Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. (English) Zbl 1130.65103

Summary: The collocation method using quartic B-splines is described for the numerical solutions of the Burgers’ equation. The effect of the quartic B-splines in the collocation method is sought. The same method is applied to the time split Burgers’ equation. Numerical comparison of results of both algorithms and some other published numerical results are done by studying three standard problems.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


[1] Bateman, H., Some recent researches on the motion of the fluids, Monthly weather rev, 43, 163-170, (1915)
[2] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Advances in applied mechanics, (1948), Academic Press New York, p. 171-99
[3] Hopf, E., The partial differential equation Ut+uux−νuxx=0, Commun pure appl math, 3, 201-230, (1950)
[4] Cole, J.D., On a quasi-linear parabolic equations occurring in aerodynamics, Quart appl math, 9, 225-236, (1951) · Zbl 0043.09902
[5] Miller EL. Predictor – corrector studies of Burgers’ model of turbulent flow. M.S. Thesis, University of Delaware, DE, Newark, 1966.
[6] Rubin, S.G.; Graves, R.A., Viscous flow solutions with a cubic spline approximation, Comput fluids, 3, 1-36, (1975) · Zbl 0347.76020
[7] Prenter, P.M., Splines and variational methods, (1975), John Wiley New York · Zbl 0344.65044
[8] Rubin, S.G.; Khosla, P.K., Higher-order numerical solutions using cubic splines, Aiaa j, 14, 851-858, (1976) · Zbl 0344.65048
[9] Davies, A.M., A numerical investigation of errors arising in applying the Galerkin method of the solution of nonlinear partial differential equations, Comput methods appl mech engrg, 11, 341-350, (1977) · Zbl 0364.65093
[10] Davies, A.M., Application of the Galerkin method to the solution of burgers’ equation, Comput methods appl mech engrg, 14, 305-321, (1978) · Zbl 0387.76014
[11] Jain, P.C.; Holla, D.N., Numerical solutions of coupled burgers’ equations, Int J non-linear mech, 13, 213-222, (1978) · Zbl 0388.76049
[12] Jain, P.C.; Lohar, B.L., Cubic spline technique for coupled nonlinear parabolic equations, Comput math appl, 5, 179-185, (1979) · Zbl 0421.65063
[13] Varoğlu, E.; Finn, W.D.L., Space-time finite elements incorporating characteristics for the burgers’ equation, Int J numer methods engrg, 16, 171-184, (1980) · Zbl 0449.76076
[14] Christie, I.; Griffiths, D.F.; Mitchell, A.R.; Sanz-Serna, J.M., Product approximation for nonlinear problems in the finite element method, IMA J numer anal, 1, 253-266, (1981) · Zbl 0469.65072
[15] Herbst, B.M.; Schoombie, S.W.; Mitchell, A.R., A moving petrov – galerkin method for transport equations, Int J numer methods engrg, 18, 1321-1336, (1982) · Zbl 0485.65093
[16] Nguyen, H.; Reynen, J., A space – time finite element approach to burgers’ equation, (), 718-728 · Zbl 0574.76053
[17] Caldwell, J., Applications of cubic splines to the nonlinear burgers’ equation, (), 253-261
[18] Ali A.H.A., Gardner L.R.T., Gardner G.A. A Galerkin approach to the solution of Burgers’ equation. University of Wales, Bangor, Math. Preprint 90.04, 1990.
[19] Kakuda, K.; Tosaka, N., The generalized boundary element approach to burgers’ equation, Int J numer methods engrg, 29, 245-261, (1990) · Zbl 0712.76070
[20] Zhang, D.S.; Wei, G.W.; Kouri, D.J.; Hoffman, D.K., Burgers’ equation with high Reynolds number, Phys fluids, 9, 1853-1855, (1997) · Zbl 1185.76843
[21] Wei, G.W.; Zhang, D.S.; Kouri, D.J.; Hoffman, D.K., A robust and reliable approach to nonlinear dynamical problems, Comput phys commun, 111, 87-92, (1998) · Zbl 0935.65108
[22] Kutluay, S.; Bahadır, A.R.; Özdeş, A., Numerical solution of one-dimensional burgers’ equation: explicit and exact-explicit finite difference methods, J comput appl math, 103, 251-261, (1999) · Zbl 0942.65094
[23] Kutluay, S.; Esen, A.; Dağ, I., Numerical solutions of the burgers’ equation by the least squares quadratic B-spline finite element method, J comput appl math, 167, 21-33, (2004) · Zbl 1052.65094
[24] Dağ, İ.; Irk, D.; Saka, B., A numerical solution of the burgers’ equation using cubic B-splines, Appl math comput, 163, 199-211, (2005) · Zbl 1060.65652
[25] Dağ, İ; Saka, B.; Boz, A., B-spline Galerkin methods for numerical solutions of the burgers’ equation, Appl math comput, 166, 506-522, (2005) · Zbl 1073.65099
[26] Ramadan, M.A.; El-Danaf, T.S.; Alaal, F., A numerical solution of the burgers’ equation using septic B-splines, Chaos, solitons & fractals, 26, 795-804, (2005) · Zbl 1075.65127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.