A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

SynopsisExistence and uniqueness theorems are proved for the solution of a Dirichlet-Neumann-Third mixed boundary value problem for the Helmholtz equation in ℝ3. The proofs make use of an equivalent system of two integral equations of the second kind.

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Electrostatic potential calculation by application of walk-on-spheres method

Cybernetics and Physics ◽

10.35470/2226-4116-2018-7-3-152-160 ◽

2018 ◽

pp. 152-160

Author(s):

Liudmila Vladimirova ◽

Irina Rubtsova ◽

Nikolai Edamenko

Keyword(s):

Integral Equation ◽

Problem Solving ◽

Boundary Value Problem ◽

Mixed Boundary ◽

Boundary Value ◽

Mixed Boundary Value Problem ◽

Equation Solving ◽

Electrostatic Potential Calculation ◽

Walk On Spheres Algorithm ◽

Boundary Form

The paper is devoted to mixed boundary-value problem solving for Laplace equation with the use of walk-on-spheres algorithm. The problem under study is reduced to finding a solution of integral equation with the kernel nonzero only at some sphere in the domain considered. Ulam-Neumann scheme is applied for integral equation solving; the appropriate Markov chain is introduced. The required solution value at a certain point of the domain is approximated by the expected value of special statistics defined on Markov paths. The algorithm presented guarantees the average Markov trajectory length to be finite and allows one to take into account boundary conditions on required solution derivative and to avoid Markov paths ending in the neighborhood of the boundaries where solution values are not given. The method is applied for calculation of electric potential in the injector of linear accelerator. The purpose of the work is to verify the applicability and effectiveness of walk-on-spheres method for mixed boundary-value problem solving with complicated boundary form and thus to demonstrate the suitability of Monte Carlo methods for electromagnetic fields simulation in beam forming systems. The numerical experiments performed confirm the simplicity and convenience of this method application for the problem considered.

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Symmetric Complex Boundary Element Scheme for 2-D Stokes Mixed Boundary Value Problem

Electronic Journal of Boundary Elements ◽

10.14713/ejbe.v13i1.1972 ◽

2016 ◽

Vol 13 (1) ◽

Author(s):

Sun-Gwon Hong

Keyword(s):

Integral Equation ◽

Boundary Integral Equation ◽

Mixed Boundary ◽

Boundary Value ◽

Boundary Integral ◽

Hypersingular Integrals ◽

Element Scheme ◽

Boundary Limit ◽

Complex Boundary ◽

Gt Element

For 2-D Stokes mixed boundary value problems we construct a boundary integral equation which couples a conventional boundary integral equation for the velocity with a hypersingular boundary integral equation for the traction. Expressing terms in the equation by complex variables, we obtain a complex boundary integral equation and realize symmetrization of boundary element scheme by Galerkin method. Applying a boundary limit method, we obtain exact calculation formulae for calculation of hypersingular boundary integrals. It is shown that all divergent terms in hypersingular integrals cancel each other out.

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