Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting

ZÜRNACI, Fatma; TANOĞLU, Gamze; SEYDAOGLU, MUAZ; GÜCÜYENEN KAYMAK, Nurcan

doi:10.3906/mat-1603-94

Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting

Fatma ZÜRNACI, (Dokuz Eylül Üniversitesi, Fen Fakültesi, Matematik Bölümü, İzmir, Türkiye)

Nurcan GÜCÜYENEN KAYMAK, (İzmir Yüksek Teknoloji Enstitüsü, Fen Fakültesi, Matematik Bölümü, İzmir, Türkiye)

Muaz SEYDAOĞLU, (Muş Alparslan Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Muş, Türkiye)

Gamze TANOĞLU (İzmir Yüksek Teknoloji Enstitüsü, Fen Fakültesi, Matematik Bölümü, Urla, İzmir, Türkiye)

Turkish Journal of Mathematics

2 0

Yıl: 2018 Cilt: 42 Sayı: 3 Sayfa Aralığı: 1471 - 1483 Metin Dili: İngilizce DOI: 10.3906/mat-1603-94 İndeks Tarihi: 06-07-2020

Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting

Öz:

In this paper, an operator splitting method is used to analyze nonlinear Benjamin–Bona–Mahony-typeequations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie–Trottersplitting is applied to the equation. The local error bounds are obtained by using the approach based on the differentialtheory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global errorestimate is obtained via Lady Windermere’s fan argument. Finally, to confirm the expected convergence order, numericalexamples are studied.

Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Al-Khaled K, Momani S, Alawneh A. Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl Math Compt 2005; 171: 281-292.
[2] Avilez-Valente P, Seabra-Santos FJ. A high-order Petrov-Galerkin finite element method for the classical Boussinesq wave model. Int J Numer Meth Fl 2009; 59: 969-1010.
[3] Avrin J. The generalized Benjamin-Bona-Mahony equation in R n with singular initial data. Nonlinear Anal-Theor 1987; 11: 139-147.
[4] Benjamin TB, Bona JL, Mahony JJ. Model equations for long waves in nonlinear dispersive systems. Philos T Roy Soc A 1972; 272: 47-78.
[5] Biswas A, Kara A. 1-soliton solution and conservation laws for nonlinear wave equation in semiconductors. Appl Math Compt 2010; 217: 4289-4292.
[6] Çiçek Y, Tanoğlu G. Strang Splitting method for Burger-Huxley equation. Appl Math Comput 2016; 276: 454-467.
[7] Dutykh D, Dias F. Dissipative Boussinesq equations. CR Mecanique 2007; 335: 559-583.
[8] Dutykh D, Katsaounis T, Mitsotakis D. Finite volume methods for unidirectional dispersive wave models. Int J Numer Meth Fl 2013; 71: 717-736.
[9] El-Wakil S, Abdou M, Hendi A. New periodic wave solutions via Exp-function method. Phys Lett A 2008; 372: 830-840.
[10] Gücüyenen N. Strang splitting method to Benjamin-Bona-Mahony type equations: analysis and application. J Comput Appl Math 2017; 318 (C): 616-623.
[11] Holden H, Karlsen K, Risebro N, Tao T. Operator splitting for the KdV equation. Math Comput 2011; 80: 821-846.
[12] Holden H, Lubich C, Risebro N. Operator splitting for partial differential equations with Burgers nonlinearity. Math Comput 2013; 82: 173-185.
[13] Jahnke T, Lubich C. Error bounds for exponential operator splitting. BIT 2000; 40: 735-744.
[14] Lubich C. On splitting methods for Schr¨odinger-Poisson and cubic nonlinear Schr¨odinger equations. Math Comput 2008; 77: 2141-2153.
[15] Nguyen HY, Dias F. A Boussinesq system for two-way propagation of interfacial waves. Physica D 2008; 237: 2365-2389.
[16] Nilsen EB. On operator splitting for the viscous Burgers’ and the Korteweg-de Vries equations. MSc, Norwegian University of Science and Technology, Trondheim, Norway 2011.
[17] Seydaoğlu M, Erdoˇgan U, Ozi¸s T. Numerical solution of Burgers’ equation with high order splitting methods. J ¨ Comput Appl Math 2016; 291: 410-421.
[18] Stanislavova M. On the global attractor for the damped Benjamin-Bona-Mahony equation. Discrete Cont Dyn S 2005; Series A: 824-832.
[19] Tari H, Ganji D. Approximate explicit solutions of nonlinear BBMB equations by He’s methods and comparison with the exact solution. Phys Lett A 2007; 367: 95-101.
[20] Trefethen LN. Spectral Methods in MATLAB. Philadelphia, PA, USA: Siam, 2000.
[21] Zürnacı F. Convergence analysis and numerical solution of the Fisher’s and Benjamin-Bona-Mahony equations by operator splitting method. MSc, ˙Izmir Institute of Technology, Urla, ˙Izmir, Turkey, 2014.

