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Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method. (English) Zbl 1242.65209
Summary: A numerical method is proposed to approximate the solution of the nonlinear Burgers’ equation. The method is based on collocation of modified cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cubic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using the SSP-RK43 or SSP-RK54. These methods need less storage space that causes less accumulation of numerical errors. The numerical approximate solutions to the Burgers’ equation are computed without transforming the equation and without using the linearization. Illustrative eleven examples are included to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method.

65M70Spectral, collocation and related methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
65L06Multistep, Runge-Kutta, and extrapolation methods
65M30Improperly posed problems (IVP of PDE, numerical methods)
Full Text: DOI
[1] Asaithambi, Asai: Numerical solution of the Burgers equation by automatic differentiation, Appl. math. Comput. 216, 2700-2708 (2010) · Zbl 1193.65154 · doi:10.1016/j.amc.2010.03.115
[2] Dogan, Abdulkadir; Galerkin, A.: Finite element approach to Burgers equation, Appl. math. Comput. 157, 331-346 (2004) · Zbl 1054.65103 · doi:10.1016/j.amc.2003.08.037
[3] Ali, A. H. A.; Gardner, G. A.; Gardner, L. R. T.: A collocation solution for Burgers equation using cubic B-spline finite elements, Comput. methods appl. Mech. eng. 100, 325-337 (1992) · Zbl 0762.65072 · doi:10.1016/0045-7825(92)90088-2
[4] Khater, A. H.; Temsah, R. S.; Hassan, M. M.: A Chebyshev spectral collocation method for solving Burgers type equations, J. comput. Appl. math. 222, 333-350 (2008) · Zbl 1153.65102 · doi:10.1016/j.cam.2007.11.007
[5] Korkmaz, Alper: Shock wave simulations using sinc differential quadrature method, Int. J. Comput. aided eng. Software 28, No. 6, 654-674 (2011) · Zbl 1284.76292
[6] Korkmaz, Alper; Dag&caron, Idris; : Polynomial based differential quadrature method for numerical solution of nonlinear Burgers equation, J. franklin inst. (2011)
[7] Korkmaz, Alper; Aksoy, A. Murat; Da&breve, Idris; G: Quartic B-spline differential quadrature method, Int. J. Nonlinear sci. 11, No. 4, 403-411 (2011)
[8] Khalifa, A. K.; Noor, Khalida Inayat; Noor, Muhammad Aslam: Some numerical methods for solving Burgers equation, Int. J. Phys. sci. 6, No. 7, 1702-1710 (2011) · Zbl 1243.49010
[9] Saka, Bülent; Dag, Idris: Quartic B-spline collocation method to the numerical solutions of the Burgers equation, Chaos solitons fractals 32, 1125-1137 (2007) · Zbl 1130.65103 · doi:10.1016/j.chaos.2005.11.037
[10] Rao, Ch. Srinivasa; Satyanarayana, Engu: Solutions of Burgers equation, Int. J. Nonlinear sci. 9, No. 3, 290-295 (2010) · Zbl 1208.35134
[11] Aksan, E. N.: Quadratic B-spline finite element method for numerical solution of the Burgers equation, Appl. math. Comput. 174, 884-896 (2006) · Zbl 1090.65108 · doi:10.1016/j.amc.2005.05.020
[12] Hesameddini, Esmaeel; Gholampour, Razieh: Soliton and numerical solutions of the Burgers equation and comparing them, Int. J. Math. anal. 4, No. 52, 2547-2564 (2010) · Zbl 1225.65115 · http://www.m-hikari.com/ijma/ijma-2010/ijma-49-52-2010/index.html
