The unified transform: A spectral collocation method for acoustic scattering

This paper employs the unified transform to present a boundary-based spectral collocation method suitable for solving acoustic scattering problems. The method is suitable for both interior and exterior scattering problems, and may be extended to three dimensions. A number of simple two-dimensional examples are presented to illustrate the versatility of this method, and, upon comparison with other spectral methods, or boundary-based methods the approach presented in this paper is shown to be very competitive

The Laplace equation for the exterior of the Hankel contour and novel identities for hypergeometric functions

By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identities for special functions. A convenient tool for deriving this type of identities is the so-called \emph{global relation}, which has appeared recently in a wide range of boundary value problems. As a concrete application, we analyze the Neumann boundary value problem for the Laplace equation in the exterior of the so-called Hankel contour, which is the contour that appears in the definition of both the gamma and the Riemann zeta functions. By utilizing the explicit solution of this problem, we derive a plethora of novel identities involving the hypergeometric function

A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions.F.P.J. de B. acknowledges the support from the National Science Foundation Grant No. EAR-1654009. M.J.C. is supported by EPSRC Grant No. EP/L016516/1. A.S.F. is supported by EPSRC, UK, via the senior fellowship

Spectral theory of some non-selfadjoint linear differential operators

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.Comment: 1 figur

Zero-dispersion limit for integrable equations on the half-line with linearisable data

We study the zero-dispersion limit for certain initial boundary value problems for the defocusing nonlinear Schrödinger (NLS) equationand for the Korteweg-de Vries (KdV)equation with dominant surface tension. These problems are formulated on the half-line and they involve linearisable boundaryconditions.Peer Reviewe

The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions

Funder: Göran Gustafssons Stiftelse för Naturvetenskaplig och Medicinsk Forskning; doi: http://dx.doi.org/10.13039/501100003426Funder: Ruth and Nils-Erik Stenbäck FoundationFunder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.</jats:p

A numerical technique for linear elliptic partial differential equations in polygonal domains.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) 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 (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.This research has been funded by the EPSRC grant, EP/H04261X/1: ‘Integrability in Multidimensions and Boundary Value Problems.’This is the final published article. It was originally published in Proceedings of the Royal Society A, 471: 20140747, DOI: 10.1098/rspa.2014.074

Solving the Initial Value Problem for the 3-Wave Interaction Equations in Multidimensions

AbstractStarting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space dimensions and one time dimension), we complexify the independent variables, thus doubling the number of real variables, and hence we work in 4+2 dimensions: $$x_1$$ x 1 , $$x_2$$ x 2 , $$y_1$$ y 1 , $$y_2$$ y 2 and $$t_1$$ t 1 , $$t_2$$ t 2 . In this paper we solve the initial value problem of the 3-wave interaction equations in 4+2 dimensions.</jats:p

The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*

Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266Funder: Foundation for Education and European Culture; Id: http://dx.doi.org/10.13039/501100005411Funder: Cambridge Trust; Id: http://dx.doi.org/10.13039/501100003343Funder: Christ's College, University of Cambridge; Id: http://dx.doi.org/10.13039/501100000590Funder: A.G. Leventis Foundation; Id: http://dx.doi.org/10.13039/501100004117Abstract: This paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic boundary conditions. First, by employing the unified transform (also known as the Fokas method) it can be shown that the solution becomes time‐periodic for large t . Second, it is shown that the coefficients of the Fourier series of the unknown boundary values can be determined explicitly in terms of the coefficients of the Fourier series of the given boundary data in a very simple, algebraic way. This approach is illustrated for second‐order linear evolution equations and also for linear evolution equations containing spatial derivatives of arbitrary order. The simple and explicit determination of the unknown boundary values is based on the “ Q ‐equation”, which for the linearized nonlinear Schrödinger equation is the linear limit of the quadratic Q ‐equation introduced by Lenells and Fokas [Proc. R. Soc. A, 471, 2015]. Regarding the latter equation, it is also shown here that it provides a very simple, algebraic way for rederiving the remarkable results of Boutet de Monvel, Kotlyarov, and Shepelsky [Int. Math. Res. Not. issue 3, 2009] for the particular boundary condition of a single exponential