scholarly journals Modeling the Diffraction of Electromagnetic Waves over Underwater Objects; the Wiener-Hopf Integral Equation

2020 ◽  
pp. 135-144
Author(s):  
Yajni Warnapala ◽  
Cole Foster
Keyword(s):  
Integral Equation ◽  
Electromagnetic Wave ◽  
Galerkin Method ◽  
Boundary Value Problems ◽  
Electromagnetic Waves ◽  
Good Convergence ◽  
Convergence Results ◽  
Lossy Media ◽  
The Galerkin Method ◽  
Airline Flight

This research, inspired by the loss of Malaysian Airline Flight 370, investigates the feasibility of obtaining good convergence results for a model of the interaction of electromagnetic waves over the surface of the Spherical Biconcave Disc. The Galerkin Method is used to numerically solve the Dirichlet and Neumann exterior boundary value problems for the Wiener-Hopf Integral Equation over the half-plane of the Spherical Biconcave Disc. This modeling accounts for the attenuation losses of the propagating electromagnetic wave as a result of absorption and scattering in lossy media with comparison to lossless propagation. The numerical results of this research nds good convergence for this model as well as limitations in the transmission of electromagnetic waves underwater.

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

scholarly journals Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

2021 ◽  
Vol 2128 (1) ◽  
pp. 012035
Author(s):  
W. Abbas ◽  
Mohamed Fathy ◽  
M. Mostafa ◽  
A. M. A Hesham
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Nonlinear Boundary ◽  
Analytic Solution ◽  
Boundary Value ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Algebraic Set ◽  
The Galerkin Method

Abstract In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

Scattering Problems: Beltrami Fields and Solvability

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Integral Equation ◽  
Electromagnetic Wave ◽  
Electromagnetic Waves ◽  
Boundary Integral ◽  
Scattering Problems ◽  
Electromagnetic Wave Scattering ◽  
Beltrami Fields ◽  
Integral Equation Methods ◽  
Time Harmonic ◽  
Surrounding Space

This chapter deals with the solvability of time-harmonic electromagnetic wave scattering by an obstacle: either the obstacle or the environment in which it is embedded, or both, is (are) occupied by a chiral material. It assumes that the scatterer and its surrounding space are homogeneous: thus allowing the use of the boundary integral equation methods for the study of the considered problems. This chapter considers two kinds of problems: first, the scattering of plane electromagnetic waves propagating in chiral space by a perfectly conducting obstacle, and second, the scattering of plane electromagnetic waves by a penetrable obstacle; either the scatterer or the surrounding space, or both, may be filled with a chiral material.

scholarly journals Efficient Solutions of Multidimensional Sixth-Order Boundary Value Problems Using Symmetric Generalized Jacobi-Galerkin Method

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Efficient Solutions ◽  
Sixth Order ◽  
Condition Numbers ◽  
Basic Functions ◽  
The Galerkin Method ◽  
Fourth Order Elliptic Equations

This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities. These algorithms are extensions to some of the algorithms proposed by Doha and Abd-Elhameed (2002) and Doha and Bhrawy (2008) for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms.

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