Method of the Riemann-Hilbert Problem for the Solution of the Helmholtz Equation in a Semi-infinite Strip

In this dissertation, a new method is developed to study BVPs of the modified Helmholtz and Helmholtz equations in a semi-infinite strip subject to the Poincare type, impedance and higher order boundary conditions. The main machinery used here is the theory of Riemann Hilbert problems, the residue theory of complex variables and the theory of integral transforms. A special kind of interconnected Laplace transforms are introduced whose parameters are related through branch of a multi-valued function. In the chapter 1 a brief review of the unified transform method used to solve BVPs of linear and non-linear integrable PDEs in convex polygons is given. Then unified transform method is applied to the BVP of the modified Helmholtz equation in a semi-infinite strip subject to the Poincare type and impedance boundary conditions. In the case of BVP of the modified Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions, two scalar RHPs are derived, then the closed form solutions of the given BVP are derived. The difficulty in application of the unified transform method to BVP of the Helmholtz equation in a semi infinite strip is discussed later on. The chapter 2 contains application of the finite integral transform (FIT) method to study the BVP for the Helmholtz equation in a semi-infinite strip subject to the Poincare type and impedance boundary conditions. In the case of the impedance boundary conditions, a series representation of the solution of the BVP for the Helmholtz equation in a semi-infinite strip is derived. The Burniston-Siewert method to find integral representations of a certain transcendental equation is presented. The roots of this equation are required for both methods, the FIT method and the RHP based method. To implement the Burniston-Siewert method, we solve a scalar RHP on several segments of the real axis. In chapter 3, we have applied the new method to study the Poincare type and impedance BVPs for the Helmholtz equation in a semi-infinite strip. In the case of the Poincare type boundary conditions an order two vector RHP is derived. In general, it is not possible to find closed form solution of an order two vector RHP. In the case of the impedance boundary conditions two scalar RHPs are derived whose closed form solutions are found. Then the series representation for solution of the BVP of the Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions, is recovered using the inverse transform operator, and the residue theory of complex variables. The numerical results are presented for various values of the parameters involved. It is observed that the FIT method and the new method generate exactly the same solution of the BVP of the Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions. In chapter 4, we have applied the new method to study the acoustic scattering from a semi-infinite strip subject to higher order boundary conditions. Two scalar RHPs are derived whose closed form solutions are found. A unique solution of the problem is obtained