Research Article

Year 2018,
Volume: 15 Issue: 1, - , 30.05.2018
### Abstract

### Keywords

### References

In this study differential quadrature method based on quintic B-spline functions is setup for numerical solutions for nonlinear viscous Burgers’ equation. After space discretization with differential quadrature and application of boundary conditions, the resultant ordinary differential equation system is integrated in time by using Runge-Kutta method of order four. The method is validated by solving two initial value problems for the Burgers’ equation. The errors of the numerical solutions are measured by using discrete maximum norm. A comparison with some earlier works also given for the problem modeling fadeout of an initial shock.

quintic B-spline Burgers’ Equation differential quadrature method shock wave sinusoidal disturbance

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- [2] M. Seydaoglu, U. Erdogan, T. Ozis¸, Numerical solution of Burgers’ equation with high order splitting methods, Journal of Computational and Applied Mathematics, 291, 410-421, 2016.
- [3] J. M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence, Adv. in App. Mech. I, 171-199, 1948.
- [4] D. C. Leslie, Developments in the Theory of Turbulence, Clarendon Press Oxford, 1973.
- [5] P. J. Olver, C. Shakiban, Applied Mathematics, 2004.
- [6] J. Billingham, A. C. King, Wave Motion, Cambridge University Press, Cambridge, 2000.
- [7] M.J., Lighthill, Viscocity effects in sound waves of finite amplitude, in Batchlor, G. K. and Davies, R. M. (Eds), Survey in Mechanics, Cambridge University Press, Cambridge, 250-351, 1956.
- [8] P. Kachroo, K. Ozbay, S. Kang, J. A. Burns, System Dynamics and Feedback Control Problem Formulations for Real Time Dynamic Traffic Routing, Mathl. Comput. Modelling, 27, Q-11, 27-49, 1998.
- [9] S. N. Gurbatov, A. I. Saichev, S. F. Shandarin, The large-scale structure of the Universe in the frame of the model equation og non-linear diffusion, Mon. Not. R. Astr. Soc., 236, 385-402, 1989.
- [10] J. L. Katz, M. L. Green, A Burgers model of interstellar dynamics, Astron. Astrophys., 161, 139-141, 1986.
- [11] L. Kofman, A. C. Raga, Modeling Structures of Knots in Jet Flows with the Burgers Equation, The Astrophysical Journal, 390, 359-364, 1992.
- [12] A. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2009.
- [13] J. D. Cole, On a Quasi-linear Parabolic Equation in Aerodynamics, Quarterly of Applied Math., 9, 225-236, 1951.
- [14] E. Hopf, The Partial Differential Equation Ut +UUx = mUxx; Comm. Pure App. Math., 3, 201-230, 1950.
- [15] E. Benton, G.W. Platzman, A table of solutions of the one-dimensional Burgers equations, Quart. Appl. Math., 30, 195-212, 1972.
- [16] W. Liao, An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Applied Mathematics and Computation, 206, 755-764, 206.
- [17] M. Sari, G. Gurarslan, A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation, Applied Mathematics and Computation, 208, 475-483, 2009.
- [18] A. Dogan, A Galerkin finite element approach to Burgers’ equation, Applied Mathematics and Computation, 157, 331- 346, 2004.
