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Year 2015, Volume: 12 Issue: 1, - , 01.05.2015

Abstract

References

  • [1] H. Bateman, Some recent researches on the motion of fluids, Montly Weather Rev. 43(4), (1915), 163-170.
  • [2] J. M. Burgers, A mathematical model illustrating the theory of turbulance, Adv. Appl. Mech. 1 (1948) , 225-236.
  • [3] J. D. Cole, On a quasi-linear parabolic equations occuring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  • [4] J. Caldwell, P. Smith, Solution of Burgers’ equation with a large Reynolds number, Appl. Math. Modelling 6 (1982), 381-385.
  • [5] S. Kutluay, A. R. Bahadır and A.Ozdes, Numerical solution of one dimensional Burgers’ equation explicit and exact-explicit finite difference method, J. Comput. Appl. Math. 103(2), (1999), 251-256.
  • [6] R. C. Mittal and P. Singhal, Numerical solution of Burgers’ equation, Comm. Numer. Methods Engrg. 9 (1993), 397-406.
  • [7] R. C. Mittal and P. Singhal and T. V. Singh, Numerical solution of periodic Burger equation, Indian J. Pure Appl. Math. 27(7), (1996), 689-700.
  • [8] M. B. Abd-el-Malek and S. M. A. El Mansi, Group theoretic methods applied to Burgers’ equation, J. Comput. Appl. Math. 115 (2000), 1-12.
  • [9] A. Dogan, A Galerkin finite element approach to Burgers’ equation, Appl. Math. Comput. 157 (2004), 331-346.
  • [10] T. Ozis, E. N. Aksan and A. Ozdes, A finite element approach for solution of Burgers’ equation, Appl. Math. Comput. 139 (2003), 417-428.
  • [11] L. R. T. Gardner, G. A. Gardner and A. Dogan, A Petrov-Galerkin finite element scheme for Burgers’ equation, Arab. J. Sci. Engrg. 22 (1997), 99-109.
  • [12] L. R. T. Gardner, G. A. Gardner and A. Dogan, A least-squares finite element scheme for Burgers’ equation, University of Wales, Bangor, Mathematics, Preprint 96.01, (1996).
  • [13] S. Kutluay, E. N. Aksan and I. Dag, Numerical solution of Burgers’ by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004), 21-33.
  • [14] H. Nguyen and J. Reynen, A space-time finite element approach to Burgers’ equation , in:C. Taylor, E. Hinton, D. R. J. Owen, E. Onate (Eds.), Numerical methods for non-linear Problems, Pineridge Publisher, Swansea, 2(1982), 718-728.
  • [15] A. H. A. Ali, G. A. Gardner and L. R. T. Gardner, A collocation solution for Burgers’ equation using cubic B-spline finite elements, Comput. Methods. Appl. Mech. Engrg. 100 (1992), 325-337.
  • [16] M. A. Ramadan, T. S. El-Danaf and Abd. Alael El, A Numerical solution of Burgers’ equation using septic Bsplines, Chaos Solitons and Fractals 26 (2005), 1249-1258.
  • [17] I. Dag, D.Irk and B. Saka, Numerical solution of Burgers’ equation using cubic B-splines, Appl. Math. Comput. 163 (2005), 199-211.
  • [18] M. A. Ramadan and T. S. El-Danaf, Numerical treatment for the modified burgers equation, Math. Comput. in Simul. 70 (2005), 90-98.
  • [19] Y. Duan , R. Liu and Y. Jiang, Lattice Boltzmann model for the modified Burgers’ equation, Appl. Math. and Comput. 202 (2008), 489–497.
  • [20] B. Saka, I. Dag and D. Irk, Numerical Solution of the Modified Burgers Equation by the Quintic B-spline Galerkin Finite Element Method, Int. J. Math. Statis. 1 (2007), 86-97.
  • [21] D. Irk, Sextic B-spline collocation method for the modified Burgers’ equation, Kybernetes, 38(9), (2009), 1599- 1620.
  • [22] T. Roshan, K. S. Bhamra, Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method, Appl. Math. Comput. 218 (2011), 3673-3679.
  • [23] R. P. Zhang, X. J. Yu and G. Z. Zhao, Modified Burgers’ equation by the local discontinuous Galerkin method, Chin. Phys. B 22(3) 2013.
  • [24] A. G. Bratsos, A fourth-order numerical scheme for solving the modified Burgers equation, Computers and Mathematics with Applications 60 (2010), 1393- 1400.
  • [25] S. L. Harris, Sonic shocks governed by the modified Burgers equation, Eur. J. Appl. Math. 6 (1996), 75-107.
  • [26] R. Bellman, B. G. Kashef and J. Casti, Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, 10 (1972), 40-52.
  • [27] R. Bellman, B. Kashef, E. S. Lee and R. Vasudevan, Differential Quadrature and Splines, Computers and Mathematics with Applications, Pergamon, Oxford, 1 (3,4), (1975), 371-376.
  • [28] J. R. Quan and C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Eng. 13 (1989a), 779-788.
  • [29] J. R. Quan and C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-II, Comput. Chem. Eng. 13 (1989b), 1017-1024.
  • [30] C. Shu and B. E. Richards, Application of generalized differential quadrature to solve two dimensional incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 15 (1992), 791-798.
  • [31] C. Shu and H. Xue, Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration 204(3), (1997), 549-555.
  • [32] H. Zhong, Spline-based differential quadrature for fourth order equations and its application to Kirchhoff plates, Applied Mathematical Modelling 28 (2004), 353-366.
  • [33] Q. Guo and H. Zhong, Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration 269 (2004), 413-420.
  • [34] H. Zhong and M. Lan, Solution of nonlinear initial-value problems by the spline-based differential quadrature method, Journal of Sound and Vibration 296 (2006), 908-918.
  • [35] J. Cheng, B. Wang and S. Du, 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 (2005), 6181-6201.
  • [36] C. Shu and Y. L. Wu, Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids 53 (2007), 969-984.
  • [37] A. G. Striz, X. Wang and C. W. Bert, Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica 111 (1995), 85-94.
  • [38] I. Bonzani, Solution of non-linear evolution problems by parallelized collocation-interpolation methods, Computers & Mathematics and Applications 34 (12), (1997), 71-79.

