NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS

Yusuf Uçar; Murat Yağmurlu; Ali Başhan

NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS

Abstract

The purpose of this work is obtain the numerical approximate solutions of the nonlinear modified Burgers’ equation (MBE) via the modified cubic B-spline (MCB) differential quadrature methods (DQMs). The accuracy and effectiveness of the methods are measured and reported by finding out error norms L_2 and L_∞. The present numerical results have been compared with some earlier studies and this comparison clearly indicates that the method is an outstanding numerical scheme for the solution of the MBE. A stability analysis has at the same time been given.

Keywords

References

[1] Bateman H.,(1915) Some recent researches on the motion of fluids, Monthly Weather Rev. 43, 163-170.
[2] Burgers J.M., (1948) A mathematical model illustrating the theory of turbulance, Adv. Appl. Mech. 1, 171-190.
[3] Haim B., Felix B., (1998) Partial differential equations in the 20th century, Adv. Math. 135, 76-144.
[4] Seydaoğlu M., (2018) An accurate approximation algorithm for Burgers equation in the presence of small viscosity, Journal of Computational and Applied Mathematics 344, 473-481.
[5] Cole J.D., (1951) On a quasi-linear parabolic equations occuring in aerodynamics, Quart. Appl. Math. 9, 225-236.
[6] Bratsos A.G., (2010) A fourth-order numerical scheme for solving the modified Burgers equation, Computers and Mathematics with Applications 60, 1393-1400.
[7] Ramadan, M.A., EI-Danaf, T.S. and Abd Alaal, F.E.I. (2005) A Numerical Solution of the Burgers Equation Using Septic B-Splines. Chaos, Solitons and Fractals 26, 1249-1258.
[8] Ramadan, M.A., EI-Danaf, T.S.,(2005) Numerical treatment for the modified burgers equation, Math. Comput. in Simul. 70, 90-98.

[9] Duan Y., Liu R. and Jiang Y., (2008) Lattice Boltzmann model for the modified Burgers’ equation, Appl. Math. and Comput. 202, 489-497.
[10] Saka B., Dag I. and Irk D., (2007) Numerical Solution of the Modified Burgers Equation by the Quintic B-spline Galerkin Finite Element Method, Int. J. Math. Statis. 1, 86-97.
[11] Irk D.,(2009) Sextic B-spline collocation method for the modified Burgers’ equation, Kybernetes 38(9), 1599-1620.
[12] Roshan T., Bhamra K.S.,(2011) Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method, Appl. Math. Comput. 218, 3673-3679.
[13] Zhang R.P., Yu X.J., Zhao G.Z., (2013) Modified Burgers’ equation by the local discontinuous Galerkin method, Chin. Phys. B, 22(3), 030210-1 030210-5.
[14] Başhan, A., Karakoç, S.B.G., Geyikli, T., (2015) B-spline Differential Quadrature Method for the Modified Burgers’ Equation, Çankaya University Journal of Science and Engineering,12(1), pp. 001-013.
[15] Karakoç, S.B.G., Başhan A. and Geyikli, T., (2014) Two Different Methods for Numerical Solution of the Modified Burgers’ Equation, 2014, Article ID 780269, 13 pages http://dx.doi.org/10.1155/2014/780269.
[16] Aswin V.S., Awasthi, A., (2018) Iterative differential quadrature algorithms for modified Burgers equation, Engineering Computations Vol. 35 No. 1, 235-250.
[17] Lakshmi C. and Awasthi A., (2018) Robust numerical scheme for nonlinear modified Burgers equation, International Journal of Computer Mathematics Vol. 95, NO. 9, 1910-1926.
[18] Bellman, R., Kashef, B.G. and Casti, J., (1972) Differential quadrature: a technique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, 10, 40-52.
[19] Shu, C., (2000) Differential Quadrature and its application in engineering, Springer-Verlag London Ltd.
[20] Başhan A., Yağmurlu N.M., Uçar Y., Esen A., (2017) An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method, Chaos, Solitons and Fractals 100, 45-56.
[21] Başhan, A., Uçar, Y., Yağmurlu, N.M., Esen, A. (2016) Numerical Solution of the Complex Modified Korteweg-de Vries Equation by DQM, Journal of Physics: Conference Series 766, 012028 doi:10.1088/1742-6596/766/1/012028.
[22] Başhan, A., (2018) An effective application of differential quadrature method based on modified cubic B-splines to numerical solutions of KdV equation, Turk J Math 42, 373-394.
[23] Başhan, A., Karakoç, S.B.G., Geyikli, T., (2015) Approximation of the KdVB equation by the quintic B-spline differential quadrature method”, Kuwait Journal of Science , 42(2), pp. 67-92.
[24] Korkmaz A. and Dağ I., (2013) Cubic B-spline differential quadrature methods and stability for Burgers’ equation, International Journal for Computer-Aided Engineering and Software Vol. 30 No.3, 320-344.
[25] Bellman, R., Kashef, B., Lee, E.S. and Vasudevan, R., (1976) Differential Quadrature and Splines, Computers and Mathematics with Applications, Pergamon, Oxford, 371-376.
[26] Quan J.R. and Chang C.T., (1989) New sightings in involving distributed system equations by the quadrature methods-I, Comput. Chem. Eng., Vol. 13, 779-788.
[27] Shu, C. and Richards B.E., (1992) Application of generalized differential quadrature to solve two dimensional incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 15, 791-798.
[28] Cheng, J., Wang, B. and Du, S. (2005) 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, 6181-6201.
[29] Shu, C. and Wu, Y.L. (2007) Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids, 53, 969-984.
[30] Striz, A.G., Wang, X. and Bert, C.W. (1995) Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica, Vol. 111, 85-94.
[31] Korkmaz, A. and Dag, I., (2011) Shock wave simulations using Sinc Differential Quadrature Method, International Journal for Computer-Aided Engineering and Software, 28 (6), 654-674.
[32] Başhan A., Uçar Y., Yağmurlu N.M., Esen A., (2018) A new perspective for quintic B-spline based Crank-Nicolson differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation, Eur. Phys. J. Plus 133: 12, 1-15.
[33] Başhan A., Yağmurlu N.M., Uçar Y., Esen A., (2018) A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method, International Journal of Modern Physics C, Vol. 29, No. 6, 1850043 (17 pages).
[34] Prenter P.M., (1975) Splines and Variational Methods, New York:John Wiley.
[35] Harris S.L., (1996) Sonic shocks governed by the modified Burgers’ equation Eur. J. Appl. Math. 6, 75-107.
[36] Mittal, R.C. and Jain, R.K. (2012) Numerical solutions of Nonlinear Burgers’ equation with modified cubic B-splines collocation method, Appl. Math. Comp.218, 7839-7855.
[37] Ketcheson, D. I. (2010) Runge–Kutta methods with minimum storage implementations, Journal of Computational Physics, 229, 1763–1773.
[38] Ketcheson, D. I. (2008) Highly efficient strong stability preserving Runge-Kutta methods with Low-Storage Implementations, SIAM J. SCI. COMPUT. 30(4), 2113–2136.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Yusuf Uçar This is me
0000-0003-1469-5002
Türkiye

