A septic B-spline collocation method for solving the generalized equal width wave equation

Seydi B. G. Karakoc, Halil Zeybek

Abstract


In this work, a septic B-spline collocation method is implemented to find the numerical solution of the generalized equalwidth (GEW) wave equation by using two different linearization techniques. Test problems including single soliton,interaction of solitons and Maxwellian initial condition are solved to verify the proposed method by calculating the errornorms L2 and L∞ and the invariants I1, I2 and I3. Applying the Von-Neumann stability analysis, the proposed method isshown to be unconditionally stable. As a result, the obtained results are found in good agreement with the some recentresults.

Keywords


Collocation method; GEW equation; septic B-spline; solitary waves; soliton.

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References


Başhan, A., Karakoç, S.B.G. & Geyikli, T. (2015). Approximation

of the KdVB equation by the quintic B-spline differential quadrature

method. Kuwait Journal of Science, 42(2):67-92.

Benjamin, T.B., Bona, J.L. & Mahony, J.J. (1972). Model equations

for long waves in non-linear dispersive systems. Philosophical

Transactions of the Royal Society of London Series A, 272:47-78.

Dag, I. & Saka, B. (2004). A cubic B-spline collocation method for

the EW equation. Mathematical and Computational Applications,

(3):381-392.

Dogan, A. (2005). Application of Galerkin’s method to equal width

wave equation. Applied Mathematics and Computation, 160(1):65-76.

Esen, A. (2005). A numerical solution of the equal width wave equation

by a lumped Galerkin method. Applied Mathematics and Computation,

(1):270-282.

Esen, A. (2006). A lumped Galerkin method for the numerical solution

of the modified equal-width wave equation using quadratic B-splines.

International Journal of Computer Mathematics, 83(5-6):449-459.

Evans, D.J. & Raslan, K.R. (2005). Solitary waves for the generalized

equal width (GEW) equation. International Journal of Computer

Mathematics, 82(4):445-455.

Fazal-i-Haq, F., Shah, I.A. & Ahmad, S. (2013). Septic B-spline

collocation method for numerical solution of the equal width wave

(EW) equation. Life Science Journal, 10(1):253-260.

Garcia-Archilla, B. (1996). A Spectral Method for the Equal Width

Equation. Journal of Computational Physics, 125(2):395-402.

Gardner, L.R.T. & Gardner, G.A. (1991). Solitary waves of the equal

width wave equation. Journal of Computational Physics, 101(1):218-

Gardner, L.R.T., Gardner, G.A., Ayoup, F.A. & Amein, N.K. (1997).

Simulations of the EW undular bore. Communications in Numerical

Methods in Engineering, 13(7):583-592.

Geyikli, T. & Karakoç, S.B.G. (2011). Septic B-spline collocation

method for the numerical solution of the modified equal width wave

equation. Applied Mathematics, 2011(2):739-749.

Geyikli, T. & Karakoç, S.B.G. (2012). Petrov-Galerkin method with

cubic B-splines for solving the MEW equation. Bulletin of the Belgian Mathematical Society-Simon Stevin, 19(2):215-227.

Hamdi, S., Enright, W.H., Schiesser, W.E., Gottlieb, J.J. & Alaal, A.

(2003). Exact solutions of the generalized equal width wave equation.

in: Proceedings of the International Conference on Computational

Science and Its Applications, LNCS, 2668:725-734.

Hamdi, S., Gottlieb, J.J. & Hansen, J.S. (2001). Numerical solutions

of the equal width wave equations using an adaptive method of lines.

In Adaptive Method of Lines, Wouver A. V., Saucez P., Schiesser W E.,

(eds). Chapman & Hall/CRC Press: Boca Raton, Florida, 65-116.

İslam, S., Haq, F. & Tirmizi, İ.A. (2010). Collocation method using

quartic B-spline for numerical solution of the modified equal width

wave equation. Journal of Applied Mathematics & Informatics, 28(3-

:611-624.

Karakoç, S.B.G. & Geyikli, T. (2012). Numerical solution of

the modified equal width wave equation. International Journal of

Differential Equations, 2012:1-15.

Karakoç, S.B.G., Ak, T. & Zeybek, H. (2014). An efficient approach

to numerical study of the MRLW equation with B-spline collocation

method. Abstract and Applied Analysis, 2014:1-15.

Karakoç, S.B.G., Zeybek, H. & Ak, T. (2014). Numerical solutions

of the Kawahara equation by the septic B-spline collocation method.

Statistics Optimization and Information Computing, 2:211-221.

Karakoç, S.B.G., Uçar, Y. & Yağmurlu, N.M. (2015). Numerical

solutions of the MRLW equation by cubic B-spline Galerkin finite

element method. Kuwait Journal of Science, 42(2):141-159.

Lu, J. (2009). He’s variational iteration method for the modified equal

width equation. Chaos, Solitons and Fractals, 39(5):2102-2109.

Panahipour, H. (2012). Numerical simulation of GEW equation using

RBF collocation method. Communications in Numerical Analysis,

:1-28.

Peregrine, D.H. (1967). Long waves on a beach. Journal of Fluid

Mechanics, 27:815-827.

Prenter, P.M. (1975). Splines and variational methods, John Wiley,

New York. Pp:323

Raslan, K.R. (2005). Collocation method using quartic B-spline for the

equal width (EW) equation. Applied Mathematics and Computation,

(2):795-805.

Raslan, K.R. (2006). Collocation method using cubic B-spline for the

generalised equal width equation. International Journal of Simulation

and Process Modelling, 2:37-44.

Roshan, T. (2011). A Petrov-Galerkin method for solving the

generalized equal width (GEW) equation. Journal of Computational

and Applied Mathematics, 235:1641-1652.

Rubin, S.G. & Graves, R.A. (1975). A cubic spline approximation for

problems in fluid mechanics. NASA TR R-436, Washington, DC.

Saka, B. (2007). Algorithms for numerical solution of the modified

equal width wave equation using collocation method. Mathematical

and Computer Modelling, 45(9-10):1096-1117.

Taghizadeh, N., Mirzazadeh, M., Akbari, M. & Rahimian, M. (2013).

Exact solutions for generalized equal width equation. Mathematical

Sciences Letters 2, 2:99-106.

Wazwaz, A.M. (2006). The tanh and the sine-cosine methods for a

reliable treatment of the modified equal width equation and its variants.

Communications in Non-linear Science and Numerical Simulation,

(2):148-160.

Zaki, S.I. (2001). Solitary waves induced by the boundary forced EW

equation. Computer Methods in Applied Mechanics and Engineering,

:4881-4887.


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