A Unified Approach to Boundary Value Problems

Fokas, A. S.

doi:10.1137/1.9780898717068

Cited by 356 publications

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“…The method is relatively new. It was discovered by A. S. Fokas in his quest to generalize the method of inverse scattering, which solves the IVP for x ∈ R for socalled soliton equations, to BVPs posed either on the half-line x ≥ 0 or on the finite interval x ∈ [0, L] [6,7,9]. It was observed immediately [8,10] that the method also produces interesting results for linear equations, which is our focus here.…”

Section: Introductionmentioning

confidence: 94%

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The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

Trogdon

Deconinck

2012

Applicable Analysis

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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.

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Section: Introductionmentioning

confidence: 94%

“…Our introduction to the method of Fokas proceeds mostly by example. More details and general arguments can be found in [9]. We start by revisiting the IVP on the whole line in the framework of the new method; see section 2.…”

Section: Introductionmentioning

confidence: 99%

The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

Trogdon

Deconinck

2012

Applicable Analysis

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“…-In changing the elliptic operator P(D), one must first construct the appropriate global relation (Fokas 2008) and repeat the proof of lemma 1.3 with Q(l) defined accordingly. This is particularly straightforward if there are no critical points on Z P , i.e.…”

Section: Generalizationsmentioning

confidence: 99%

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

Ashton

2012

Proc. R. Soc. A.

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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.

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“…In fact, too many initial-boundary conditions are usually required for the study of initial-boundary value problems (see, e.g., [5,19,27,58] and references therein), and the problem becomes overdetermined. Thus, a crucial step here is the reduction of the initialboundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Evolution of Weyl Functions and Initial-Boundary Value Problems

Sakhnovich

2016

Math. Model. Nat. Phenom.

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This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well.

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A Unified Approach to Boundary Value Problems

Cited by 356 publications

References 0 publications

The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

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

Evolution of Weyl Functions and Initial-Boundary Value Problems

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