Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function. (English) Zbl 1140.65069

This paper present a numerical solution for the one-dimensional Burgers equation using a reproducing kernel function. The numerical solution and experiments are quite interesting. The convergence and stability are compared with the exact solutions. It is also shown that the results obtained can be easily implemented and are quite effective.


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
Full Text: DOI


[1] Aronszajn, Z., Theory of reproducing kernels, Trans. amer. math. soc., 68, 337-404, (1950) · Zbl 0037.20701
[2] Bateman, H., Some recent researches on the motion of fluids, Monthly weather rev., 43, 163-170, (1915)
[3] Benton, E.; Platzman, G.W., A table of solutions of the one-dimensional Burgers equations, Quart. appl. math., 30, 195-212, (1972) · Zbl 0255.76059
[4] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-199, (1948)
[5] Cole, J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. appl. math., 9, 225-236, (1951) · Zbl 0043.09902
[6] Daǧ, İ.; Irk, D.; Sahin, A., \(B\)-spline collocation methods for numerical solutions of the Burgers equation, Math. problems in eng., 5, 521-538, (2005) · Zbl 1200.76141
[7] Fletcher, C.A., A comparison of finite element and difference solutions of the one and two dimensional burgers’ equations, J. comput. phys., 51, 159-188, (1983) · Zbl 0525.65077
[8] Kakuda, K.; Tosaka, N., The generalized boundary element approach to burgers’ equation, Internat. J. numer. methods eng., 29, 245-261, (1990) · Zbl 0712.76070
[9] Kutluay, S.; Bahadir, A.R.; Özdes, 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
[10] Kutluay, S.; Esen, A.; Daǧ, İ., 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
[11] E.L. Miller, Predictor – corrector studies of Burgers’ model of turbulent flow, M.S. Thesis, University of Delaware, Newark, DE, 1966.
[12] Mittal, R.C.; Singhal, P., Numerical solution of burgers’ equation, Comm. numer. methods eng., 9, 397-406, (1993) · Zbl 0782.65147
[13] Mittal, R.C.; Singhal, P., Numerical solution of periodic burgers’ equation, Indian J. pure appl. math., 27, 689-700, (1996) · Zbl 0859.76053
[14] S.-S. Xie, New numerical methods for some initial – boundary value problems and applications, Ph.D. Thesis, Shandong University, PR China, 1996.
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.