Y. Zhang, Exact solutions for a new coupled MKdV equations and a coupled KdV equations, Phys. Lett. A 297 (2002), pp. 68–74. doi: 10.1016/S0375-9601(02)00376-6
   Web of Science ®Google Scholar
• J.L. Bona, V.A. GDougalis, and O.A. Karakashian, Fully discrete Galerkin methods for the Korteweg–de Vries equation, Comput. Math. Appl. 7 (1986), pp. 859–884. doi: 10.1016/0898-1221(86)90031-3
   Web of Science ®Google Scholar
• A. Can***var, M. Sari, and I. Dag, A Taylor–Galerkin finite element method for the KdV equation using cubic B-splines, Physica B 405 (2010), pp. 3376–3383. doi: 10.1016/j.physb.2010.05.008
   Web of Science ®Google Scholar
• Y. Chen and H. An, Homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative, Appl. Math. Comput. 204 (2008), pp. 764–772. doi: 10.1016/j.amc.2008.07.018
   Web of Science ®Google Scholar
• I. Dag and Y. Dereli, Numerical solutions of KdV equation using radial basis functions, Appl. Math. Model. 32 (2008), pp. 535–546. doi: 10.1016/j.apm.2007.02.001
   Web of Science ®Google Scholar
• E. Fan, Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A 282 (2001), pp. 18–22. doi: 10.1016/S0375-9601(01)00161-X
   Web of Science ®Google Scholar
• L.R.T. Gardner, G.A. Gardner, and T. Geyikli, Solitary wave solutions of the MKdV equation, Comput. Methods Appl. Mech. Eng. 124 (1995), pp. 321–333. doi: 10.1016/0045-7825(94)00755-C
   Web of Science ®Google Scholar
• T. Geyikli and D. Kaya, An application for a modified KdV equation by the decomposition method and finite element method, Appl. Math. Comput. 169 (2005), pp. 971–981. doi: 10.1016/j.amc.2004.11.017
   Web of Science ®Google Scholar
• T. Geyikli and D. Kaya, Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation, Appl. Math. Comput. 169 (2005), pp. 146–156. doi: 10.1016/j.amc.2004.10.045
   Web of Science ®Google Scholar
• S. Guo and Y. Zhou, Auxiliary equation method for the mKdV equation with variable coefficients, Appl. Math. Comput. 217 (2010), pp. 1476–1483. doi: 10.1016/j.amc.2009.06.017
   Web of Science ®Google Scholar
• A.A. Halim and S.B. Leble, Analytical and numerical solution of coupled KdV–MKdV system, Chaos Solitons Fractals 19 (2004), pp. 99–108. doi: 10.1016/S0960-0779(03)00085-7
   Web of Science ®Google Scholar
• S. Hao, S. Xie, and S. Yi, The Galerkin method for the KdV equation using a new basis of smooth piecewise cubic polynomials, Appl. Math. Comput. 218 (2012), pp. 8659–8671. doi: 10.1016/j.amc.2012.02.027
   Web of Science ®Google Scholar
• B. Hong, New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation, Appl. Math. Comput. 215 (2009), pp. 2908–2913. doi: 10.1016/j.amc.2009.09.035
   Web of Science ®Google Scholar
• M. Inc, Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method, Chaos Solitons Fractals 34 (2007), pp. 1075–1081. doi: 10.1016/j.chaos.2006.04.069
   Web of Science ®Google Scholar
• D. Irk, I. Dag, and B. Saka, A small time solutions for the Korteweg–de Vries equation using spline approximation, Appl. Math. Comput. 173 (2006), pp. 834–846. doi: 10.1016/j.amc.2005.04.018
   Web of Science ®Google Scholar
• P.C. Jain, R. Shankar, and D. Bhardwaj, Numerical solution of the Korteweg–de Vries equation, Chaos Solitons Fractals 8 (1997), pp. 943–951. doi: 10.1016/S0960-0779(96)00135-X
   Web of Science ®Google Scholar
• A.J. Khattak, S.I.A. Tirmizi, and Siraj-ul-Islam, Application of mesh free collocation method to a class of nonlinear partial differential equations, Eng. Anal. Bound. Elem. 33 (2009), pp. 661–667. doi: 10.1016/j.enganabound.2008.10.001
   Web of Science ®Google Scholar
• A.G. Johnpillai and C.M. Khalique, Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 1207–1215. doi: 10.1016/j.cnsns.2010.06.025
   Web of Science ®Google Scholar
• F. Kangalgil and F. Ayaz, Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos Solitons Fractals 41 (2009), pp. 464–472. doi: 10.1016/j.chaos.2008.02.009
   Web of Science ®Google Scholar
• A.H. Khater, R.S. Temsah, and D.K. Callebaut, Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method, Math. Comput. Model. 48 (2008), pp. 1237–1253. doi: 10.1016/j.mcm.2008.02.001
   Web of Science ®Google Scholar
• A.J. Khattak and Siraj-ul-Islam, A comparative study of numerical solutions of a class of KdV equation, Appl. Math. Comput. 199 (2008), pp. 425–434. doi: 10.1016/j.amc.2007.10.002
   Web of Science ®Google Scholar
• D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), pp. 422–443.