- [19] C. G. Zhu, R. H. Wang, Numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation, Applied Mathematics and Computation, 208, 260-272, 2009.
- [20] R. C. Mittal, R. K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method, Applied Mathematics and Computation, 218, 7839-7855, 2012.
- [21] A. A. Soliman, A Galerkin Solution for Burgers’ Equation Using Cubic B-Spline Finite Elements, Abstract and Applied Analysis, 2012, Article ID 527467, 1-15, 2012.
- [22] A. H. A. Ali, L. R. T. Gardner, G. A. Gardner, A Galerkin Approach to the Solution of Burgers’ Equation, Maths Preprint Series, no. 90.04, University College of North Wales, Bangor, 1990.
- [23] L. R. T. Gardner, G. A. Gardner, B-spline Finite Elements, U.C.N.W. Maths Preprint, 91.10.
- [24] B. Saka, I. Dag, Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation, Chaos, Solitons and Fractals, 32, 1125-1137, 2007.
- [25] I. Dag, D. Irk, A. Sahin, B-spline collocation methods for numerical solutions of the Burgers’ equation, Mathematical Problems in Engineering, 5, 521-538, 2005.
- [26] A. Korkmaz, I. Dag, Cubic B-spline differential quadrature methods and stability for Burgers’ equation, Engineering Computations, 30, 3, 320-344, 2013.
- [27] A. Korkmaz, A. M. Aksoy, I. Dag, Quartic B-spline differential quadrature method, International Journal of Nolinear Science, 11, 4, 403-411, 2011.
- [28] A. Korkmaz, I. Dag, Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation, Journal of the Franklin Institute, 348, 10, 2863-2875, 2011.
- [29] R. Bellman, B. G. Kashef, J. Casti, Differential Quadrature: A Tecnique for the Rapid Solution of Nonlinear Differential Equations, Journal of Computational Physics 10, 40-52, 1972.
- [30] R. Bellman , Kashef Bayesteh, Lee E. S., Vasudevan R., Differential quadrature and splines, Computers and mathematics with applications, pp. 371-376. Pergamon, Oxford, 1976.
- [31] C. Shu, Y.L. Wu, Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids 2007; 53:969–984.
- [32] J. R. Quan, C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Engrg., Vol 13, 779-788, 1989.
- [33] J. R. Quan, C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-II, Comput. Chem. Engrg., Vol 13, 1017-1024, 1989.
- [34] C. Shu, H. Xue, Explicit Computation ofWeighting Coefficients in the Harmonic Differential Quadrature, Journal of Sound and Vibration,204, 3, 549-555, 1997.
- [35] Q. Guo, H. Zhong, Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration, 269, 413-420, 2004.
- [36] H. Zhong, Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates, Applied Mathematical Modelling, 28, 353-366, 2004.
- [37] H. Zhong, M. Lan, Solution of nonlinear initial-value problems by the spline-based differential quadrature method, Journal of Sound and Vibration, 296, 908-918, 2006.
- [38] P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1989.
- [39] H. Nguyen, J. Reynen, A space-time finite element approach to Burgers equation, in Taylor, C., Hinton, E., Owen, D.R.J. and Onate, E. (Eds), Numerical Methods for Non-linear Problems, Vol. 2, Pineridge Publisher, Swansea, 718-728,1982.
- [40] B. V. R. Kumar, M. Mehra, Wavelet-Taylor Galerkin Method for the Burgers Equation, BIT Numerical Mathematics, 45, 543-560, 2005.

