An Integral Equation Formulation of a Mixed Boundary Value Problem on a Sphere

1975 ◽  
Vol 6 (2) ◽  
pp. 417-426 ◽  
Author(s):  
H. L. Johnson
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem ◽  
Equation Formulation ◽  
Integral Equation Formulation

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.

scholarly journals Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

2021 ◽  
Vol 13 (1) ◽  
pp. 85-97
Author(s):  
Elnur Hasan ogly Khalilov ◽  
Mehpara Nusrat kyzy Bakhshaliyeva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Laplace Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem

On the integral equation method for the plane mixed boundary value problem of the Laplacian

1979 ◽  
Vol 1 (3) ◽  
pp. 265-321 ◽  
Author(s):  
W. L. Wendland ◽  
E. Stephan ◽  
G. C. Hsiao ◽  
E. Meister
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Equation Method ◽  
Equation Method ◽  
Mixed Boundary Value Problem

scholarly journals Localized boundary-domain integral equation formulation for mixed type problems

2010 ◽  
Vol 17 (3) ◽  
pp. 469-494 ◽  
Author(s):  
Otar Chkadua ◽  
Sergey E. Mikhailov ◽  
David Natroshvili
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Integral Equations ◽  
Variable Coefficient ◽  
Mixed Type ◽  
Mixed Boundary ◽  
Integral Operators ◽  
Mixed Boundary Value Problem ◽  
Domain Integral ◽  
Integral Equation Formulation

Abstract Some modifed direct localized boundary-domain integral equations (LBDIEs) systems associated with the mixed boundary value problem (BVP) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed. The main results established in the paper are the LBDIEs equivalence to the original variable-coefficient BVPs and the invertibility of the corresponding localized boundary-domain integral operators in appropriately chosen function spaces.

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