Fokas Method Research Articles

On the Initial‐Boundary Value Problem for the Linearized Boussinesq Equation

This work is concerned with the initial‐boundary value problem for the Boussinesq equation. By employing the unified transform method of Fokas, novel solution formulae for the linearized “good” Boussinesq equation on the half‐line with various initial and boundary conditions are obtained. Moreover, these solution formulae are numerically illustrated in the case of concrete data. Finally, Boussinesq's original physical derivation of the so‐called “bad” Boussinesq equation is provided.

Heat conduction on the ring: Interface problems with periodic boundary conditions

The classical problem of heat conduction in one dimension on a composite ring is examined. The problem is formulated using the heat equation with periodic boundary conditions. We provide an explicit solution of this problem using the Method of Fokas. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumption of continuity at the interface is imposed.

A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation

We use the Fokas method to analyze the derivative nonlinear Schrödinger (DNLS) equation iqt(x,t)=−qxx(x,t)+(rq2)x on the interval [0, L]. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit (x,t) dependence, and it has jumps across {ξ ∈ℂ| Imξ4=0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ),b(ξ)},{A(ξ),B(ξ)}, and {A(ξ),B(ξ)}, which in turn are defined in terms of the initial data q0(x)=q(x,0), the boundary data g0(t)=q(0,t),g1(t)=qx(0,t), and another boundary values f0(t)=q(L,t),f1(t)=qx(L,t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

A Perturbative Approach for the Asymptotic Evaluation of the Neumann Value Corresponding to the Dirichlet Datum of a Single Periodic Exponential for the NLS

Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists of a single periodic exponential, namely, a exp(iωt), a, ω real, it has been shown in [2–4] that if one assumes that the Neumann boundary value is given for large t by c exp(iωt), then c can be computed explicitly in terms of a and ω. Here, using the perturbative approach introduced in [16], it is shown that for typical initial conditions, it is indeed the case that at least up to third order in a perturbative expansion the Neumann boundary value is given by c exp(iωt) and the value of c is at least up to this order the value found in [2–4].

Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity

Motivated by proliferation-diffusion mathematical models for studying highly diffusive brain tumors, that also take into account the heterogeneity of the brain tissue, in the present work we consider a multi-domain linear reaction-diffusion equation with a discontinuous diffusion coefficient. For the solution of the problem at hand we implement Fokas transform method by directly following, and extending in this way, our recent work for a white-gray-white matter brain model pertaining to high grade gliomas. Fokas's novel approach for the solution of linear PDE problems, yields novel integral representations of the solution in the complex plane that, for appropriately chosen integration contours, decay exponentially fast and converge uniformly at the boundaries. Combining these method-inherent advantages with simple numerical quadrature rules, we produce an efficient method, with fast decaying error properties, for the solution of the discontinuous reaction-diffusion problem.

The Fokas Method: The Dirichlet to Neumann Map for the Sine‐Gordon Equation

We study initial boundary value problems for the sine‐Gordon equation on the half‐line via the Fokas method, known as an extension of the inverse scattering transform. The method is based on the simultaneous analysis of both parts of the Lax pair and the global algebraic relation that couples known and unknown boundary values. One of most difficult steps of the method is to characterize the unknown boundary values that appear in the spectral functions. We derive the Dirichlet to Neumann map by using the global relation and the asymptotics of the eigenfunctions. Furthermore, employing perturbation expansion, we present an effective characterizations of the unknown boundary value in terms of the given initial and boundary values, and we then derive the first few terms of the expansions of the Neumann boundary value up to the third order.

Education

Separation of variables... Fourier transforms... Green's functions... The Eastern cut-off... There are many techniques in the undergraduate canon for solving linear partial differential equations (PDEs) that have been popularized in excellent introductory texts. I confess that “The Eastern Cut-off” is not a mathematical technique, but rather, it was a well-regarded method for the high jump. By coincidence, the early 1800s saw both the first publication of Joseph Fourier's investigations into solutions to the heat equation using trigonometric series and the introduction of the high jump as an athletic event. For more than 150 years, the Eastern cut-off and a collection of other approaches were “known” to be the best high jump techniques until Richard Fosbury shocked the track and field community by turning away from the bar and leaping over it backwards with better results than other athletes could achieve. The lesson is clear: Success can discourage innovation by leading one to believe that the current workable solution is the best and only solution. This section of SIAM Review addresses this issue by injecting new ideas into traditional courses, and “The Method of Fokas for Solving Linear Partial Differential Equations,” by Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, is ideally suited to this purpose. Fokas's method is a relatively new approach that should supplement the existing basket of methods and techniques, including separation of variables, Fourier transform, Laplace transforms, and so forth. In the context of this tutorial, a solution is an explicit expression that solves the PDE and satisfies the given boundary and initial conditions. The explicit expression typically takes the form of integrals or summations involving these boundary and initial conditions. (An explicit expression composed of information that is not provided in the statement of the problem, such as Dirichlet in place of Neumann boundary data, would not be considered a solution.) The new approach is to transform the linear PDE into a local relation of the form \[ \frac\partial \rho\partial t + \frac\partial j\partial x = 0, \rho(x,t,k) = e^-ikx + ømega(k)t q(x,t), \] where $q(x,t)$ is the dependent variable in the PDE, $\omega(k)$ is the dispersion relation, and $j(x,t,k)$ is deduced from the PDE. To solve the PDE, one applies Green's Theorem to the local relation over the $(x,t)$ domain with boundaries that fit the initial and boundary data. The majority of the effort involves mapping the resulting line integrals into the available information along the boundaries of the domain. In a broader sense, more traditional integral transform techniques are suited to certain types of boundary data, while the method of Fokas is a route to an appropriate integral transform by mathematical inspection and manipulation. As such, it can be a more general approach to a broad category of problems. There are advantages and disadvantages to covering this approach in the undergraduate curriculum. The method of Fokas requires students to have a solid background in complex analysis in order to know which manipulations and symmetries will map the existing integral expressions to the initial and boundary information posed in the problem. The traditional methods in a first course in PDEs require little background beyond a course in ODEs, but at the same time these methods are limited to a small set of problems. For an undergraduate PDEs course in which students have had some complex analysis beforehand, or for a graduate course, the method of Fokas is an outstanding supplement that provides a fresh perspective on integral transforms and where one transform might be more useful than another. This particular tutorial is limited to constant coefficient evolution equations with one spatial dimension, but there are many extensions and potential student projects available by digging into the literature. Similar to the “Fosbury Flop,” the method of Fokas approaches familiar problems from a new direction, providing students and instructors with new insights into linear PDEs.

