July  2013, 33(7): 3171-3188. doi: 10.3934/dcds.2013.33.3171

Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation

1. 

Department of Mathematics, Pennsylvania State University, State College, PA, 16802, United States

2. 

Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA 98195

Received  March 2012 Revised  August 2012 Published  January 2013

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary-value problem is found to be ill posed.
Citation: Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171
References:
[1]

A. S. Fokas, "A Unified Approach to Boundary Value Problems,", SIAM: CBMS-NSF Regional Conference Series in Applied Mathematics, (2008).  doi: 10.1137/1.9780898717068.  Google Scholar

[2]

B. Deconinck, T. Trogdon and V. Vasan, The method of Fokas for solving linear partial differential equations,, Accepted for publication (SIAM Review), (2012), 1.   Google Scholar

[3]

A. S. Fokas, Boundary-value problems for linear PDEs with variable coefficients,, Proc. R. Soc. Lond, 460 (2004), 1131.  doi: 10.1098/rspa.2003.1208.  Google Scholar

[4]

P. A. Treharne and A. S. Fokas, Initial-boundary value problems for linear PDEs with variable coefficients,, Math. Proc. Camb. Phil. Soc., 143 (2007), 221.  doi: 10.1017/S0305004107000084.  Google Scholar

[5]

P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations,, IMA J. Applied Math., 69 (2004), 539.  doi: 10.1093/imamat/69.6.539.  Google Scholar

[6]

A. S. Fokas and B. Pelloni, Generalized Dirichlet to Neumann Map for moving boundary value problems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2405405.  Google Scholar

[7]

K. Kalimeris and A. S. Fokas, The heat equation in the interior of an equilateral triangle,, Studies in Applied Math., 124 (2010), 283.  doi: 10.1111/j.1467-9590.2009.00471.x.  Google Scholar

[8]

S. A. Smitheman, E. A. Spence and A. S. Fokas, A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon,, IMA J. Num. Anal., 30 (2010), 1184.  doi: 10.1093/imanum/drn079.  Google Scholar

[9]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. Roy. Soc. London. Ser. A, 272 (1972), 47.   Google Scholar

[10]

A. S. Fokas, On a class of physically important integrable equations,, Physica D, 87 (1995), 145.  doi: 10.1016/0167-2789(95)00133-O.  Google Scholar

[11]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems,, Math. Phys. Anal. Geom., 8 (2005), 59.  doi: 10.1007/s11040-004-1650-6.  Google Scholar

[12]

J. M.-K. Hong, J. Wu and J.-M. Yuan, A new solution representation for the BBM equation in a quarter plane and the eventual periodicity,, Nonlinearity, 22 (2009), 1927.  doi: 10.1088/0951-7715/22/8/009.  Google Scholar

[13]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems,, Proc. Camb. Phil. Soc., 73 (1973), 391.   Google Scholar

show all references

References:
[1]

A. S. Fokas, "A Unified Approach to Boundary Value Problems,", SIAM: CBMS-NSF Regional Conference Series in Applied Mathematics, (2008).  doi: 10.1137/1.9780898717068.  Google Scholar

[2]

B. Deconinck, T. Trogdon and V. Vasan, The method of Fokas for solving linear partial differential equations,, Accepted for publication (SIAM Review), (2012), 1.   Google Scholar

[3]

A. S. Fokas, Boundary-value problems for linear PDEs with variable coefficients,, Proc. R. Soc. Lond, 460 (2004), 1131.  doi: 10.1098/rspa.2003.1208.  Google Scholar

[4]

P. A. Treharne and A. S. Fokas, Initial-boundary value problems for linear PDEs with variable coefficients,, Math. Proc. Camb. Phil. Soc., 143 (2007), 221.  doi: 10.1017/S0305004107000084.  Google Scholar

[5]

P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations,, IMA J. Applied Math., 69 (2004), 539.  doi: 10.1093/imamat/69.6.539.  Google Scholar

[6]

A. S. Fokas and B. Pelloni, Generalized Dirichlet to Neumann Map for moving boundary value problems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2405405.  Google Scholar

[7]

K. Kalimeris and A. S. Fokas, The heat equation in the interior of an equilateral triangle,, Studies in Applied Math., 124 (2010), 283.  doi: 10.1111/j.1467-9590.2009.00471.x.  Google Scholar

[8]

S. A. Smitheman, E. A. Spence and A. S. Fokas, A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon,, IMA J. Num. Anal., 30 (2010), 1184.  doi: 10.1093/imanum/drn079.  Google Scholar

[9]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Phil. Trans. Roy. Soc. London. Ser. A, 272 (1972), 47.   Google Scholar

[10]

A. S. Fokas, On a class of physically important integrable equations,, Physica D, 87 (1995), 145.  doi: 10.1016/0167-2789(95)00133-O.  Google Scholar

[11]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems,, Math. Phys. Anal. Geom., 8 (2005), 59.  doi: 10.1007/s11040-004-1650-6.  Google Scholar

[12]

J. M.-K. Hong, J. Wu and J.-M. Yuan, A new solution representation for the BBM equation in a quarter plane and the eventual periodicity,, Nonlinearity, 22 (2009), 1927.  doi: 10.1088/0951-7715/22/8/009.  Google Scholar

[13]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems,, Proc. Camb. Phil. Soc., 73 (1973), 391.   Google Scholar

[1]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[2]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[3]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017

[4]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781

[5]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[6]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[7]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[8]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[9]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[10]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[11]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[12]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[13]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[14]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

[15]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[16]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]