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The modified Helmholtz equation in a semi-strip

Published online by Cambridge University Press:  24 February 2005

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW. e-mail:


We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a $2\times 2$-matrix Riemann–Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the $2\times 2$ RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.

Research Article
2005 Cambridge Philosophical Society

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