The Method of Fokas for Solving Linear Partial Differential Equations

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.

The three-wave equation on the half-line

The Fokas method is used to analyze the initial–boundary value problem for the three-wave equationpij,t−bi−bjai−ajpij,x+∑k(bk−bjak−aj−bi−bkai−ak)pikpkj=0,i,j,k=1,2,3, on the half-line. Assuming that the solution pij(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann–Hilbert problem formulated in the plane of the complex spectral parameter λ.

On the rigorous foundations of the Fokas method for linear elliptic partial differential equations

In this paper, we address some of the rigorous foundations of the Fokas method, confining attention to boundary value problems for linear elliptic partial differential equations on bounded convex domains. The central object in the method is the global relation, which is an integral equation in the spectral Fourier space that couples the given boundary data with the unknown boundary values. Using techniques from complex analysis of several variables, we prove that a solution to the global relation provides a solution to the corresponding boundary value problem, and that the solution to the global relation is unique. The result holds for any number of spatial dimensions and for a variety of boundary value problems.

Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.

On the Global Relation and the Dirichlet-to-Neumann Correspondence

The recently developed Fokas method for solving two-dimensional Boundary Value Problems (BVP) via the use of global relations is utilized to solve axisymmetric problems in three dimensions. In particular, novel integral representations for the interior and exterior Dirichlet and Neumann problems for the sphere are derived, which recover and improve the already known solutions of these problems. The BVPs considered in this paper can be classically solved using either the finite Legendre transform or the Mellin-sine transform (which can be derived from the classical Mellin transform in a way similar to the way that the sine transform can be derived from the Fourier transform). The Legendre transform representation is uniformly convergent at the boundary, but it involves a series that is not useful for many applications. The Mellin-sine transform involves of course an integral but it is not uniformly convergent at the boundary. In this paper: (a) The Legendre transform representation is rederived in a simpler approach using algebraic manipulations instead of solving ODEs. (b) An integral representation, different that the Mellin-sine transform representation is derived which is uniformly convergent at the boundary. Furthermore, the derivation of the Fokas approach involves only algebraic manipulations, instead of solving an ordinary differential equation.

EXPLICIT SOLITON ASYMPTOTICS FOR THE NONLINEAR SCHRÖDINGER EQUATION ON THE HALF-LINE

There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which is as efficient as the analogous formalism for the Cauchy problem. Here, we employ this formalism for the analysis of several concrete initial-boundary value problems for the nonlinear Schrödinger equation. This includes problems involving initial conditions of a hump type coupled with boundary conditions of Robin type.

The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case.We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation.We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation i q t + q x x = i ( | q | 2 q ) x on the half-line using the Fokas method. Assuming that the solution q ( x , t ) exists, we show that it can be represented in terms of the solution of a matrix Riemann–Hilbert problem formulated in the plane of the complex spectral parameter ζ . The jump matrix has explicit x , t dependence and is given in terms of the spectral functions a ( ζ ) , b ( ζ ) (obtained from the initial data q 0 ( x ) = q ( x , 0 ) ) as well as A ( ζ ) , B ( ζ ) (obtained from the boundary values g 0 ( t ) = q ( 0 , t ) and g 1 ( t ) = q x ( 0 , t ) ). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values { q 0 ( x ) , g 0 ( t ) , g 1 ( t ) } such that there exist spectral functions satisfying the global relation, we show that the function q ( x , t ) defined by the above Riemann–Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.

What non-linear methods offered to linear problems? The Fokas transform method

In 1750 D’ Alembert demonstrated how a linear partial differential equation can be solved via separation of variables, a method that decomposes a PDE into a set of ODEs. This method was the basis for the development of many branches of contemporary analysis, from function spaces to spectral analysis of operators and the theory of special functions. A condition for the method of separation of variables to work is the existence of a coordinate system that fits the boundary of the fundamental domain and at the same time it separates the PDE. It is remarkable that two and a half centuries later a generalization is introduced that has its origin in the analysis of non-linear integrable equations. In the present work, this promising new transform method is outlined and applied to particular boundary value problems. A crucial part of the method is the introduction of a global relation which, if properly used, can provide the missing boundary data in a very elegant and effective way. We show how this can be used to generate separable solutions of partial differential equations even when no system, that fits the geometry of the fundamental domain, is available. This is shown for the case of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle. Furthermore, the connection of the Fokas method to the classical moment problem is investigated. It is shown that, in this case, the global relation is decomposed into a sequence of global relations, directly associated with the Fourier coefficients of the Dirichlet and Neumann boundary values.