# Approximation of the KdVB equation by the quintic B-spline differential quadrature method

## Keywords:

KdVB equation, differential quadrature method, quintic B-splines, partial differential equation, stability.## Abstract

In this paper, the Korteweg-de Vries-Burgers’ (KdVB) equation is solved numerically by anew differential quadrature method based on quintic B-spline functions. The weightingcoefficients are obtained by semi-explicit algorithm including an algebraic system with fivebandcoefficient matrix. The L2 and L∞ error norms and lowest three invariants 1 2 I , I and3 I have computed to compare with some earlier studies. Stability analysis of the method isalso given. The obtained numerical results show that the present method performs better thanthe most of the methods available in the literature.## References

Ali, A. H. A., Gardner, L. R. T. & Gardner, G. A. 1993. Numerical study of the KdVB equation using B-spline finite elements, J. Math.Phys. Sci. 27: 37-53.

Ali, A. H. A., Gardner, G. A. Gardner & L. R. T. 1992. A collocation solution for Burgers’ equation using cubic B-spline finite elements, Computer Methods in Applied Mechanics and Engineering, 100: 325-337.

Arora, G. & Singh, B. K. 2013. Numerical solution of Burgers equation with modified cubic B-spline differential quadrature method, Appl. Math. Comput., 224: 166–177.

Bellman, R., Kashef, B. & Casti, J. 1972. Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, 10: 40-52.

Bellman, R., Kashef, B., Lee, E. S. & Vasudevan, R. 1976. Differential Quadrature and Splines, Computers and Mathematics with Applications, Pergamon, Oxford, 1: 371-376.

Bonzani, I. 1997. Solution of non-linear evolution problems by parallelized collocation-interpolation methods, Computers & Mathematics and Applications, 34: 71-79.

Canosa, J. & Gazdag, J. 1977. The Korteweg–de Vries–Burgers’ equation, J. Comp. Phys. 23: 393-403.

Cheng, J., Wang, B. & Du, S. 2005. A theoretical analysis of piezoelectric/composite laminate with larger-amplitude deflection effect, Part II: hermite differential quadrature method and application, International Journal of Solids and Structures, 42: 6181-6201.

Civalek, O. 2004. Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, An International Journal

of Engineering Structures, 26: 171-186.

Civalek, O. 2006. Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation,

Journal of Sound and Vibration , 294: 966-980.

Demiray, H. 2004. A travelling wave solution to the KdV–Burgers equation, Appl. Math. Comput. 154: 665-670.

Dağ, I. & Dereli, Y. 2008. Numerical solutions of KdV equation using radial basis functions, Applied Mathematical Modelling, 32: 535-546.

Gardner, L. R. T., Gardner, G. A. & Ali, A. H. A. 1991. Simulations of solitons using quadratic spline finite elements, Comput. Methods Appl. Mech. Engrg. 92: 231.

Grad, H. & Hu, P. N. 1967. Unified shock profile in a plasma, Phys. Fluids, 10: 2596-2601.

Guo, Q. & Zhong, H. 2004. Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration , 269: 413-420.

Helal, M. A. & Mehanna, M. S. 2006. A comparison between two different methods for solving KdV–Burgers equation, Chaos Soliton. Fract., 28: 320-326.

Jain, M. K. 1983. Numerical Solution of Differential Equations, 2nd ed.,Wiley, New York, NY.

Johnson, R. S. 1970. A non-linear equation incorporating damping and dispersion, J. Fluid Mech. 42: 49-60.

Johnson, R. S. 1972. Shallow water waves in a viscous fluid, the undular bore, Phys. Fluids 15: 1958-1988.

Kaya, D. 1999. On the solution of a Korteweg–de Vries like equation by the decomposition method, Int. J. Comput. Math., 72: 531-539.

Kaya, D. 2004. An application of the decomposition method for the KdVB equation, Appl. Math.Comput., 152: 279-288.

Korkmaz, A. (2010a). Numerical algorithms for solutions of Korteweg-de Vries Equation, Numerical methods for partial differential equations, 26: 1504-1521. Korkmaz, A. & Dağ, I. 2009. Solitary wave simulations of complex modified Korteweg-de Vries equation using differential quadrature method, omputers Physics Communications, 180: 1516-1523.

Korkmaz, A. & Dağ, I. 2010b. Numerical solutions of some one dimensional partial differential equations using B-spline differential quadrature method, Doctoral Dissertation, Eskisehir Osmangazi University.

