The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

Sifalakis, A. G.; Fokas, A. S.; Fulton, Scott R.; Saridakis, Y. G.

doi:10.1016/j.cam.2007.07.012

Cited by 48 publications

(51 citation statements)

References 12 publications

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“…Just as both the classical transform method and Green's integral representation give rise to numerical methods, the new method has also given rise to novel techniques. Such techniques for linear evolution PDEs, linear elliptic PDEs, and nonlinear evolution PDEs are presented in [35], in [97] and [50] (following on from [52] and [94]), and in [114], respectively.…”

Section: Extensions Of the Method And Connections With Other Techniqmentioning

confidence: 99%

Synthesis, as Opposed to Separation, of Variables

Fokas¹,

Spence²

2012

SIAM Rev.

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Abstract. Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-selfadjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the "synthesis" as opposed to the "separation" of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.

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Section: Extensions Of the Method And Connections With Other Techniqmentioning

confidence: 99%

Synthesis, as Opposed to Separation, of Variables

Fokas¹,

Spence²

2012

SIAM Rev.

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“…It is also noted that in other studies of boundary value problems by the same unified transform method, previous authors have proposed alternative representations of unknown boundary data in terms of Fourier [11], Chebyshev [12] and Legendre [13] expansions. We emphasize that our own choice (2.36) is motivated by the specific demands of the problem, i.e.…”

Section: (C) Spectral Analysismentioning

confidence: 99%

Solving Wiener–Hopf problems without kernel factorization

Crowdy

Luca

2014

Proc. R. Soc. A.

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A new approach to solving problems of Wiener-Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener-Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

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“…such that the constraint (20) is satisfied, and where in the above integrand x = x j (s), y = y j (s) satisfy the parameterization (34). The above expression can be rewritten in the more convenient form…”

Section: A Global Relation For the Laplace Helmholtz And Modified Hmentioning

confidence: 99%

“…. , n, with (λ 1 , λ, μ) such that the constraint (20) is satisfied, and where in the above integrand x = x j (s), y = y j (s) satisfy the parameterization (34).…”

Section: A Global Relation For the Laplace Helmholtz And Modified Hmentioning

confidence: 99%

“…(vi) The first steps have been taken toward extending the unified transform to three dimensions, see, for example [11,14]. (vi) The analysis of the global relations yields a novel numerical technique for the numerical solution of the generalized Dirichlet-to-Neumann map, i.e., for the determination of the unknown boundary values in terms of the prescribed boundary data [15][16][17][18][19][20][21][22][23]. In particular, substantial progress in this direction was made by Fornberg and coauthors [16,17], as well as in [24].…”

Section: Introductionmentioning

confidence: 99%

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Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Hitzazis

Fokas

2017

Stud Appl Math

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Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations 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 (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three-dimensional cylindrical domain with a polygonal cross-section.

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The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation

Cited by 48 publications

References 12 publications

Synthesis, as Opposed to Separation, of Variables

Synthesis, as Opposed to Separation, of Variables

Solving Wiener–Hopf problems without kernel factorization

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

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