Murat Yağmurlu This is me
0000-0003-1593-0254
Türkiye

Ali Başhan This is me
0000-0001-8500-493X
Palestine

Publication Date

March 1, 2019

Submission Date

January 8, 2018

Acceptance Date

July 15, 2018

Published in Issue

Year 2019 Volume: 37 Number: 1

IZ

https://izlik.org/JA39EC84CB

Cite

RIS / Bibtex

APA

Uçar, Y., Yağmurlu, M., & Başhan, A. (2019). NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS. Sigma Journal of Engineering and Natural Sciences, 37(1), 129-142. https://izlik.org/JA39EC84CB

AMA

1.Uçar Y, Yağmurlu M, Başhan A. NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS. SIGMA. 2019;37(1):129-142. https://izlik.org/JA39EC84CB

Chicago

Uçar, Yusuf, Murat Yağmurlu, and Ali Başhan. 2019. “NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS”. Sigma Journal of Engineering and Natural Sciences 37 (1): 129-42. https://izlik.org/JA39EC84CB.

EndNote

Uçar Y, Yağmurlu M, Başhan A (March 1, 2019) NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS. Sigma Journal of Engineering and Natural Sciences 37 1 129–142.

IEEE

[1]Y. Uçar, M. Yağmurlu, and A. Başhan, “NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS”, SIGMA, vol. 37, no. 1, pp. 129–142, Mar. 2019, [Online]. Available: https://izlik.org/JA39EC84CB

ISNAD

Uçar, Yusuf - Yağmurlu, Murat - Başhan, Ali. “NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS”. Sigma Journal of Engineering and Natural Sciences 37/1 (March 1, 2019): 129-142. https://izlik.org/JA39EC84CB.

JAMA

1.Uçar Y, Yağmurlu M, Başhan A. NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS. SIGMA. 2019;37:129–142.

MLA

Uçar, Yusuf, et al. “NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS”. Sigma Journal of Engineering and Natural Sciences, vol. 37, no. 1, Mar. 2019, pp. 129-42, https://izlik.org/JA39EC84CB.

Vancouver

1.Yusuf Uçar, Murat Yağmurlu, Ali Başhan. NUMERICAL SOLUTIONS AND STABILITY ANALYSIS OF MODIFIED BURGERS EQUATION VIA MODIFIED CUBIC B-SPLINE DIFFERENTIAL QUADRATURE METHODS. SIGMA [Internet]. 2019 Mar. 1;37(1):129-42. Available from: https://izlik.org/JA39EC84CB