Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

Dujardin, Guillaume

doi:10.1098/rspa.2009.0194

Cited by 23 publications

(24 citation statements)

References 11 publications

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“…In this case, equations (5.39) and (5.41) simplify as follows: 1 (x, λ)) 11 , (Φ 1 (x, λ)) 21 ) and ((Φ 3 (x, λ)) 11 , (Φ 3 (x, λ)) 21 ) satisfy the same symmetry properties. Hence, computing these vectors at x = 0, we obtain…”

Section: The Boundary Conditions (535)mentioning

confidence: 86%

Chapter 5: Elliptic Problems: Nonlinear

Pelloni¹

2014

Unified Transform for Boundary Value Problems

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The elliptic sine-Gordon equation is the nonlinear PDE q x x + q yy = sin q, q = q(x, y), x, y ∈ .( 5 . 1 ) This equation appears as a mathematical model of several physical phenomena. For example, it was derived in the context of the theory of Josephson effects, superconductors, and spin waves in ferromagnets; see [35]. From a strictly mathematical point of view, the elliptic sine-Gordon equation is the simplest and most fundamental example of an nonlinear elliptic integrable PDE, in the sense of admitting a Lax pair formulation. Therefore the tools developed in connection with the so-called inverse scattering transform can be used for its analysis, at least in principle. The fundamental problems with using inverse scattering approach is that the problems posed for this PDE are always boundary value problems, and hence one has to turn instead to the unified transform approach, introduced by Fokas in the late 1990s.The inverse scattering analysis for this integrable equation had been considered in [7] for a problem posed on 2 with prescribed periodic behavior at infinity, and in [35] for the problem on a half plane y ≥ 0, with all boundary values prescribed, but constrained by a certain nonlinear relation. Due to the limitations of the inverse scattering approach when dealing with boundary value problems, in these works the solution is not constructed effectively. Special exact solutions for the problem posed in the whole of 2 were found in the 1980s by various authors (see the references in [35]).Recent progress was made using the unified transform [43,45,29,30]. In this series of works, the equation was considered on a half plane, quarter plane, and semistrip. Particular cases of boundary conditions that yield an explicit representation for the solution were identified and analyzed. In this paper, we summarize this recent progress to

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Section: The Boundary Conditions (535)mentioning

confidence: 86%

Chapter 5: Elliptic Problems: Nonlinear

Pelloni¹

2014

Unified Transform for Boundary Value Problems

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“…whereq 0 (λ) is defined by (26), G 0 (t, λ) by (59) and f (t) is defined by (6). Our aim is to invert the global relation (61), and obtain an equation for f (t) in terms of the known data.…”

Section: The Dirichlet Problemmentioning

confidence: 99%

“…The function f (t) given by (6) is characterized as the solution of the following Volterra integral equation:…”

Section: Introductionmentioning

confidence: 99%

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

Fokas

Pelloni

2012

Stud Appl Math

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We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

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“…For λ 1 in the lower half λ 1 -plane the term sin[(μ 1 + √ 3λ 1 ) l 4 ] is dominated by e i √ 3λ 1 l 4 . Furthermore, the definition ofq(λ 1 , μ 1 , t) in (17) implies that this term behaves like e −iλ 1 η 1 with − l √ 3 < η 1 < l 2 √ 3 . Hence the integrand of (38) with respect to λ 1 behaves like…”

Section: The Symmetric Dirichlet Problemmentioning

confidence: 99%

The Heat Equation in the Interior of an Equilateral Triangle

Kalimeris

Fokas

2010

Studies in Applied Mathematics

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We present the solution of the classical problem of the heat equation formulated in the interior of an equilateral triangle with Dirichlet boundary conditions. This solution is expressed as an integral in the complex Fourier space, i.e., the complex k 1 and k 2 planes, involving appropriate integral transforms of the Dirichlet boundary conditions. By choosing Dirichlet data so that their integral transforms can be computed explicitly, we show that the solution is expressed in terms of an integral whose integrand decays exponentially as |k| → ∞. Hence, it is possible to evaluate this integral numerically in an efficient and straightforward manner. Other types of boundary value problems, including the Neumman and Robin problems, can be solved similarly.

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Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

Cited by 23 publications

References 11 publications

Chapter 5: Elliptic Problems: Nonlinear

Chapter 5: Elliptic Problems: Nonlinear

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

The Heat Equation in the Interior of an Equilateral Triangle

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