Using the Galerkin method to construct an analytical solution for the integral equation of a certain diffraction problem

1988 ◽  
Vol 31 (11) ◽  
pp. 974-979 ◽  
Author(s):  
I. M. Braver ◽  
Kh. L. Garb ◽  
P. Sh. Fridberg ◽  
I. M. Yakover
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Galerkin Method ◽  
Diffraction Problem ◽  
The Galerkin Method

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 Algorithm for Numerical Solution of Diffraction Problem on the Joint of Two Open Three-Layer Waveguides

2018 ◽  
Vol 186 ◽  
pp. 01010
Author(s):  
Dmitriy Divakov ◽  
Anastasiia Tiutiunnik ◽  
Anton Sevastianov
Keyword(s):  
Galerkin Method ◽  
Numerical Solution ◽  
Dimensional Space ◽  
Diffraction Problem ◽  
Liouville Problem ◽  
Planar Waveguides ◽  
Problem Solution ◽  
Scalar Diffraction ◽  
Open Waveguide ◽  
The Galerkin Method

This paper describes the algorithm for the numerical solution of the diffraction problem of waveguide modes at the joint of two open planar waveguides. For planar structures under consideration, we can formulate a scalar diffraction problem, which is a boundary value problem for the Helmholtz equation with a variable coefficient in two-dimensional space. The eigenmode problem for an open three-layer waveguide is the Sturm-Liouville problem for a second-order operator with piecewise constant potential on the axis, where the potential is proportional to the refractive index. The described problem is singular and has a mixed spectrum and therefore the Galerkin method can not be used in this definition. One way to adapt the Galerkin method for the problem solution is to artificially limit the area, which is equivalent to placing the open waveguide in question in a hollow closed waveguide whose boundaries are remote from the real boundaries of the waveguide layer of the open waveguide. Thus, we obtain a diffraction problem on a finite interval and with a discrete spectrum, which can be solved by the projection method, as done in this paper.

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.

Non-Newtonian oscillatory layer flow, approximate solution

1979 ◽  
Vol 44 (10) ◽  
pp. 2908-2914 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Galerkin Method ◽  
Horizontal Plane ◽  
Oscillatory Flow ◽  
Oscillatory Boundary Layer ◽  
Layer Flow ◽  
Pseudoplastic Liquid ◽  
The Galerkin Method

The problem of the oscillatory flow of pseudoplastic liquid in vicinity of the infinitely long horizontal plane is formulated in stresses. For Re i.e. for conditions of oscillatory boundary layer the problem is solved approximately by the Galerkin method.

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