×

A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin-Bona-Mahony-Burgers equation. (English) Zbl 1397.65208

Summary: In this paper, a new method based on hybridization of Lucas and Fibonacci polynomials is developed for approximate solutions of 1D and 2D nonlinear generalized Benjamin-Bona-Mahony-Burgers equations. Firstly time discretization is made by using finite difference approaches. After that unknown function and its derivatives are expanded to Lucas series. Based on these series expansion, differentiation matrices are derived by utilizing Fibonacci polynomials. By doing so, the solution of the mentioned equations is reduced to the solution of an algebraic system of equations. By solving this system of equations the Lucas series coefficients are obtained. Then substituting these coefficients into Lucas series expansion approximate solutions can be constructed successively. The main goal of this paper is to indicate that Lucas polynomial based method is appropriate for 1D and 2D nonlinear problems. Efficiency and performance of the proposed method are judged on six test problems which consists of the 1D and 2D version of mentioned equation by calculating \(L_2\) and \(L_\infty\) error norms. Feasibility of the method is verified by obtained accurate results.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

Software:

Matplotlib
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zheng, S., Nonlinear Evolution Equations (Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Math), (2004), CRC Press · Zbl 1085.47058
[2] Soliman, A. A.; Abdou, M. A., Numerical solutions of nonlinear evolution equations using variational iteration method, J. Comput. Appl. Math., 207, 1, 111-120, (2007) · Zbl 1120.65111
[3] Ganji, Z. Z.; Ganji, D. D.; Bararnia, H., Approximate general and explicit solutions of nonlinear BBMB equations by exp-function method, Appl. Math. Model., 33, 1836-1841, (2009) · Zbl 1205.35250
[4] Gómez, C. A.; Salas, A. H.; Frias, B. A., New periodic and soliton solutions for the generalized BBM and BBM-Burgers equations, Appl. Math. Comput., 217, 1430-1434, (2010) · Zbl 1203.35221
[5] Guo, C.; Fang, S., Optimal decay rates of solutions for a multi-dimensional generalized Benjamin-Bona-Mahony equation, Nonlinear Analysis: Theory, Methods & Applications, 75, 3385-3392, (2012) · Zbl 1235.35245
[6] Omrani, K., The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony (BBM) equation, Appl. Math. Comput., 180, 614-621, (2006) · Zbl 1103.65101
[7] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions, Comput. Math. Appl., 68, 212-237, (2014) · Zbl 1369.65126
[8] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate, J. Comput. Appl. Math., 286, 211-231, (2015) · Zbl 1315.65086
[9] Yousefi, S. A.; Behroozifar, M.; Dehghan, M., Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math. Model., 36, 945-963, (2012) · Zbl 1243.65127
[10] Eslahchi, M. R.; Dehghan, M.; Amani, S., The third and fourth kinds of Chebyshev polynomials and best uniform approximation, Math. Comput. Modelling, 55, 1746-1762, (2012) · Zbl 1255.41015
[11] Zhao, T.; Zhang, X.; Huo, J.; Su, W.; Liu, Y.; Wu, Y., Optimal error estimate of Chebyshev-Legendre spectral method for the generalised Benjamin-Bona-Mahony-Burgers equations, Abstr. Appl. Anal., 2012, (2012) · Zbl 1246.65176
[12] Dehghan, M.; Saray, B. N.; Lakestani, M., Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers-Huxley equation, Math. Comput. Modelling, 55, 1129-1142, (2012) · Zbl 1255.65182
[13] Dehghan, M.; Aryanmehr, S.; Eslahchi, M. R., A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method, J. Comput. Appl. Math., 244, 125-139, (2013) · Zbl 1268.65092
[14] Fornberg, B., A Practical Guide To Pseudospectral Methods, (1996), Cambridge University Press New York · Zbl 0844.65084
[15] Boyd, J. P., Chebyshev and Fourier Spectral Methods, (2000), Dover New York
[16] Sezer, M.; Kesan, C., Polynomial solutions of certain differential equations, Int. J. Comput. Math, 76, 1, 93-104, (2000) · Zbl 0973.65055
[17] Bhrawy, A. H.; A-Shomrani, M. M., A shifted Jacobi-Gauss-lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals, Bound. Value Probl, 62, 2012, (2012) · Zbl 1280.65079
