An exponential cubic B-spline algorithm for solving the nonlinear Coupled Burgers’ equation

Document Type : Research Paper


1 Mathematics and Computer Science Department, Science and Letters Faculty, Eskisehir Osmangazi University, 26480, Eskisehir, Turkey.

2 Idris Dag Computer Engineering Department, Engineering and Architecture Faculty, Eskisehir Osmangazi University, 26480, Eskisehir, Turkey.


The collocation method based on the exponential cubic B-splines (ECB-splines) together with the Crank Nicolson formula is used to solve nonlinear coupled Burgers’ equation (CBE). This method is tested by studying three different problems. The proposed scheme is compared with some existing methods. It produced accurate results with the suitable selection of the free parameter of the ECB-spline function. It produces accurate results. Stability of the fully discretized CBE is investigated by the Von Neumann analysis.


  • [1]          A. Ali, A. Islam, and S. Haq, A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 10 (2009), 406–422.
  • [2]          A. M. Aksoy, Numerical solutions of some partial differential equations using the Taylor collocation-extended cubic B-spline functions, Department of Mathematics, Doctoral Disser- tation, 2012.
  • [3]          I. Dag and O. Ersoy, The exponential cubic B-spline algorithm for Fisher equation, Chaos Solitons $ Fractals, 86 (2015), 101–106.
  • [4]          I. Dag and O. Ersoy, Numerical solution of generalized Burgers-Fisher equation by exponential cubic B-spline collocation Method, Aip Conference Proceedings, 1648 (2015), 370008.
  • [5]          O. Ersoy, A. Korkmaz, and I. Dag, Exponential B-splines for numerical solutions to some Boussinesq systems for water waves, Mediterranean Journal of Mathematics 13 (2016), 4975– 4994.
  • [6]          O. Ersoy and I. Dag, Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms, Open Physics, 13 (2015), 414–425.
  • [7]          O. Ersoy, I. Dag, and N. Adar, Exponential twice continuously differentiable B-spline algorithm for Burgers’ equation, Ukrainian Mathematical Journal, 70 (2018), 906–921.
  • [8]          O. E. Hepson, Generation of the exponential cubic B-spline collocation solutions for some partial differential equation systems, Department of Mathematics, Doctoral Dissertation, 2015.
  • [9]          O. E. Hepson, A. Korkmaz, and I. Dag, Exponential B-spline collocaiton solutions to the Gard- ner equation, International Journal of Computer Mathematics, 97 (2020), 837–850.
  • [10]        S. Islam, S. Haq and M. Uddin, A mesh free interpolation method for the numerical solution of the coupled nonlinear partial differential equations, Engineering Analysis with Boundary Elements, 33 (2009), 399–409.
  • [11]        A. H. Khater, R. S. Temsah, and M. M. Hassan, A chebyshev spectral collocation method for solving Burgers’-type equations, Journal of Computational and Applied Mathematics, 22 (2008), 333–350.
  • [12]        A. Korkmaz, O. Ersoy, and I. Dag, Motion of patterns modeled by the Gray-Scott autocatalsis system in one dimension, MATCH Communications in Mathematical and in Computer Chem- istry, 77 (2017), 507–521.
  • [13]        M. Kumar and S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers’ equation, Computer Physics Communications, 185 (2014), 809–817.
  • [14]        S. Kutluay and Y. Ucar, Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method, Mathematical Methods in the Applied Sciences, 36 (2013), 2403–2415.
  • [15]        J. Liu and G. Hou, Numerical solutions of the space-and time-fractional coupled Burgers equa- tions by generalized differential transform method, Applied Mathematics and Computation, 217 (2011), 7001–7008.
  • [16]        B. J. McCartin, Computation of exponential splines, SIAM Journal on Scientific and Statistical Computing, 11 (1990), 242–262.
  • [17]        B. J. McCartin. Theory of exponential splines. Journal of Approximation Theory, 66 (1991),  1–23.
