A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

• Authors: Eid Doha , Ali Bhrawy , Mohamed Abdelkawy , Ramy Hafez
• Languages of publication: EN

Abstracts

EN

This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers’ equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers’ equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.

Keywords

EN

nonlinear coupled viscous Burgers’ equation; Jacobi quadrature rule; pseudospectral scheme; implicit Runge-Kutta-Nyström scheme

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2014
• Volume: 12
• Issue: 2
• Pages: 111-122

Dates

published

1 - 2 - 2014

online

15 - 2 - 2014

Contributors

author

Eid Doha

• Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

author

Ali Bhrawy

author

Mohamed Abdelkawy

• Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

author

Ramy Hafez

• Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt

References

• [1] S. E. Esipov, Phys. Rev. E 52, 3711 (1995) http://dx.doi.org/10.1103/PhysRevE.52.3711[Crossref]
• [2] J. D. Cole, Q. Appl. Math. 9, 225 (1951)
• [3] J. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics (Academic Press, New York, 1948)
• [4] M. A. Abdou, A. A. Soliman, J. Comput. Appl. Math. 181, 245 (2005) http://dx.doi.org/10.1016/j.cam.2004.11.032[Crossref]
• [5] H. Aminikhah, Appl. Math. Model. 37, 5979 (2013) http://dx.doi.org/10.1016/j.apm.2012.12.013[Crossref]
• [6] A. H. Khater, R. S. Temsah, M. M. Hassan, J. Comput. Appl. Math. 222, 33 (2008) http://dx.doi.org/10.1016/j.cam.2007.11.007[Crossref]
• [7] D. Kaya, IJMMS 27, 675 (2001)
• [8] I. Kucuk, I. Sadek, Appl. Math. Comput. 210, 126 (2009) http://dx.doi.org/10.1016/j.amc.2008.12.056[Crossref]
• [9] M. T. Darvishi, S. Kheybari, F. Khani, Commun. Nonlinear Sci. Numer. Simul. 13, 2091 (2008) http://dx.doi.org/10.1016/j.cnsns.2007.05.023[Crossref]
• [10] M. Javidi, A. Golbabai, J. Math. Anal. Appl. 333, 1119 (2007) http://dx.doi.org/10.1016/j.jmaa.2006.12.018[Crossref]
• [11] E. Tohidi, A. H. Bhrawy, K. Erfani, Appl. Math. Model. 37, 4283 (2013) http://dx.doi.org/10.1016/j.apm.2012.09.032[Crossref]
• [12] Y. -Y. Kwan, Appl. Numer. Math. 59, 170 (2009) http://dx.doi.org/10.1016/j.apnum.2008.01.003[Crossref]
• [13] F. Chen, J. Shen, J. Comput. Phys. 231, 5016 (2012) http://dx.doi.org/10.1016/j.jcp.2012.03.001[Crossref]
• [14] F. Ghoreishi, S. Yazdani, Comput. Math. Appl. 61, 30 (2011) http://dx.doi.org/10.1016/j.camwa.2010.10.027[Crossref]
• [15] E. H. Doha, A. H. Bhrawy, R. M. Hafez, Commun. Nonlinear Sci. Numer. Simul. 17, 3802 (2012) http://dx.doi.org/10.1016/j.cnsns.2012.02.027[Crossref]
• [16] T. Kattelans, W. Heinrichs, J. Comput. Phys. 228, 4649 (2009) http://dx.doi.org/10.1016/j.jcp.2009.03.015[Crossref]
• [17] B. W. Li, S. Tian, Y. S. Sun, Z. M. Hu, J. Comput. Phys. 229, 1198 (2010) http://dx.doi.org/10.1016/j.jcp.2009.10.025[Crossref]
• [18] K. Parand, M. Shahini, M. Dehghan, J. Comput. Phys. 228, 8830 (2009) http://dx.doi.org/10.1016/j.jcp.2009.08.029[Crossref]
• [19] M. Dehghan, F. F. Izadi, Math. Comput. Model. 53, 1865 (2011) http://dx.doi.org/10.1016/j.mcm.2011.01.011[Crossref]
• [20] A. H. Bhrawy, Appl. Math. Comput. 222, 255 (2013) http://dx.doi.org/10.1016/j.amc.2013.07.056[Crossref]
• [21] Y. Ordokhani, M. Razzaghi, Appl. Math. Lett. 21, 4 (2008) http://dx.doi.org/10.1016/j.aml.2007.02.007[Crossref]
• [22] K. Zhang, J. Li, H. Song, Appl. Math. Comput. 218, 10848 (2012) http://dx.doi.org/10.1016/j.amc.2012.04.045[Crossref]
• [23] Y. Jiang, J. Ma, J. Comput. Appl. Math. 244, 115 (2013) http://dx.doi.org/10.1016/j.cam.2012.10.033[Crossref]
• [24] S. Yüzbasi, M. Sezer, B. Kemanci, Appl. Math. Model. 37, 2086 (2013) http://dx.doi.org/10.1016/j.apm.2012.05.012[Crossref]
• [25] A. H. Bhrawy, M. M. Tharwat, A. Yildirim, Appl. Math. Model. 37, 4245 (2013) http://dx.doi.org/10.1016/j.apm.2012.08.022[Crossref]
• [26] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, New York, 2006)
• [27] A. H. Bhrawy, A. S. Alofi, Commun. Nonlinear Sci. Numer. Simul. 17, 62 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.04.025[Crossref]
• [28] B. Bialecki, A. Karageorghis, J. Comput. Phys. 227, 8588 (2008) http://dx.doi.org/10.1016/j.jcp.2008.06.009[Crossref]
• [29] S. Esmaeili, M. Shamsi, Y. Luchko, Comput. Math. Appl. 62, 918 (2011) http://dx.doi.org/10.1016/j.camwa.2011.04.023[Crossref]
• [30] E. H. Doha, A. H. Bhrawy, R. M. Hafez, Math. Comput. Model. 53, 1820 (2011) http://dx.doi.org/10.1016/j.mcm.2011.01.002[Crossref]
• [31] D. Gottlieb, C. -W. Shu, SIAM Rev. 29, 644 (1997) http://dx.doi.org/10.1137/S0036144596301390[Crossref]
• [32] D. Tchiotsop, D. Wolf, V. Louis-Dorr, R. Husson, ECG data compression using Jacobi polynomials, Proceedings of the 29th Annual International Conference of the IEEE EMBS, August 23–26, 2007, Lyon, France (Engineering in Medicine and Biology Society, France, 2007) 1863
• [33] E. H. Doha, A. H. Bhrawy, Numer. Meth. Part. D. E. 25, 712 (2009) http://dx.doi.org/10.1002/num.20369[Crossref]
• [34] M. El-Kady, J. Korean Math. Soc. 49, 99 (2012) http://dx.doi.org/10.4134/JKMS.2012.49.1.099[Crossref]
• [35] G. Szegö, Orthogonal Polynomials, Colloquium Publications, XXIII (American Mathematical Society, Rhode Islands, 1939)
• [36] R. C. Mittal, G. Arora, Commun. Nonlinear Sci. Numer. Simul. 16, 1304 (2011) http://dx.doi.org/10.1016/j.cnsns.2010.06.028[Crossref]
• [37] A. Rashid, A. I. B. Ismail, Comput. Methods App. Math. 9, 412 (2009)
• [38] R. A. Van Gorder, Acta Mech. 216, 345 (2011) http://dx.doi.org/10.1007/s00707-010-0360-3[Crossref]

Identifiers

DOI

10.2478/s11534-014-0429-z