NONPOLYNOMIAL CUBIC SPLINE APPROXIMATION FOR THE EQUAL WIDTH EQUATION

Year 2015, Volume: 3 Issue: 2, 17 - 32, 01.10.2015

Ali Sahın Levent Akyuz

Abstract

In this paper, we investigate the numerical solutions of the equal width (EW) equation via the nonpolynomial cubic spline functions. Crank- Nicolson formulas are used for time discretization of the target equation. A linearization technique is also employed for the numerical purpose. Accuracy of the method is observed by the pointwise rate of convergence. Stability of the suggested method is investigated via the von-Neumann analysis. Six numerical examples related to single solitary wave, interaction of two, three and opposite waves, wave undulation and the Maxwell wave are considered as the test problems. The accuracy and the eciency of the purposed method are measured by L1 and L2 error norms and conserved constants. The obtained results are compared with the possible analytical values and those in some earlier studies.

Keywords

EW equation; Nonpolynomial spline; Solitary waves

References

• [1] Rubin S.G. and Graves R.A., Cubic spline approximation for problems in fuid mechanics, Nasa TR R-436, Washington, DC, (1975).
• [2] Morrison P.J., Meiss J.D., Carey J.R., Scattering of RLW solitary waves, Physica 11D (1981) 324{36.
• [3] Gardner L.R.T., Gardner G.A., Solitary waves of the equal width wave equation, J Comput Phys 101 (1992) 218{23.
• [4] Garcia-Archilla B., A spectral method for the equal width equation, J Comput Phys 125 (1996) 395{402.
• [5] Zaki S.I., A least-squares nite element scheme for the EW equation, Comput Meth Appl Mech Eng 189 (2000) 587{94.
• [6] Saka B., Dag I., Dogan A., A Galerkin method for the numerical solution of the RLWequation using quadratic B-splines, Int J Comput Math 81 (2004) 727{739.
• [7] Dag I., Saka B., A cubic B-spline collocation method for the EW equation. Math Comput Appl 9 (2004) 381{392.
• [8] Dogan A., Application of Galerkin's method to equal width wave equation. Appl Math Comput 160 (2005;) 65{76.
• [9] Esen A., A numerical solution of the equal width wave equation by a lumped Galerkin method, Appl Math Comput 168 (2005) 270{282.
• [10] Raslan K.R., Collocation method using quartic B-spline for the equal width (EW) equation, Int J Comput Math 81 (2004) 63{72.
• [11] Raslan K.R., A computational method for the equal width equation, Appl Math Comput 168 (2005) 795{805.
• [12] Ramos J.I., Explicit nite dierence methods for the EW and RLW equations, Appl MathComput 179 (2006) 622{638.
• [13] Ramos J.I., Solitary waves of the EW and RLW equations, Chaos Solitons and Fractals 34 (2007) 1498{1518.
• [14] Saka B., Dag I., Dereli Y., Korkmaz A., Three different methods for numerical solution of the EW equation, Engineering Analysis with Boundary Elements 32 (2008) 556-566.
• [15] Rashidinia J., Mohammadi R., Non-polynomial cubic spline methods for the solution of parabolic equations, Int. J. Comput. Math. 85:5 (2008) 843-850.
• [16] Griewanka A., El-Danaf T.S., Ecient accurate numerical treatment of the modied Burgers' equation, Applicable Analysis 88 (2009) 75-87.
• [17] Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation, Comput. Phys. Commun. 181 (2010) 78{91.
• [18] Jalilian, R.: Non-polynomial spline method for solving Bratu's problem, Comput. Phys. Commun. 181 (2010) 1868{1872.
• [19] Roshan T., A Petrov{Galerkin method for equal width equation, Applied Mathematics and Computation 218 (2011) 2730{2739.
• [20] El-Danaf T.S., Ramadan M.A., Abd Alaal F.E.I, Numerical studies of the cubic non-linear Schrodinger equation, Nonlinear Dyn. 67 (2012) 619-627.
• [21] Chegini N.G., Salaripanah A., Mokhtari R., Isvand D., Numerical solution of the regularized long wave equation using nonpolynomial splines, Nonlinear Dyn. 69 (2012) 459-471.
• [22] Dereli Y., Schaback R., The meshless kernel-based method of lines for solving the equal width equation, Applied Mathematics and Computation 219 (2013) 5224-5232.