   Google Scholar
• B.V.R. Kumar and M. Mehra, Time-accurate solutions of Korteweg–de Vries equation using wavelet Galerkin method, Appl. Math. Comput. 162 (2005), pp. 447–460. doi: 10.1016/j.amc.2003.12.104
   Web of Science ®Google Scholar
• R.M. Miura, Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), pp. 1202–1204. doi: 10.1063/1.1664700
   Web of Science ®Google Scholar
• R. Mokhtari and M. Mohseni, A meshless method for solving mKdV equation, Comput. Phys. Commun. 183 (2012), pp. 1259–1268. doi: 10.1016/j.cpc.2012.02.006
   Web of Science ®Google Scholar
• P.M. Prenter, Splines and Variational Methods, Wiley, New York, 1975.
   Google Scholar
• A.H. Salas, Exact solutions to mKdV equation with variable coefficients, Appl. Math. Comput. 216 (2010), pp. 2792–2798. doi: 10.1016/j.amc.2010.03.129
   Web of Science ®Google Scholar
• J. Sarma, Exact solutions for modified Korteweg–de Vries equation, Chaos Solitons Fractals 42 (2009), pp. 1599–1603. doi: 10.1016/j.chaos.2009.03.041
   Web of Science ®Google Scholar
• M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC, Boca Raton, FL, 1996.
   Google Scholar
• T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55(2) (1984), pp. 192–202. doi: 10.1016/0021-9991(84)90002-0
   Web of Science ®Google Scholar
• H. Triki and M.S. Ismail, Solitary wave solutions for a coupled pair of mKdV equations, Appl. Math. Comput. 217(4) (2010), pp. 1540–1548. doi: 10.1016/j.amc.2009.06.047
   Web of Science ®Google Scholar
• A. Wazwaz, New sets of solitary wave solutions to the KdV, mKdV, and the generalized KdV equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 331–339. doi: 10.1016/j.cnsns.2006.03.013
   Web of Science ®Google Scholar
• A. Wazwaz, A completely integrable system of coupled modified KdV equations, JNOPM 19(1) (2010), pp. 145–151. doi: 10.1142/S0218863510005030
   Web of Science ®Google Scholar
• Z. Yan, Approximate Jacobi elliptic function solutions of the modified KdV equation via the decomposition method, Appl. Math. Comput. 166 (2005), pp. 571–583. doi: 10.1016/j.amc.2004.07.004
   Web of Science ®Google Scholar
• J. Zhu, New explicit exact solutions of the mKdV equation using the variational iteration method combined with Exp-function method, Chaos Solitons Fractals 40 (2009), pp. 952–957. doi: 10.1016/j.chaos.2007.08.051
   Web of Science ®Google Scholar
• J. Zuo and Y. Zhang, Application of the G′/G-expansion method to solve coupled MKdV equations and coupled Hirota–Satsuma coupled KdV equations, Appl. Math. Comput. 217 (2011), pp. 5936–5941. doi: 10.1016/j.amc.2010.12.104
   Google Scholar