B-spline Differential Quadrature Method for the Modified Burgers' Equation

Year 2015, Volume: 12 Issue: 1, - , 01.05.2015

Abstract

In this study, the Quintic B-spline Differential Quadrature method (QBDQM) is applied to find
the numerical solution of the modified Burgers’ equation (MBE). The efficiency and accuracy of the method
are measured by calculating the maximum error norm L∞ and the discrete root mean square error L2. The
obtained numerical results are compared with published numerical results and the comparison shows that the
method is an effective numerical scheme to solve the MBE. A rate of convergence analysis is also given.

References

  • [1] H. Bateman, Some recent researches on the motion of fluids, Montly Weather Rev. 43(4), (1915), 163-170.
  • [2] J. M. Burgers, A mathematical model illustrating the theory of turbulance, Adv. Appl. Mech. 1 (1948) , 225-236.
  • [3] J. D. Cole, On a quasi-linear parabolic equations occuring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  • [4] J. Caldwell, P. Smith, Solution of Burgers’ equation with a large Reynolds number, Appl. Math. Modelling 6 (1982), 381-385.
  • [5] S. Kutluay, A. R. Bahadır and A.Ozdes, Numerical solution of one dimensional Burgers’ equation explicit and exact-explicit finite difference method, J. Comput. Appl. Math. 103(2), (1999), 251-256.
  • [6] R. C. Mittal and P. Singhal, Numerical solution of Burgers’ equation, Comm. Numer. Methods Engrg. 9 (1993), 397-406.
  • [7] R. C. Mittal and P. Singhal and T. V. Singh, Numerical solution of periodic Burger equation, Indian J. Pure Appl. Math. 27(7), (1996), 689-700.
  • [8] M. B. Abd-el-Malek and S. M. A. El Mansi, Group theoretic methods applied to Burgers’ equation, J. Comput. Appl. Math. 115 (2000), 1-12.
  • [9] A. Dogan, A Galerkin finite element approach to Burgers’ equation, Appl. Math. Comput. 157 (2004), 331-346.
  • [10] T. Ozis, E. N. Aksan and A. Ozdes, A finite element approach for solution of Burgers’ equation, Appl. Math. Comput. 139 (2003), 417-428.
  • [11] L. R. T. Gardner, G. A. Gardner and A. Dogan, A Petrov-Galerkin finite element scheme for Burgers’ equation, Arab. J. Sci. Engrg. 22 (1997), 99-109.
  • [12] L. R. T. Gardner, G. A. Gardner and A. Dogan, A least-squares finite element scheme for Burgers’ equation, University of Wales, Bangor, Mathematics, Preprint 96.01, (1996).
  • [13] S. Kutluay, E. N. Aksan and I. Dag, Numerical solution of Burgers’ by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004), 21-33.
  • [14] H. Nguyen and J. Reynen, A space-time finite element approach to Burgers’ equation , in:C. Taylor, E. Hinton, D. R. J. Owen, E. Onate (Eds.), Numerical methods for non-linear Problems, Pineridge Publisher, Swansea, 2(1982), 718-728.
  • [15] A. H. A. Ali, G. A. Gardner and L. R. T. Gardner, A collocation solution for Burgers’ equation using cubic B-spline finite elements, Comput. Methods. Appl. Mech. Engrg. 100 (1992), 325-337.
  • [16] M. A. Ramadan, T. S. El-Danaf and Abd. Alael El, A Numerical solution of Burgers’ equation using septic Bsplines, Chaos Solitons and Fractals 26 (2005), 1249-1258.
  • [17] I. Dag, D.Irk and B. Saka, Numerical solution of Burgers’ equation using cubic B-splines, Appl. Math. Comput. 163 (2005), 199-211.
  • [18] M. A. Ramadan and T. S. El-Danaf, Numerical treatment for the modified burgers equation, Math. Comput. in Simul. 70 (2005), 90-98.
  • [19] Y. Duan , R. Liu and Y. Jiang, Lattice Boltzmann model for the modified Burgers’ equation, Appl. Math. and Comput. 202 (2008), 489–497.
  • [20] B. Saka, I. Dag and D. Irk, Numerical Solution of the Modified Burgers Equation by the Quintic B-spline Galerkin Finite Element Method, Int. J. Math. Statis. 1 (2007), 86-97.
  • [21] D. Irk, Sextic B-spline collocation method for the modified Burgers’ equation, Kybernetes, 38(9), (2009), 1599- 1620.
  • [22] T. Roshan, K. S. Bhamra, Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method, Appl. Math. Comput. 218 (2011), 3673-3679.
  • [23] R. P. Zhang, X. J. Yu and G. Z. Zhao, Modified Burgers’ equation by the local discontinuous Galerkin method, Chin. Phys. B 22(3) 2013.
  • [24] A. G. Bratsos, A fourth-order numerical scheme for solving the modified Burgers equation, Computers and Mathematics with Applications 60 (2010), 1393- 1400.
  • [25] S. L. Harris, Sonic shocks governed by the modified Burgers equation, Eur. J. Appl. Math. 6 (1996), 75-107.
  • [26] R. Bellman, B. G. Kashef and J. Casti, Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, 10 (1972), 40-52.
  • [27] R. Bellman, B. Kashef, E. S. Lee and R. Vasudevan, Differential Quadrature and Splines, Computers and Mathematics with Applications, Pergamon, Oxford, 1 (3,4), (1975), 371-376.
  • [28] J. R. Quan and C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Eng. 13 (1989a), 779-788.
  • [29] J. R. Quan and C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-II, Comput. Chem. Eng. 13 (1989b), 1017-1024.
  • [30] C. Shu and B. E. Richards, Application of generalized differential quadrature to solve two dimensional incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 15 (1992), 791-798.
  • [31] C. Shu and H. Xue, Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration 204(3), (1997), 549-555.
  • [32] H. Zhong, Spline-based differential quadrature for fourth order equations and its application to Kirchhoff plates, Applied Mathematical Modelling 28 (2004), 353-366.
  • [33] Q. Guo and H. Zhong, Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration 269 (2004), 413-420.
  • [34] H. Zhong and M. Lan, Solution of nonlinear initial-value problems by the spline-based differential quadrature method, Journal of Sound and Vibration 296 (2006), 908-918.
  • [35] J. Cheng, B. Wang and S. Du, 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 (2005), 6181-6201.
  • [36] C. Shu and Y. L. Wu, Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids 53 (2007), 969-984.
  • [37] A. G. Striz, X. Wang and C. W. Bert, Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica 111 (1995), 85-94.
  • [38] I. Bonzani, Solution of non-linear evolution problems by parallelized collocation-interpolation methods, Computers & Mathematics and Applications 34 (12), (1997), 71-79.
There are 38 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Ali Başhan This is me

