Reduced surface integral equations for Laplacian fields in the presence of layered bodies

2006 ◽  
Vol 84 (12) ◽  
pp. 1049-1061 ◽  
Author(s):  
I R Ciric
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Source Region ◽  
Normal Derivative ◽  
Surface Integral ◽  
Field Problem ◽  
Layered Bodies ◽  
High Computational Efficiency ◽  
Reduced Surface

Laplacian potential fields in stratified media are usually analyzed using an integral equation for an unknown function over the union of all the interfaces between regions with different homogeneous materials. In this paper, the field problem is solved using a reduced integral equation involving a single unknown function over only the boundary of the source region. The new integral equation is derived by introducing surface operators to express the potential and its normal derivative on each interface in terms of a single unknown function over the same interface. These operators and the corresponding single functions are obtained recursively, from one interface to the next. Thus, a substantial decrease in the amount of necessary numerical computation and computer memory is achieved especially for systems containing identical layered bodies where the reduction operators are only constructed for one of the bodies. The purpose of this paper is to derive reduced integral equations by directly applying the interface conditions and to show their high computational efficiency for systems of layered bodies.PACS Nos.: 02.30.Rz, 02.70.Pt, 41.20.Cv

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1193
Author(s):  
Suzan Cival Buranay ◽  
Mehmet Ali Özarslan ◽  
Sara Safarzadeh Falahhesar
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Input Data ◽  
Linear Equations ◽  
Numerical Approach ◽  
Volterra Integral Equations ◽  
Test Problems ◽  
The Stability ◽  
Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

scholarly journals The effect of near-zone preconditioning on electromagnetic integral equations of first and second kind

2013 ◽  
Vol 11 ◽  
pp. 61-65 ◽  
Author(s):  
O. Wiedenmann ◽  
T. F. Eibert
Keyword(s):  
Integral Equation ◽  
Electric Field ◽  
Integral Equations ◽  
Fast Multipole Method ◽  
Surface Integral ◽  
Electric Field Integral Equation ◽  
Integral Methods ◽  
Near Zone ◽  
Field Integral ◽  
Linear Equation Systems

Abstract. The linear equation systems which arise from the discretization of surface integral equations are conveniently solved with iterative methods because of the possibility to employ fast integral methods like the Multilevel Fast Multipole Method. However, especially integral equations of the first kind often lead to very ill-conditioned systems, which require the usage of effective preconditioners. In this paper, the regularization property of near-zone preconditioning operators on the Electric Field Integral Equation is demonstrated and investigated for problems of different size. Furthermore, comparisons are drawn to second-kind integral equations such as the Combined Field Integral Equation.

Solving the Volterra Integral Equations with Weakly Singular Kernel by Taylor Expansion Methods

2012 ◽  
Vol 220-223 ◽  
pp. 2129-2132
Author(s):  
Li Huang ◽  
Yu Lin Zhao ◽  
Liang Tang
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Taylor Expansion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we propose a Taylor expansion method for solving (approximately) linear Volterra integral equations with weakly singular kernel. By means of the nth-order Taylor expansion of the unknown function at an arbitrary point, the Volterra integral equation can be converted approximately to a system of equations for the unknown function itself and its n derivatives. This method gives a simple and closed form solution for the integral equation. In addition, some illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

On the evaluation of hyper-singular double normal derivative kernels in surface integral equation methods

2013 ◽  
Vol 37 (2) ◽  
pp. 205-210 ◽  
Author(s):  
A.G. Polimeridis ◽  
S. Järvenpää ◽  
P. Ylä-Oijala ◽  
L.J. Gray ◽  
S.P. Kiminki ◽  
...  
Keyword(s):  
Integral Equation ◽  
Normal Derivative ◽  
Surface Integral ◽  
Surface Integral Equation ◽  
Integral Equation Methods

Efficient Jacobian Matrix Determination for H2 Representations of Nonlinear Electrostatic Surface Integral Equations

2021 ◽  
Vol 35 (11) ◽  
pp. 1264-1265
Author(s):  
John Young ◽  
Robert Adams ◽  
Stephen Gedney
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Linear Algebra ◽  
Jacobian Matrix ◽  
Nonlinear Behavior ◽  
Electrochemical Reactions ◽  
Efficient Computation ◽  
Surface Integral ◽  
Surface Integral Equation ◽  
Fast Solution

In this paper, a nonlinear electrostatic surface integral equation is presented that is suitable for predicting corrosion-related fields. Nonlinear behavior arises due to electrochemical reactions at polarized surfaces. Hierarchical H2 matrices are used to compress the discretized integral equation for the fast solution of large problems. A technique based on randomized linear algebra is discussed for the efficient computation of the Jacobian matrix required at each iteration of a nonlinear solution.

Lanczos Biconjugate A-Orthonormalization Methods for Surface Integral Equations in Electromagnetism

PIERS Online ◽  
2010 ◽  
Vol 6 (4) ◽  
pp. 335-339 ◽  
Author(s):  
Bruno Carpentieri ◽  
Yan-Fei Jing ◽  
Tingzhu Huang
Keyword(s):  
Integral Equations ◽  
Surface Integral ◽  
Surface Integral Equations
Sign in / Sign up

Export Citation Format

Share Document