Solving a mixed boundary value problem via an integral equation with adjoint generalized Neumann kernel in bounded multiply connected regions

In this paper, we solve the mixed boundary value problem on unbounded multiply connected region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test regions are presented.

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A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

Journal of Applied Mathematics ◽

10.1155/2012/254123 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 5

Author(s):

Mohamed M. S. Nasser ◽

Ali H. M. Murid ◽

Samer A. A. Al-Hatemi

Keyword(s):

Integral Equation ◽

Boundary Integral Equation ◽

Mixed Boundary Condition ◽

Mixed Boundary ◽

Boundary Integral ◽

Numerical Examples ◽

Multiply Connected ◽

Generalized Neumann Kernel ◽

Multiply Connected Regions ◽

Neumann Kernel

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two-dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method.

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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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