Abstract. The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method also allows for the equally explicit solution of problems for which no classical approach exists. In addition, it is possible to elucidate which boundary-value problems are well posed and which are not. We provide examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, and others do not. For the former, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.

A new method is proposed to recover the water-wave surface elevation from pressure data obtained at the bottom of the fluid. The new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. The solvability properties of the nonlocal equation are rigorously analyzed using the Implicit Function Theorem.

The inverse water wave problem of bathymetry detection is the problem of deducing the bottom topography of the seabed from measurements of the water wave surface. In this paper, we present a fully nonlinear method to address this problem in the context of the Euler equations for inviscid irrotational fluid flow with no further approximation. Given the water wave height and its first two time derivatives, we demonstrate that the bottom topography may be reconstructed from the numerical solution of a set of two coupled non-local equations. Owing to the presence of growing hyperbolic functions in these equations, their numerical solution is increasingly difficult if the length scales involved are such that the water is sufficiently deep. This reflects the ill-posed nature of the inverse problem. A new method for the solution of the forward problem of determining the water wave surface at any time, given the bathymetry, is also presented.

Abstract. We compare the effectiveness of solving Dirichlet-Neumann problems via the Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit formulation, the dual AFM formulation (AFM * ), a boundary integral collocation method (BIM), and the transformed field expansion (TFE) method. The first three methods involve highly ill-conditioned intermediate calculations that we show can be overcome using multiple-precision arithmetic. The latter two methods avoid catastrophic cancellation of digits in intermediate results, and are much better suited to numerical computation.For the Craig-Sulem expansion, we explore the cancellation of terms at each order (up to 150th) for three types of wave profiles, namely band-limited, realanalytic, or smooth. For the AFM and AFM * methods, we present an example in which representing the Dirichlet or Neumann data as a series using the AFM basis functions is impossible, causing the methods to fail. The example involves band-limited wave profiles of arbitrarily small amplitude, with analytic Dirichlet data. We then show how to regularize the AFM and AFM * methods by over-sampling the basis functions and using the singular value decomposition or QR-factorization to orthogonalize them. Two additional examples are used to compare all five methods in the context of water waves, namely a largeamplitude standing wave in deep water, and a pair of interacting traveling waves in finite depth.

We have derived exact analytic expressions for the following matrix elements for a Morse oscillator: 〈m‖exp[γ(r−r0)]‖n〉, 〈m‖(r−r0)exp[−a(r−r0)]‖n〉, and 〈m‖(r−r0)exp[−2a(r−r0)]‖n〉, where a is the Morse range parameter and γ is an arbitrary constant. We have found that several of the commonly used expressions for various diagonal matrix elements of the Morse oscillator involve considerable roundoff error. Due to near cancellation of terms the result may be many orders of magnitude lower than any of the terms. We obtain simple yet accurate asymptotic approximations for the diagonal elements which avoid these problems. These approximations are obtained for 〈m‖(r−r0)‖m〉, 〈m‖(r−r0)2‖m〉, and the cases mentioned above.