Invertibility of Helmholtz operators for nonhomogeneous medias

The guided modes of Woods' magnetohydrodynamic waveguide for a uniform, cylindrical plasma are shown to satisfy a homogeneous wave equation whose differential operator is the product of eight Helmholtz operators. The propagation constants of the Helmholtz operators are the characteristic roots of an 8 × 8 matrix which is derived and written down explicitly. This reformulated theory is extended to include localized sources which excite the guided modes. For certain cases, the Green's functions for the differential operators can be represented by Dini expansions in terms of modal eigenfunctions, which manifestly satisfy the boundary conditions. For the case of MHD waves excited by an azimuthally symmetric current source in a resistive, pressureless, inviscid, fully ionized plasma, a detailed solution is obtained which is in good qualitative agreement with experiments.

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Helmholtz operators and symmetric space duality

Inventiones mathematicae ◽

10.1007/s002220050158 ◽

1997 ◽

Vol 129 (1) ◽

pp. 63-74 ◽

Cited By ~ 1

Author(s):

Thomas Branson ◽

Gestur Ólafsson

Keyword(s):

Symmetric Space ◽

Helmholtz Operators

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The Method of Potential Operators for Anisotropic Helmholtz Operators on Domains with Smooth Unbounded Boundaries

Recent Trends in Operator Theory and Partial Differential Equations - Operator Theory: Advances and Applications ◽

10.1007/978-3-319-47079-5_11 ◽

2017 ◽

pp. 229-256 ◽

Cited By ~ 1

Author(s):

Vladimir Rabinovich

Keyword(s):

Potential Operators ◽

Helmholtz Operators

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Lp-theory of boundary integral operators for domains with unbounded smooth boundary

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0049 ◽

2016 ◽

Vol 23 (4) ◽

pp. 595-614

Author(s):

Vladimir Rabinovich

Keyword(s):

Smooth Boundary ◽

Integral Operators ◽

Variable Coefficients ◽

Boundary Integral ◽

Index Formula ◽

Robin Problem ◽

Neumann Problems ◽

Boundary Integral Operators ◽

Helmholtz Operators ◽

Lp Theory

AbstractThe paper is devoted to the ${L^{p}}$-theory of boundary integral operators for boundary value problems described by anisotropic Helmholtz operators with variable coefficients in unbounded domains with unbounded smooth boundary. We prove the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Bessel-potential spaces ${H^{s,p}(\partial D)}$, ${p\in(1,\infty)}$, and the Besov spaces ${B_{p,q}^{s}(\partial D)}$, ${p,q\in[1,\infty]}$. We prove also the Fredholmness of the Robin problem in these spaces and give the index formula.

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The Euler and Helmholtz operators on fibered manifolds with oriented bases

Colloquium Mathematicum ◽

10.4064/cm105-2-1 ◽

2006 ◽

Vol 105 (2) ◽

pp. 171-177

Author(s):

J. Kurek ◽

W. M. Mikulski

Keyword(s):

Helmholtz Operators ◽

Fibered Manifolds

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Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

Mathematical Problems in Engineering ◽

10.1155/2015/246159 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12

Author(s):

Chia-Cheng Tsai

Keyword(s):

Present Method ◽

Thick Plate ◽

Mindlin Plate ◽

Shear Forces ◽

Helmholtz Operator ◽

Particular Solutions ◽

Product Operator ◽

Plate Models ◽

Operator Decomposition ◽

Helmholtz Operators

Analytical particular solutions of the polyharmonic multiquadrics are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.

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