Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Abstract

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.

Preface: Special Issues in Memory of David Benney (Part II)

“…They employ “uniform expansions” whose equations encompass those of the approximations to be matched. In , Hitzazis and Fokas extended the unified transform method to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three‐dimensional cylindrical domain with a polygonal cross section. In , Newell and Venkataramani connected the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns, and built a multiscale universe inspired by patterns in which the short spatial and temporal scales are given by a nearly periodic microstructure.…”