[13] Güraslan, G.; Sari, M.: Numerical solutions of linear and nonlinear diffusion equations by a differential quadrature method (DQM), Int. J. Numer. methods biomed. Eng. 27, 69-77 (2011) · Zbl 1210.65175 · doi:10.1002/cnm.1292
[14] Da&breve, I.; G; Irk, D.; Sahin, A.: B-spline collocation methods for numerical solutions of the Burgers equation, Math. probl. Eng. 5, 521-538 (2005) · Zbl 1200.76141
[15] Hassanien, I. A.; Salama, A. A.; Hosham, H. A.: Fourth-order finite difference method for solving Burgers equation, Appl. math. Comput. 170, 781-800 (2005) · Zbl 1084.65078 · doi:10.1016/j.amc.2004.12.052
[16] Kaysar Rahman, Nurmamat Helil, Rahmatjan Yimin, Some New Semi-Implicit Finite Difference Schemes for Numerical Solution of Burgers Equation, International Conference on Computer Application and System Modeling (ICCASM 2010), 978-1-4244-7237-6/10/$26.00 \copyright 20l0 IEEE V14-451.$ · Zbl 1324.65112
[17] Altıparmak, Kemal: Numerical solution of Burgers equation with factorized diagonal Padé´ approximation, Int. J. Numer. methods heat fluid flow 21, No. 3, 310-319 (2011) · Zbl 1231.65139 · doi:10.1108/09615531111108486
[18] Raslan, K. R.: A collocation solution for Burgers equation using quadratic B-spline finite elements, Int. J. Comput. math. 80, No. 7, 931-938 (2003) · Zbl 1037.65103 · doi:10.1080/0020716031000079554
[19] Ramadan, M. A.; El-Danaf, T. S.; Alaal, F. E. I. Abd: Application of the non-polynomial spline approach to the solution of the Burgers equation, Open appl. Math. J. 1, 15-20 (2007) · Zbl 1322.65086
[20] Cecchi, M. Morandi; Nociforo, R.; Grego, P. Patuzzo: Space-time finite elements numerical solution of Burgers problems, Le matematiche Li (Fasc. I), 43-57 (1996) · Zbl 0904.35081
[21] Xu, Min; Wang, Ren-Hong; Zhang, Ji-Hong; Fang, Qin: A novel numerical scheme for solving Burgers equation, Appl. math. Comput. 217, 4473-4482 (2011) · Zbl 1207.65111 · doi:10.1016/j.amc.2010.10.050
[22] Mittal, R. C.; Singhal, P.: Numerical solution of burger’s equation, Commun. numer. Methods eng. 9, 397-406 (1993) · Zbl 0782.65147 · doi:10.1002/cnm.1640090505
[23] Mittal, R. C.; Singhal, P.: Numerical solution of periodic burger equation, Ind. J. Pure appl. Math. 27, No. 7, 689-700 (1996) · Zbl 0859.76053
[24] Kutulay, S.; Bahadir, A. R.; Özdes, A.: Numerical solution of the one-dimensional Burgers equation: explicit and exact-explicit finite difference methods, J. comput. Appl. math. 103, 251-261 (1999) · Zbl 0942.65094 · doi:10.1016/S0377-0427(98)00261-1
[25] Kutulay, S.; Esen, A.; Dag, 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 · doi:10.1016/j.cam.2003.09.043
[26] Xie, Shu-Sen; Heo, Sunyeong; Kim, Seokchan; Woo, Gyungsoo; Yi, Sucheol: Numerical solution of one-dimensional Burgers equation using reproducing kernel function, J. comput. Appl. math. 214, 417-434 (2008) · Zbl 1140.65069 · doi:10.1016/j.cam.2007.03.010
[27] Özis, T.; Esen, A.; Kutluay, S.: Numerical solution of Burgers equation by quadratic B-spline finite elements, Appl. math. Comput. 165, 237-249 (2005) · Zbl 1070.65097 · doi:10.1016/j.amc.2004.04.101
[28] Liao, Wenyuan: An implicit fourth-order compact finite difference scheme for one-dimensional Burgers equation, Appl. math. Comput. 206, 755-764 (2008) · Zbl 1157.65438 · doi:10.1016/j.amc.2008.09.037
[29] Jiang, Ziwu; Wang, Renhong: An improved numerical solution of Burgers equation by cubic B-spline quasi-interpolation, J. inform. Comput. sci. 7, No. 5, 1013-1021 (2010)