  • [18]        B. J. McCartin and A. Jameson, Numerical solution of nonlinear hyperbolic conservation laws using exponential splines, Computational Mechanics, 6 (1990), 77-91.
  • [19]        R. C. Mittal and G. Arora, Numerical solution of the coupled viscous Burgers equation, Com- munications in Nonlinear Science and Numerical Simulation, 16 (2011), 1304–1313.
  • [20]        R. C. Mittal and R. Rohila, Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method, Chaos Solitons $ Fractals, 92 (2016), 9–18.
  • [21]        R. C. Mittal and Ram Jiwari, A differential quadrature method for numerical solutions of Burgers’-type equations, International Journal of Numerical Methods for Heat & Fluid Flow, 22 (2012), 880–895.
  • [22]        R. C. Mittal and A. Tripathi, A collocation method for numerical solutions of coupled Burg- ers’ equations, International Journal for Computational Methods in Engineering Science and Mechanics, 15 (2014), 457–471.
  • [23]        R. Mohammadi, Exponential B-spline solution of convection-diffusion Equations, Applied Mathematics, 4 (2013), 933–944.
  • [24]        R. Mokhtari, A. S. Toodar, and N. G. Chegini, Application of the generalized differential quad- rature method in solving Burgers’ equations, Communications in Theoretical Physics, 56 (2011), 1009–1015.
  • [25]        A. T. Onarcan and O. E. Hepson, Higher order trigonometric B-spline algorithms to the solution of coupled Burgers’ equation, Aip Conference Procedeings, 1926 (2018), 020044.
  • [26]        D. Radunovic, Multiresolution exponential B-splines and singularly perturbed boundary problem, Numerical Algorithms, 47 (2008), 191–210.
  • [27]        S. F. Radwan, On the fourth-order accurate compact ADI scheme for solving the unsteady nonlinear coupled Burgers’ equations, Journal of Nonlinear Mathematical Physics, 6 (1999), 13–34.
  • [28]        S. C. S. Rao and M. Kumar, Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems, Applied Numerical Mathematics, 58 (2008), 1572–1581.
  • [29]        A. Rashid and A. I. B. MD. I˙smail, A fourier pseudospectral method for solving coupled viscous Burgers equations, Computational Methods in Applied Mathematics, 9 (2009), 412-420.
  • [30]        K. R. Raslan, T. S. El-Danaf, and K. K. Ali, Collocation method with cubic trigonometric B-spline algorithm for solving coupled Burgers’ equations, Far East Journal of Applied Mathe- matics, 95 (2016), 109–123.
  • [31]        I. Sadek and I. Kucuk, A robust technique for solving optimal control of coupled Burgers’ equations, IMA Journal of Mathematical Control and Information, 28 (2011), 239–250.
  • [32]        H. M. Salih, L. N. M. Tawfiq, and Z. R. Yahya, Numerical solution of the coupled viscous burgers equation via cubic trigonometric b-spline approach, Mathematics and Statistics, 2 (2016), 1–13.
  • [33]        M. Sakai and R. A. Usmani, A class of simple exponential B-splines and their application to numerical solution to singular perturbation problems, Numerical Mathematics 55 (1989), 493- 500.
  • [34]        M. A. Shallal, K. K. Ali,  K.  R.  Raslan,  and  A.  H.  Taqi,  Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations, Arab Journal of Basic and Applied Sciences University of Bahrain, 26 (2019), 331–341.
  • [35]        S. Singh, S. Singh, and R. Arora, Numerical solution of second-order one-dimensional hyperbolic equation by exponential B-spline collocation method, Numerical Analysis and Applications, 10 (2017), 164-176.
  • [36]        V. K. Srivastava, M. K. Awasthi, and M. Tamsir, A fully implicit finite-difference solution to one dimensional coupled nonlinear Burgers’ equations, International Journal of Mathematical, Computational, Physical and Quantum Engineering, 7 (2013), 417–422.
  • [37]        V. K. Srivastava, M. Tamsir, M. K. Awasthi, and S. Sing, One-dimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method, Aip Advances, 4 (2014), 037119.