American Institute of Mathematical Sciences

Advanced
July  2005, 13(4): 921-940. doi: 10.3934/dcds.2005.13.921

Comparison of quarter-plane and two-point boundary value problems: the BBM-equation

Jerry Bona 1, , Hongqiu Chen 2, , Shu Ming Sun 3, and  B.-Y. Zhang 4,

1. 

Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 60607-7045
2. 

Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, and Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, Illinois, United States
3. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, United States
4. 

Department of Mathematics, University of Cincinnati, Cincinnati, Ohio, United States

Received  February 2005 Revised  May 2005 Published  August 2005

The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.
Keywords: two-point boundary value problems, BBM equation, comparison principles., quarter-plane problems, Nonlinear dispersive wave equations.
Mathematics Subject Classification: Primary: 35Q53, 35Q80, 76B03, 76B15; Secondary: 45G10, 65M99.
Citation: Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921
[1]

Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465
[2]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[3]

Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31
[4]

V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
[5]

Jerry Bona, Jiahong Wu. Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1141-1168. doi: 10.3934/dcds.2009.23.1141
[6]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624
[7]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[8]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
[9]

Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019
[10]

Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149
[11]

Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
[12]

Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
[13]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[14]

Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377
[15]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[16]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[17]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[18]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[19]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[20]

J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy