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 and Continuous Dynamical Systems, 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, Philadelphia, 2008. doi: 10.1137/1.9780898717068.

[2]

B. Deconinck, T. Trogdon and V. Vasan, The method of Fokas for solving linear partial differential equations, Accepted for publication (SIAM Review), (2012), 1-24.

[3]

A. S. Fokas, Boundary-value problems for linear PDEs with variable coefficients, Proc. R. Soc. Lond, 460 (2004), 1131-1151. doi: 10.1098/rspa.2003.1208.

[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), 221-242. doi: 10.1017/S0305004107000084.

[5]

P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations, IMA J. Applied Math., 69 (2004), 539-555. doi: 10.1093/imamat/69.6.539.

[6]

A. S. Fokas and B. Pelloni, Generalized Dirichlet to Neumann Map for moving boundary value problems, J. Math. Phys., 48 (2007), 013502 doi: 10.1063/1.2405405.

[7]

K. Kalimeris and A. S. Fokas, The heat equation in the interior of an equilateral triangle, Studies in Applied Math., 124 (2010), 283-305. doi: 10.1111/j.1467-9590.2009.00471.x.

[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), 1184-1205. doi: 10.1093/imanum/drn079.

[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), 47-78.

[10]

A. S. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O.

[11]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom., 8 (2005), 59-96. doi: 10.1007/s11040-004-1650-6.

[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), 1927-1944. doi: 10.1088/0951-7715/22/8/009.

[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), 391-405.

show all references

References:
[1]

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

[2]

B. Deconinck, T. Trogdon and V. Vasan, The method of Fokas for solving linear partial differential equations, Accepted for publication (SIAM Review), (2012), 1-24.

[3]

A. S. Fokas, Boundary-value problems for linear PDEs with variable coefficients, Proc. R. Soc. Lond, 460 (2004), 1131-1151. doi: 10.1098/rspa.2003.1208.

[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), 221-242. doi: 10.1017/S0305004107000084.

[5]

P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations, IMA J. Applied Math., 69 (2004), 539-555. doi: 10.1093/imamat/69.6.539.

[6]

A. S. Fokas and B. Pelloni, Generalized Dirichlet to Neumann Map for moving boundary value problems, J. Math. Phys., 48 (2007), 013502 doi: 10.1063/1.2405405.

[7]

K. Kalimeris and A. S. Fokas, The heat equation in the interior of an equilateral triangle, Studies in Applied Math., 124 (2010), 283-305. doi: 10.1111/j.1467-9590.2009.00471.x.

[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), 1184-1205. doi: 10.1093/imanum/drn079.

[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), 47-78.

[10]

A. S. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O.

[11]

A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom., 8 (2005), 59-96. doi: 10.1007/s11040-004-1650-6.

[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), 1927-1944. doi: 10.1088/0951-7715/22/8/009.

[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), 391-405.

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