Abstract
The purpose of the present study is to obtain numerical solutions of the modified Korteweg–de Vries equation (mKdV) by using mixed Crank–Nicolson scheme and differential quadrature method based on quintic B-spline basis functions. In order to control the effectiveness and accuracy of the present approximation, five well-known test problems, namely, single soliton, interaction of double solitons, interaction of triple solitons, Maxwellian initial condition and tanh initial condition, are used. Furthermore, the error norms \(L_{2}\) and \( L _{\infty }\) are calculated for single soliton solutions to measure the efficiency and the accuracy of the present method. At the same time, the three lowest conservation quantities are calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants are calculated and presented. After all these processes, the newly obtained numerical results are compared with results of some of the published articles.
Similar content being viewed by others
References
W X Ma and Y Zhou, J. Diff. Equ. 264, 2633 (2018)
W X Ma, J. Geom. Phys. 133, 10 (2018)
W X Ma, X Young and H Q Zhang, Comput. Math. Appl. 75, 289 (2018)
J Y Yang, W X Ma and Z Qin, Anal. Math. Phys. 8, 427 (2018)
J Y Yang, W X Ma and Z Qin, East Asia J. Appl. Math. 8, 224 (2018)
R M Miura, SIAM Rev. 18, 412 (1976)
W Hereman and A Nuseir, Math. Comput. Simul. 43, 13 (1997)
M J Ablowitz and P A Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1991)
M J Ablowitz and H Segur, Solitons and inverse scattering transform (SIAM, Philadelphia, 1981)
R Bullough and P Caudrey, Solitons: Topics in current physics (Springer, Berlin, 1980) Vol. 17
P G Drazin and R S Johnson, Solitons: An introduction (Cambridge University Press, Cambridge, 1996)
R M Miura, J. Math. Phys. 9, 1202 (1968)
T Nagatani, Physica A 264, 581 (1999)
J Zhou, Z K Shi and J L Cao, Physica A 396, 77 (2014)
C S Gardner, J M Greene, M D Kruskal and M R Miura, Phys. Lett. A 19, 1095 (1967)
C H Su and C S Gardner, J. Math. Phys. 10, 536 (1969)
M Salahuddin, Plasma Phys. Control. Fusion 32, 33 (1990)
W X Ma and Y You, Trans. Am. Math. Soc. 357(5), 1753 (2004)
A Başhan, Turkish J. Math. 42, 373 (2018)
O E Hepson, A Korkmaz and I Dag, Pramana – J. Phys. 91: 59 (2018)
A Başhan, Y Uçar, N M Yağmurlu and A Esen, J. Phys. Conf. Ser. 766, 012028 (2016)
A Wazwaz, Commun. Nonlinear Sci. Numer. Simul. 13, 331 (2008)
A H Salas, Appl. Math. Comput. 216, 2792 (2010)
D Kaya, Commun. Nonlinear Sci. Numer. Simul. 10, 693 (2000)
L R T Gardner, G A Gardner and T Geyikli, Comput. Methods Appl. Mech. Eng. 124, 321 (1995)
A Biswas and K R Raslan, Phys. Wave Phenom. 19(2), 142 (2011)
T Ak, S B G Karakoç and A Biswas, Sci. Iran B 24(3), 1148 (2017)
T Ak, S B G Karakoç and A Biswas, Iran. J. Sci. Technol. Trans. Sci. 41, 1109 (2017)
T Geyikli, Finite element studies of the modified KdV equation, Doctoral dissertation (University College of North Wales, Bangor, UK, 1994)
G A Gardner, A H A Ali and L R T Gardner, Numer. Methods Eng. 1, 590 (1990)
R Bellman, B G Kashef and J Casti, J. Comput. Phys. 10, 40 (1972)
R Bellman, B G Kashef, E S Lee and R Vasudevan, Computers and mathematics with applications (Pergamon, Oxford, 1976) Vol. 1, p. 371
J Cheng, B Wang and S Du, Int. J. Solids Struct. 42, 6181 (2005)
C Shu and Y L Wu, Int. J. Numer. Methods Fluids 53, 969 (2007)
A G Striz, X Wang and C W Bert, Acta Mech. 111, 85 (1995)
I Bonzani, Comput. Math. Appl. 34, 71 (1997)
A Korkmaz and I Dağ, Int. J. Comput.-Aided Eng. Softw. 28(6), 654 (2011)
A Başhan, S B G Karakoç and T Geyikli, Kuwait J. Sci. 42(2), 67 (2015)
A Başhan, Y Uçar, N M Yağmurlu and A Esen, Eur. Phys. J. Plus 133, 12 (2018)
S B G Karakoç, A Başhan and T Geyikli, Sci. World J. 2014, 1 (2014)
R C Mittal and R K Jain, Appl. Math. Comput. 218, 7839 (2012)
A Başhan, N M Yağmurlu, Y Uçar and A Esen, Chaos Solitons Fractals 100, 45 (2017)
A Başhan, N M Yağmurlu, Y Uçar and A Esen, Int. J. Mod. Phys. C 29(6), 1850043 (2018)
P M Prenter, Splines and variational methods (John Wiley, New York, 1975)
S G Rubin and R A Graves, A cubic spline approximation for problems in fluid mechanics, Technical Report (National Aeronautics and Space Administration, Washington, 1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Başhan, A. A novel approach via mixed Crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation. Pramana - J Phys 92, 84 (2019). https://doi.org/10.1007/s12043-019-1751-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-019-1751-1