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

#### Abstract

#### Keywords

#### Full Text:

PDF#### 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.

### Refbacks

- There are currently no refbacks.