APA	ZÜRNACI F, GÜCÜYENEN KAYMAK N, SEYDAOGLU M, TANOĞLU G (2018). Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. , 1471 - 1483. 10.3906/mat-1603-94
Chicago	ZÜRNACI Fatma,GÜCÜYENEN KAYMAK Nurcan,SEYDAOGLU MUAZ,TANOĞLU Gamze Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. (2018): 1471 - 1483. 10.3906/mat-1603-94
MLA	ZÜRNACI Fatma,GÜCÜYENEN KAYMAK Nurcan,SEYDAOGLU MUAZ,TANOĞLU Gamze Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. , 2018, ss.1471 - 1483. 10.3906/mat-1603-94
AMA	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. . 2018; 1471 - 1483. 10.3906/mat-1603-94
Vancouver	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. . 2018; 1471 - 1483. 10.3906/mat-1603-94
IEEE	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G "Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting." , ss.1471 - 1483, 2018. 10.3906/mat-1603-94
ISNAD	ZÜRNACI, Fatma vd. "Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting". (2018), 1471-1483. https://doi.org/10.3906/mat-1603-94

APA	ZÜRNACI F, GÜCÜYENEN KAYMAK N, SEYDAOGLU M, TANOĞLU G (2018). Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. Turkish Journal of Mathematics, 42(3), 1471 - 1483. 10.3906/mat-1603-94
Chicago	ZÜRNACI Fatma,GÜCÜYENEN KAYMAK Nurcan,SEYDAOGLU MUAZ,TANOĞLU Gamze Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. Turkish Journal of Mathematics 42, no.3 (2018): 1471 - 1483. 10.3906/mat-1603-94
MLA	ZÜRNACI Fatma,GÜCÜYENEN KAYMAK Nurcan,SEYDAOGLU MUAZ,TANOĞLU Gamze Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. Turkish Journal of Mathematics, vol.42, no.3, 2018, ss.1471 - 1483. 10.3906/mat-1603-94
AMA	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. Turkish Journal of Mathematics. 2018; 42(3): 1471 - 1483. 10.3906/mat-1603-94
Vancouver	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting. Turkish Journal of Mathematics. 2018; 42(3): 1471 - 1483. 10.3906/mat-1603-94
IEEE	ZÜRNACI F,GÜCÜYENEN KAYMAK N,SEYDAOGLU M,TANOĞLU G "Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting." Turkish Journal of Mathematics, 42, ss.1471 - 1483, 2018. 10.3906/mat-1603-94
ISNAD	ZÜRNACI, Fatma vd. "Convergence analysis and numerical solution of the Benjamin–Bona–Mahony equation by Lie–Trotter splitting". Turkish Journal of Mathematics 42/3 (2018), 1471-1483. https://doi.org/10.3906/mat-1603-94