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]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems & Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737

[2]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[3]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[4]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[5]

Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315

[6]

Zhengyong Zhou, Bo Yu. A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 977-989. doi: 10.3934/jimo.2011.7.977

[7]

Francisco J. Ibarrola, Ruben D. Spies. A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity. Inverse Problems & Imaging, 2017, 11 (2) : 247-262. doi: 10.3934/ipi.2017012

[8]

Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems & Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875

[9]

Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109

[10]

Ouayl Chadli, Gayatri Pany, Ram N. Mohapatra. Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019034

[11]

Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003

[12]

Alexander Komech. Attractors of Hamilton nonlinear PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6201-6256. doi: 10.3934/dcds.2016071

[13]

Jun-Jie Miao, Sara Munday. Derivatives of slippery Devil's staircases. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 353-365. doi: 10.3934/dcdss.2017017

[14]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039

[15]

Enrico Valdinoci. Contemporary PDEs between theory and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : i-i. doi: 10.3934/dcds.2015.35.12i

[16]

Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411

[17]

Matthias Eller. Loss of derivatives for hyperbolic boundary problems with constant coefficients. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1347-1361. doi: 10.3934/dcdsb.2018154

[18]

Gyula Csató, Bernard Dacorogna. An identity involving exterior derivatives and applications to Gaffney inequality. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 531-544. doi: 10.3934/dcdss.2012.5.531

[19]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 695-708. doi: 10.3934/dcdss.2020038

[20]

Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]