Seydi Battal Gazi Karakoc

Turabi Geyikli This is me

Publication Date May 1, 2015
Published in Issue Year 2015 Volume: 12 Issue: 1

Cite

APA Başhan, A., Karakoc, S. B. G., & Geyikli, T. (2015). B-spline Differential Quadrature Method for the Modified Burgers’ Equation. Cankaya University Journal of Science and Engineering, 12(1).
AMA Başhan A, Karakoc SBG, Geyikli T. B-spline Differential Quadrature Method for the Modified Burgers’ Equation. CUJSE. May 2015;12(1).
Chicago Başhan, Ali, Seydi Battal Gazi Karakoc, and Turabi Geyikli. “B-Spline Differential Quadrature Method for the Modified Burgers’ Equation”. Cankaya University Journal of Science and Engineering 12, no. 1 (May 2015).
EndNote Başhan A, Karakoc SBG, Geyikli T (May 1, 2015) B-spline Differential Quadrature Method for the Modified Burgers’ Equation. Cankaya University Journal of Science and Engineering 12 1
IEEE A. Başhan, S. B. G. Karakoc, and T. Geyikli, “B-spline Differential Quadrature Method for the Modified Burgers’ Equation”, CUJSE, vol. 12, no. 1, 2015.
ISNAD Başhan, Ali et al. “B-Spline Differential Quadrature Method for the Modified Burgers’ Equation”. Cankaya University Journal of Science and Engineering 12/1 (May 2015).
JAMA Başhan A, Karakoc SBG, Geyikli T. B-spline Differential Quadrature Method for the Modified Burgers’ Equation. CUJSE. 2015;12.
MLA Başhan, Ali et al. “B-Spline Differential Quadrature Method for the Modified Burgers’ Equation”. Cankaya University Journal of Science and Engineering, vol. 12, no. 1, 2015.
Vancouver Başhan A, Karakoc SBG, Geyikli T. B-spline Differential Quadrature Method for the Modified Burgers’ Equation. CUJSE. 2015;12(1).