Numerical Solution of Modified Forms of Camassa-Holm and Degasperis-Procesi Equations via Quartic B-Spline Collocation Method
In this paper, a collocation finite difference scheme based on Quartic B-spline function is developed for solving non-linear modified Camassa-Holm and Degasperis-Procesi equations. A finite difference scheme and Quartic B-spline function are used to discretize the time and spatial derivatives, respectively. The obtained numerical results are compared with the exact analytical solutions and some methods existing in literature. The numerical solutions of proposed non-linear equations are acquired without any linearization technique. The convergence of the method is proved of order \((\Delta t + h^2)\). The efficiency of the proposed scheme is demonstrated through illustrative examples. The presented scheme is realized to be a very reliable alternate method to some existing schemes for such physical problems.
M. Abbas, A.A. Majid, A.I. Ismail and A. Rashid, Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system, PLOS ONE 9 (1) (2014), e83265, DOI: 10.1371/journal.pone.0083265.
M. Abbas, A.A. Majid, A.I. Ismail and A. Rashid, Numerical method using cubic trigonometric B-spline technique for non-classical diffusion problem, Abstr. Appl. Anal. Article ID 849682 (2014), 10 pages.
M. Abbas, A.A. Majid and A.I. Ismail, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Appl. Math. Comp. 239 (2014), 74 – 88.
S. Abbasbandy, Solitary wave solution to the modified form Camassa-Holm equation, Cha. Sol. Frac. 39 (2009), 428 – 435.
C. Clavero, J.C. Jorge and F. Lisbona, Uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math. 154 (2003), 415 – 429.
C. DeBoor, On the convergence of odd degree spline interpolation, J. Approx. Theo. 1 (1968), 452 – 463.
D.D. Ganji, E.M.M. Sadeghi and M.G. Rahmat, Modified forms of Degaperis-Procesi and Camassa-Holm equations solved by Adomian’s Decomposition method and comparison with HPM and exact solution, Acta. Appl. Math. 104 (2008), 303 – 311.
C.A. Hall, On error bounds for spline interpolation, J. Approx. Theo. 1 (1968), 209 – 218.
J. Manafian, R. Shahabi, M. Asaspour, I. Zamanpour and J. Jalali, Construction of exact solutions to the modified forms of DP and CH equations by analytical methods, Stat. Opt. Info. Comp. 3 (2015), 336 – 347.
R. Mohanty, M. Jain and K. George, Fourth order approximation at first time level, linear stability analysis and the numerical solution of multidimenional second-order non-linear hyperbolic equations in polar coordinates, J. Comp. Appl. Math. 93 (1998), 1 – 12.
T. Nazir, M. Abbas, A.A. Majid, A.I. Ismail and A. Rashid, The numerical solution of convectiondiffusion problem with cubic trigonometric B-splines, Appl. Math. Model. 40 (2016), 4586 – 4611.
W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill Inc., p. 218 (1976).
A.M. Wazawaz, Solitary wave solutions for modified forms of Degaperis-Procesi and Camassa-Holm equations, Phy. Lett. A. 352 (6) (2006), 500 – 504.
A.M.Wazwaz, New solitary wave solutions to the modified forms of Degaperis-Procesi and Camassa-Holm equations, Appl. Math. 186 (1) (2007), 130 – 141.
M. Yaseen, M. Abbas, A.I. Ismail and T. Nazir, A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Appl. Math. Comp. 293 (2017), 311 – 319.
A. Yildirim, Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations, Int. J. Num. meth. Biomed. Engi. 26 (2010), 266 – 272.
M.A. Yousif, B.A. Mahmood and F.H. Easif, A new analytical study of modified Camassa-Holm and Degaperis-Procesi equations, Amr. Jour. Comp. Math. 5 (2015), 267 – 273.
E. Yusufoglu, New solitary solutions for modified forms of DP and CH equations using Exp-function method, Cha. Sol. 39 (2009), 2442 – 2447.
B. Zhang, S. Li and Z. Liu, Homotopy Perturbation method for modified Camassa-Holm and Degaperis-Procesi equations, Phy. Lett. A 372 (2008), 1867 – 1872.
L. Zhang and X.T. Wang, A note on solitary wave solutions of the non-linear generalized Camassa-Holm equation, Int. J. Anal. 2013 (2013), Article ID 723698, 6 pages, DOI: 10.1155/2013/723698.
B. Zhang, Z. Liu and J. Mao, New exact solutions for MCH and MDP equations by auxiliary equation method, App. Math. Comp. 217 (2010), 1306 – 1314.
S.M. Zin, A.A. Majid, A.I. Ismail and M. Abbas, Application of hybrid cubic B-spline collocation approach for solving a generalized non-linear Klien-Gordon equation, Math. Prob. Engin. 2014, Article ID 108560 (2014), 10 pages.
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