Solution of the Dirichlet Problem for the Laplace equation

Medková, Dagmar

doi:10.1023/a:1022209421576

Cited by 12 publications

(6 citation statements)

References 33 publications

(21 reference statements)

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“…Using the homogeneous identity (17), we check that µ 0 (s) satisfies conditions (5). Furthermore, acting on the homogeneous identity (17) with a singular operator with the kernel (s − t) −1 , we find that µ 0 (s) satisfies the homogeneous equation (13). Consequently, µ 0 (s) satisfies the homogeneous equations (11).…”

Section: The Fredholm Integral Equation and The Solution Of The Problmentioning

confidence: 84%

“…to an infinite algebraic system of equations [20], while the Dirichlet problem in domains bounded by closed curves was reduced to the Fredholm equation of the second kind [13]- [17]. The combination of these methods in the case of domains bounded by closed curves and cuts leads to an integral equation, which is algebraic or hypersingular on cuts, while it is an equation of the second kind with compact integral operators on the closed curves.…”

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confidence: 99%

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On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Крутицкий

2007

Quart. Appl. Math.

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Abstract. The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated. Introduction. The boundary of a 2-D cracked domain consists of both closed curves and open arcs (cuts).Open arcs model cracks in solids and screens or wings in fluids. Different physical processes in cracked domains can be described by boundary value problems for the Laplace equation, for example, distribution of stationary heat and electric fields in cracked solids, electric flow in cracked semiconductors, flow of an ideal fluid over several obstacles and wings, etc. Appropriate boundary conditions must be specified on the total boundary, i.e., on both closed curves and open arcs (cracks). The Neumann boundary condition reflects the nonflow (of fluid, electric current, etc.) through the boundary. The Dirichlet boundary condition corresponds to the given temperature in heat theory, fluid pressure in hydrodynamics, electric potential in electrostatics, etc.Boundary value problems with mixed boundary conditions were not treated in cracked domains by rigorous mathematical methods before. Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [2], [13]- [17] and problems in the exterior of cuts (cracks) [14,16], [18]-[20] were treated separately, because different methods were used in their analysis. Previously the Neumann problem in the exterior of a cut was reduced to a hypersingular integral equation [14,16,18,19] or

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Section: The Fredholm Integral Equation and The Solution Of The Problmentioning

confidence: 84%

mentioning

confidence: 99%

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Крутицкий

2007

Quart. Appl. Math.

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“…Different techniques have been proposed in order to solve the mentioned class of differential problems both from a theoretical and numerical point of view (e.g., representing the solution by means of boundary layer techniques [4], solving the corresponding boundary integral equation by iterative methods [5], approximating the relevant Green function by means of the least squares fitting technique [6], solving the linear system relevant to an elliptic partial differential equation by means of relaxation methods [7]). However, none of the contributions already available in the scientific literature deals with the classical Fourier projection method [8] which has been extended in recent papers [9][10][11][12][13][14][15][16] in order to address boundary-value problems (BVPs) in simply connected starlike domains whose boundaries may be regarded as an anisotropically stretched unit circle or sphere centered at the origin.…”

Section: Introductionmentioning

confidence: 99%

Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

Caratelli¹,

Gielis²,

Tavkhelidze³

et al. 2013

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The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called "superformula" introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica © is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.

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“…It includes a detailed study of the corresponding boundary layer potentials and ends up with a boundary integral equation method reducing (1) to a system of uniquely solvable second kind Fredholm boundary integral equations. We construct a solution of the integral equations for a non-cracked domain in the form of a Neumann series, where we extend the method used in [7] for the Dirichlet problem of the Laplace equation. Then we apply this result for the construction of a solution for the boundary value problem of the Stokes system in case of a cracked domain.…”

Section: Introductionmentioning

confidence: 99%

Boundary value problems for the Stokes equations with jumps in open sets

Medková

Varnhorn

2008

Applicable Analysis

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A boundary value problem for the Stokes system is studied in a cracked domain in R n , n 4 2, where the Dirichlet condition is specified on the boundary of the domain. The jump of the velocity and the jump of the stress tensor in the normal direction are prescribed on the crack. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence, a maximum modulus estimate for the Stokes system is proved.

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Solution of the Dirichlet Problem for the Laplace equation

Cited by 12 publications

References 33 publications

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

Boundary value problems for the Stokes equations with jumps in open sets

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