# American Institute of Mathematical Sciences

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.

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##### 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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