Year 2018,
Volume: 15 Issue: 1, - , 30.05.2018
### Abstract

### References

- [1] H. Bateman, Some Recent Researches on the Motion of Fluids, Mon.Weather Rev., 43, 163-170, 1915.
- [2] M. Seydaoglu, U. Erdogan, T. Ozis¸, Numerical solution of Burgers’ equation with high order splitting methods, Journal of Computational and Applied Mathematics, 291, 410-421, 2016.
- [3] J. M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence, Adv. in App. Mech. I, 171-199, 1948.
- [4] D. C. Leslie, Developments in the Theory of Turbulence, Clarendon Press Oxford, 1973.
- [5] P. J. Olver, C. Shakiban, Applied Mathematics, 2004.
- [6] J. Billingham, A. C. King, Wave Motion, Cambridge University Press, Cambridge, 2000.
- [7] M.J., Lighthill, Viscocity effects in sound waves of finite amplitude, in Batchlor, G. K. and Davies, R. M. (Eds), Survey in Mechanics, Cambridge University Press, Cambridge, 250-351, 1956.
- [8] P. Kachroo, K. Ozbay, S. Kang, J. A. Burns, System Dynamics and Feedback Control Problem Formulations for Real Time Dynamic Traffic Routing, Mathl. Comput. Modelling, 27, Q-11, 27-49, 1998.
- [9] S. N. Gurbatov, A. I. Saichev, S. F. Shandarin, The large-scale structure of the Universe in the frame of the model equation og non-linear diffusion, Mon. Not. R. Astr. Soc., 236, 385-402, 1989.
- [10] J. L. Katz, M. L. Green, A Burgers model of interstellar dynamics, Astron. Astrophys., 161, 139-141, 1986.
- [11] L. Kofman, A. C. Raga, Modeling Structures of Knots in Jet Flows with the Burgers Equation, The Astrophysical Journal, 390, 359-364, 1992.
- [12] A. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2009.
- [13] J. D. Cole, On a Quasi-linear Parabolic Equation in Aerodynamics, Quarterly of Applied Math., 9, 225-236, 1951.
- [14] E. Hopf, The Partial Differential Equation Ut +UUx = mUxx; Comm. Pure App. Math., 3, 201-230, 1950.
- [15] E. Benton, G.W. Platzman, A table of solutions of the one-dimensional Burgers equations, Quart. Appl. Math., 30, 195-212, 1972.
- [16] W. Liao, An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Applied Mathematics and Computation, 206, 755-764, 206.
- [17] M. Sari, G. Gurarslan, A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation, Applied Mathematics and Computation, 208, 475-483, 2009.
- [18] A. Dogan, A Galerkin finite element approach to Burgers’ equation, Applied Mathematics and Computation, 157, 331- 346, 2004.
- [19] C. G. Zhu, R. H. Wang, Numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation, Applied Mathematics and Computation, 208, 260-272, 2009.
- [20] R. C. Mittal, R. K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method, Applied Mathematics and Computation, 218, 7839-7855, 2012.
- [21] A. A. Soliman, A Galerkin Solution for Burgers’ Equation Using Cubic B-Spline Finite Elements, Abstract and Applied Analysis, 2012, Article ID 527467, 1-15, 2012.
- [22] A. H. A. Ali, L. R. T. Gardner, G. A. Gardner, A Galerkin Approach to the Solution of Burgers’ Equation, Maths Preprint Series, no. 90.04, University College of North Wales, Bangor, 1990.
- [23] L. R. T. Gardner, G. A. Gardner, B-spline Finite Elements, U.C.N.W. Maths Preprint, 91.10.
- [24] B. Saka, I. Dag, Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation, Chaos, Solitons and Fractals, 32, 1125-1137, 2007.
- [25] I. Dag, D. Irk, A. Sahin, B-spline collocation methods for numerical solutions of the Burgers’ equation, Mathematical Problems in Engineering, 5, 521-538, 2005.
- [26] A. Korkmaz, I. Dag, Cubic B-spline differential quadrature methods and stability for Burgers’ equation, Engineering Computations, 30, 3, 320-344, 2013.
- [27] A. Korkmaz, A. M. Aksoy, I. Dag, Quartic B-spline differential quadrature method, International Journal of Nolinear Science, 11, 4, 403-411, 2011.
- [28] A. Korkmaz, I. Dag, Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation, Journal of the Franklin Institute, 348, 10, 2863-2875, 2011.
- [29] R. Bellman, B. G. Kashef, J. Casti, Differential Quadrature: A Tecnique for the Rapid Solution of Nonlinear Differential Equations, Journal of Computational Physics 10, 40-52, 1972.
- [30] R. Bellman , Kashef Bayesteh, Lee E. S., Vasudevan R., Differential quadrature and splines, Computers and mathematics with applications, pp. 371-376. Pergamon, Oxford, 1976.
- [31] C. Shu, Y.L. Wu, Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids 2007; 53:969–984.
- [32] J. R. Quan, C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Engrg., Vol 13, 779-788, 1989.
- [33] J. R. Quan, C. T. Chang, New sightings in involving distributed system equations by the quadrature methods-II, Comput. Chem. Engrg., Vol 13, 1017-1024, 1989.
- [34] C. Shu, H. Xue, Explicit Computation ofWeighting Coefficients in the Harmonic Differential Quadrature, Journal of Sound and Vibration,204, 3, 549-555, 1997.
- [35] Q. Guo, H. Zhong, Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration, 269, 413-420, 2004.
- [36] H. Zhong, Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates, Applied Mathematical Modelling, 28, 353-366, 2004.
- [37] H. Zhong, M. Lan, Solution of nonlinear initial-value problems by the spline-based differential quadrature method, Journal of Sound and Vibration, 296, 908-918, 2006.
- [38] P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1989.
- [39] H. Nguyen, J. Reynen, A space-time finite element approach to Burgers equation, in Taylor, C., Hinton, E., Owen, D.R.J. and Onate, E. (Eds), Numerical Methods for Non-linear Problems, Vol. 2, Pineridge Publisher, Swansea, 718-728,1982.
- [40] B. V. R. Kumar, M. Mehra, Wavelet-Taylor Galerkin Method for the Burgers Equation, BIT Numerical Mathematics, 45, 543-560, 2005.

There are 40 citations in total.

Primary Language | English |
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Journal Section | Articles |

Authors | |

Publication Date | May 30, 2018 |

Published in Issue | Year 2018 Volume: 15 Issue: 1 |