Korkmaz, A. & DağI. 2011a. Shock wave simulations using Sinc Differential Quadrature Method, International Journal for Computer-Aided Engineering and Software, 28: 654-674.

Korkmaz, A. & Dağ, I. 2011b. Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation, Journal of the Franklin Institute, 348: 2863-2875.

Korkmaz, A. & Dağ, I. 2012. Cubic B-spline differential quadrature methods for the advection-diffusion equation, International Journal of Numerical Methods for Heat & Fluid Flow, 22: 1021-1036.

Korkmaz, A. & Dağ, I. 2013a. Cubic B-spline differential quadrature methods and stability for Burgers’ equation, International Journal for Computer-Aided Engineering and Software, 30: 320-344.

Korkmaz, A. & Dağ, I. 2013b. Numerical Simulations of Boundary-Forced RLW Equation with Cubic B-spline based differential quadrature methods , Arab. J. Sci. Eng., 38: 1151-1160.

Lee, T. S., Hu, G. S. & Shu, C. 2004. Application of GDQ method for study of mixed convection in horizontal eccentric annuli, International Journal of Computational Fluid Dynamics, 18: 71-79.

Mittal, R. C. & Jiwari, R. 2009. Differential quadrature method for two-dimensional Burgers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10: 450-459.

Mittal, R. C. & Jiwari, R. 2011. Numerical solution of two-dimensional reaction–diffusion Brusselator system, Appl. Math. Comput. 217: 5404-5415.

Mittal, R. C. & Jiwari, R. 2012. A differential quadrature method for numerical solutions of Burgerstype equations, International Journal of Numerical Methods for Heat & Fluid Flow, 22: 880-895.

Sahu, B. & Roychoudhury, R. 2003. Travelling wave solution of Korteweg-de Vries–Burger’s equation, Czech. J. Phys. 53: 517-527.

Saka, B., Dağ, I. Dereli, Y. & Korkmaz, A. 2008. Three different methods for numerical solutions of the EW equation, Engineering Analysis with Boundary Elements, 32: 556-566.

Saka, B. & Dağ, I. 2009. Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation, Appl. Math. Comput. 215: 746-758.

Saka, B. & Dağ, I. 2007. Quartic B-spline collocation method to the numerical solutions of the Burgers equation, Chaos Soliton. Fract., 32: 1125-1137.

Shu, C. & Xue, H. 1997. Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration, 204: 549-555.

Shu, C. & Richards, B. E. 1992. Application of generalized differential quadrature to solve two dimensional incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 15: 791-798.

Shu, C. & Wu, Y. L. 2007. Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids, 53: 969-984. Soliman, A. A. 2004. Collocation solution of the Korteweg-de Vries Equation using septic splines, Int. J. Comput. Math., 81: 325-331.

Striz, A. G., Wang, X. & Bert, C. W. 1995. Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica,111: 85-94.

Su, C. H. & Gardner, C. S. 1969. Derivation of the Korteweg-de Vries and Burgers equation, J. Math. Phys., Vol. 10: 536-539.

Quan, J. R. & Chang, C. T. 1989a. New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Eng., 13: 779-788.

Quan, J. R. & Chang, C. T. 1989b. New sightings in involving distributed system equations by the quadrature methods-II, Comput. Chem. Eng., 13: 717-724.

Tomasiello, S. 2010. Numerical solutions of the Burger-Huxley equation by the IDQ method, Int. J. Comput. Math., 87: 129-140.

Zaki, S. I. 2000a. A quintic B-spline finite elements scheme for the KdVB equation, Comput. Meth. Appl. Mech. Engrg. 188: 121-134.

Zaki, S. I. 2000b. Solitary waves of the Korteweg-de Vries–Burgers’ equation, Comput. Phys. Commun.,126: 207-218.

Zhong, H. 2004. Spline-based differential quadrature for fourth order equations and its application to Kirchhoff plates, Applied Mathematical Modelling , 28: 353-366.

Zhong, H. & Lan, M. 2006. Solution of nonlinear initial-value problems by the spline-based differential

quadrature method, Journal of Sound and Vibration, 296: 908-918.

Zhu, Y. D., Shu, C., Qiu, J. & Tani, J. 2004. Numerical simulation of natural convection between two elliptical cylinders using DQ method, International Journal of Heat and Mass Transfer, 47: 797-808.