An improved integral equation for the numerical solution of two-dimensional diffraction problems

1970 ◽  
Vol 48 (8) ◽  
pp. 954-963 ◽  
Author(s):  
Lotfollah Shafai
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Electromagnetic Fields ◽  
Current Distribution ◽  
Conformal Transformation ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Square Prism

Using a conformal transformation, the singularities of electromagnetic fields near a sharp edge are related to the geometry of the scatterer. The results are then used to treat the singularities of the current in the integral equations for the current distribution. The properties of the improved integral equations and the procedure for their evaluation is discussed by considering two-dimensional scattering by a conducting square prism.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

Integral Equation Formulations and Related Numerical Solution Methods in Some Biomedical Applications of Electromagnetic Fields

2020 ◽  
pp. 249-267
Author(s):  
Dragan Poljak ◽  
Mario Cvetković ◽  
Vicko Dorić ◽  
Ivana Zulim ◽  
Zoran Đogaš ◽  
...  
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Nerve Fiber ◽  
Electromagnetic Fields ◽  
Biomedical Applications ◽  
Current Source ◽  
Myelinated Nerve Fiber ◽  
Surface Integral ◽  
Myelinated Nerve ◽  
Solution Methods

The paper reviews certain integral equation approaches and related numerical methods used in studies of biomedical applications of electromagnetic fields pertaining to transcranial magnetic stimulation (TMS) and nerve fiber stimulation. TMS is analyzed by solving the set of coupled surface integral equations (SIEs), while the numerical solution of governing equations is carried out via Method of Moments (MoM) scheme. A myelinated nerve fiber, stimulated by a current source, is represented by a straight thin wire antenna. The model is based on the corresponding homogeneous Pocklington integro-differential equation solved by means of the Galerkin Bubnov Indirect Boundary Element Method (GB-IBEM). Some illustrative numerical results for the TMS induced fields and intracellular current distribution along the myelinated nerve fiber (active and passive), respectively, are presented in the paper.

Integral Equation Formulations and Related Numerical Solution Methods in Some Biomedical Applications of Electromagnetic Fields

2018 ◽  
Vol 9 (1) ◽  
pp. 65-84 ◽  
Author(s):  
Dragan Poljak ◽  
Mario Cvetković ◽  
Vicko Dorić ◽  
Ivana Zulim ◽  
Zoran Đogaš ◽  
...  
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Nerve Fiber ◽  
Electromagnetic Fields ◽  
Biomedical Applications ◽  
Current Source ◽  
Wire Antenna ◽  
Myelinated Nerve Fiber ◽  
Myelinated Nerve ◽  
Numerical Solution Methods

The paper reviews certain integral equation approaches and related numerical methods used in studies of biomedical applications of electromagnetic fields pertaining to transcranial magnetic stimulation (TMS) and nerve fiber stimulation. TMS is analyzed by solving the set of coupled surface integral equations (SIEs), while the numerical solution of governing equations is carried out via Method of Moments (MoM) scheme. A myelinated nerve fiber, stimulated by a current source, is represented by a straight thin wire antenna. The model is based on the corresponding homogeneous Pocklington integro-differential equation solved by means of the Galerkin Bubnov Indirect Boundary Element Method (GB-IBEM). Some illustrative numerical results for the TMS induced fields and intracellular current distribution along the myelinated nerve fiber (active and passive), respectively, are presented in the paper.

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