Initial and boundary value problems in two and three dimensions

Kalimeris, Konstantinos (2010-04-13)

Thesis

This thesis: (a) presents the solution of several boundary value problems (BVPs) for the Laplace and the modified Helmholtz equations in the interior of an equilateral triangle; (b) presents the solution of the heat equation in the interior of an equilateral triangle; (c) computes the eigenvalues and eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions; (d) discusses the solution of several BVPs for the non-linear Schrödinger equation on the half line. In 1967 the Inverse Scattering Transform method was introduced; this method can be used for the solution of the initial value problem of certain integrable equations including the celebrated Korteweg-de Vries and nonlinear Schrödinger equations. The extension of this method from initial value problems to BVPs was achieved by Fokas in 1997, when a unified method for solving BVPs for both integrable nonlinear PDEs, as well as linear PDEs was introduced. This thesis applies "the Fokas method" to obtain the results mentioned earlier. For linear PDEs, the new method yields a novel integral representation of the solution in the spectral (transform) space; this representation is not yet effective because it contains certain unknown boundary values. However, the new method also yields a relation, known as "the global relation", which couples the unknown boundary values and the given boundary conditions. By manipulating the global relation and the integral representation, it is possible to eliminate the unknown boundary values and hence to obtain an effective solution involving only the given boundary conditions. This approach is used to solve several BVPs for elliptic equations in two dimensions, as well as the heat equation in the interior of an equilateral triangle. The implementation of this approach: (a) provides an alternative way for obtaining classical solutions; (b) for problems that can be solved by classical methods, it yields novel alternative integral representations which have both analytical and computational advantages over the classical solutions; (c) yields solutions of BVPs that apparently cannot be solved by classical methods. In addition, a novel analysis of the global relation for the Helmholtz equation provides a method for computing the eigenvalues and the eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions. Finally, for the nonlinear Schrödinger on the half line, although the global relation is in general rather complicated, it is still possible to obtain explicit results for certain boundary conditions, known as "linearizable boundary conditions". Several such explicit results are obtained and their significance regarding the asymptotic behavior of the solution is discussed.