Thesis (M.S.) University of Alaska Fairbanks, 2019Our goal is to study the linear Klein-Gordon equation in matrix form, with initial conditions originating on a curve. This equation has applications to the Cross-Sectionally Averaged Shallow Water equations, i.e. a system of nonlinear partial differential equations used for modeling tsunami waves within narrow bays, because the general Carrier-Greenspan transform can turn the Cross-Sectionally Averaged Shallow Water equations (for shorelines of constant slope) into a particular form of the matrix Klein-Gordon equation. Thus the matrix Klein-Gordon equation governs the run-up of tsunami waves along shorelines of constant slope. If the narrow bay is U-shaped, the Cross-Sectionally Averaged Shallow Water equations have a known general solution via solving the transformed matrix Klein-Gordon equation. However, the initial conditions for our Klein-Gordon equation are given on a curve. Thus our goal is to solve the matrix Klein-Gordon equation with known conditions given along a curve. Therefore we present a method to extrapolate values on a line from conditions on a curve, via the Taylor formula. Finally, to apply our solution to the Cross-Sectionally Averaged Shallow Water equations, our numerical simulations demonstrate how Gaussian and N-wave profiles affect the run-up of tsunami waves within various U-shaped bays.Chapter 1: Introduction and overview -- 1.1 Introduction -- 1.2 Overview -- Chapter 2: Preliminaries -- 2.1 Wave equation -- 2.2 Wave equation initial value problem -- 2.3 Wave equation for spatially-variable speed -- 2.4 The 1D Klein-Gordon Equation -- 2.5 Initial conditions long an arbitrary curve -- Chapter 3: Statement of the problem -- Chapter 4: Solution to the problem -- 4.1 Projection of initial conditions -- 4.2 Solutions to the Klein-Gordon Equation with specific A and B -- Chapter 5: Applications to the shallow water wave equation -- 5.1 Cross-sectionally averaged shallow water equations -- 5.2 U-shaped Bays -- 5.3 Numerical simulations -- Chapter 6: Conclusions -- References