[18] Gulsu, M.; Gurbuz, B.; Ozturk, Y.; Sezer, M., Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput, 217, 6765-6776, (2011) · Zbl 1211.65166
[19] Abbasbandy, S.; Shirzadi, A., The first integral method for modified Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul., 15, 1759-1764, (2010) · Zbl 1222.35166
[20] Abdollahzadeh, M.; Hosseini, M.; Ghanbarpour, M.; Kashani, S., Exact travelling solutions for Benjamin-Bona-Mahony-Burgers equations by G/Ǵ -expansion method, International Journal of Applied Mathematics and Computation, 3, 70-76, (2011)
[21] Yin, H.; Hu, J., Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equations, Nonlinear Analysis: Theory, Methods & Applications, 73, 1729-1738, (2010) · Zbl 1196.35195
[22] Qinghua, X.; Zheng, C. Z., Degenerate boundary layer solutions to the generalized Benjamin-Bona-Mahony-Burgers equation, Acta Mathematica Scientia, 32, 1743-1758, (2012) · Zbl 1274.35340
[23] Xiao, Q.; Zhao, H., Nonlinear stability of generalized Benjamin-Bona-Mahony-Burgers shock profiles in several dimensions, J. Math. Anal. Appl., 406, 165-187, (2013) · Zbl 1306.35118
[24] Tari, H.; Ganji, D. D., Approximate explicit solutions of nonlinear BBMB equations by he’s methods and comparison with the exact solution, Phys. Lett. A, 367, 95-101, (2007) · Zbl 1209.65117
[25] Al-Khaled, K.; Momani, S.; Alawneh, A., Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations, Appl. Math. Comput., 171, 281-292, (2005) · Zbl 1084.65097
[26] Omrani, K.; Ayadi, M., Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation, Numer. Methods Partial Differential Equations, 24, 1, 239-248, (2008) · Zbl 1133.65067
[27] Zarebnia, M.; Parvaz, R., On the numerical treatment and analysis of Benjamin-Bona-Mahony-Burgers equation, Appl. Math. Comput., 284, 79-88, (2016) · Zbl 1410.65406
[28] Çetin, M.; Sezer, M.; Güler, C., Lucas polynomial approach for system of high-order linear differential equations and residual error estimation, Math. Probl. Eng., (2015), Article ID 625984, 14 pages doi:10.1155/2015/625984 · Zbl 1394.65061
[29] Filipponi, P.; Horadam, A. F., Derivative sequences of Fibonacci and Lucas polynomials, (Applications of Fibonacci Numbers, Vol. 4, (1991), Kluwer Dordrecht), 99-108 · Zbl 0744.11013
[30] Filipponi, P.; Horadam, A. F., Second derivative dequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 31, 3, 194-204, (1993) · Zbl 0779.11009
[31] Koç, A. B.; Çakmak, M.; Kurnaz, A.; Uslu, K., A new Fibonacci type collocation procedure for boundary value problems, Advances in Difference Equations, 2013, 1, 1-11, (2013) · Zbl 1378.65153
[32] Hunter, J. D., Matplotlib: A 2D graphics environment, Computing in Science & Engineering, 9, 3, 90-95, (2007)
[33] Rubin, S. G.; Graves, R. A., Cubic Spline Approximation for Problems in Fluid Mechanics, (1975), NASA TR R-436 Washington, DC
[34] Oruç, Ö.; Bulut, F.; Esen, A., Numerical solutions of regularized long wave equation by Haar wavelet method, Mediterranean Journal of Mathematics, 13, 5, 3235-3253, (2016), https://link.springer.com/article/10.1007/s00009-016-0682-z · Zbl 1354.65194
[35] Kutluay, S.; Esen, A., A finite difference solution of the regularized long-wave equation, Math. Probl. Eng., 2006, 1-14, (2006) · Zbl 1200.76131
[36] Dogan, A., Numerical solution of RLW equation using linear finite elements within galerkins method, Appl. Math. Model., 26, 771-783, (2002) · Zbl 1016.76046
[37] Dag, I.; Özer, M. N., Approximation of RLW equation by least square cubic B-spline finite element method, Appl. Math. Model., 25, 221-231, (2001) · Zbl 0990.65110
[38] Gardner, L. R.T.; Gardner, G. A.; Dogan, A., A least-squares finite element scheme for the RLW equation, Commun. Numer. Methods. Eng., 12, 795-804, (1996) · Zbl 0867.76040
[39] Saka, B.; Dağ, I., Quartic B-spline collocation algorithms for numerical solution of the RLW equation, Numer. Methods Partial Differential Equations, 23, 3, 731-751, (2007) · Zbl 1114.65122
[40] Dag, I.; Korkmaz, A.; Saka, B., Cosine expansion-based differential quadrature algorithm for numerical solution of the RLW equation, Numer. Methods Partial Differential Equations, 26, 3, 544-560, (2010) · Zbl 1189.65236
[41] Olver, P. J., Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc, 85, 143-160, (1979) · Zbl 0387.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.