On the Treatment of a Dirichlet–Neumann Mixed Boundary Value Problem for Harmonic Functions by an Integral Equation Method

1977 ◽  
Vol 8 (3) ◽  
pp. 504-517 ◽  
Author(s):  
R. González ◽  
R. Kress
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Harmonic Functions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Equation Method ◽  
Equation Method ◽  
Mixed Boundary Value Problem

The treatment of a Neumann boundary value problem for force-free fields by an integral equation method

1978 ◽  
Vol 82 (1-2) ◽  
pp. 71-86 ◽  
Author(s):  
Rainer Kress
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Vector Fields ◽  
Integral Formula ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Integral Equation Method ◽  
Equation Method ◽  
Fredholm Alternative ◽  
Neumann Boundary Value Problem

SynopsisA Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.

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

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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