Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three-dimensional cylindrical domain with a polygonal cross-section.
The initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval of the spatial variable is considered. Using the method of simultaneous spectral analysis of the associated Lax pair, this problem is mapped into a Riemann-Hilbert problem formulated in the complex plane of the spectral parameter, but with explicit dependence on the space-time variables appearing in the Korteweg-de Vries equation. It is shown that, under certain conditions, the solution of this Riemann-Hilbert is uniquely determined by certain functions of the spectral parameter, which are defined by the initial and boundary data of the original problem. In turn, the solution of the Riemann-Hilbert problem provides the solution of the initial-boundary value problem for the Korteweg-de Vries equation, for which an integral representation is derived. From the latter system, one immediately deduces that Y 0 ͑x͒, Ỹ 1 ͑x͒ ª iY 1 ͑x͒, Z 0 ͑x͒ ª iZ 0 ͑x͒, and Z 1 ͑x͒ are real-valued functions. Finally, using ͑2.37͒, one arrives at
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