On the Solution of an Ill-Posed Non-Linear Fredholm Integral Equation Connected with an Inverse Problem of Thin Film Optics

1993 ◽  
Vol 12 (2) ◽  
pp. 319-326
Author(s):  
H. Schachtzabel ◽  
H.-A. Braunss ◽  
B. Hofmann
Keyword(s):  
Thin Film ◽  
Inverse Problem ◽  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Non Linear ◽  
Thin Film Optics ◽  
Ill Posed

scholarly journals ON THE VIBRATIONS OF INHOMOGENEOUS PIEZO DISC

2019 ◽  
Vol 81 (3) ◽  
pp. 369-380
Author(s):  
A.O. Vatulyan ◽  
Yu.N. Zubkov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Radial Displacement ◽  
Canonical System ◽  
Cauchy Problems ◽  
Resonance Frequencies ◽  
Finite Set ◽  
Inhomogeneous Properties ◽  
Set Of Points

In the framework of the model of coupled electroelasticity of inhomogeneous bodies, the problem of steady-state oscillations of a thin piezodisc with inhomogeneous properties is considered, in particular, in the presence of radial polarization. The necessary simplifications are made within the framework of traditional hypotheses, the formulated boundary-value problem is reduced to a canonical system of first-order differential equations with respect to dimensionless components of radial displacement and radial stress with corresponding boundary conditions. The direct problem of oscillations of an inhomogeneous disk is solved numerically based on the shooting method by numerically analyzing auxiliary Cauchy problems. The analysis of the amplitude-frequency characteristics and resonance frequencies depending on various laws of variation of the inhomogeneous properties of the piezodisc is performed, which in the presented model are characterized by two functions, one of which characterizes the change in the elastic modulus, the second changes in the piezomodule. The inverse problem is formulated in the first statement, in which the laws of variation of the piezodisc heterogeneity (two functions) are restored from the values of the functions characterizing the radial displacement and stress, known in a finite set of points. The results of computational experiments on solving the inverse problem in the first formulation are presented, various aspects of reconstruction are discussed. The second formulation of the inverse problem is formulated to determine the piezoelectric characteristics of the disk, where a function that describes the laws of change in the elastic characteristics of the disk and the amplitude-frequency characteristic is considered known. To solve the inverse problem, in this formulation, the Fredholm integral equation of the first kind with a smooth kernel is formulated. The results of numerical experiments on solving the Fredholm integral equation of the first kind using the Tikhonov regularizing method are presented, various aspects of reconstruction are discussed.

scholarly journals Numerical Solution of Nonlinear Fredholm Integral Equations by Using NKM and ADM

2017 ◽  
Vol 65 (1) ◽  
pp. 61-66
Author(s):  
MM Hasan ◽  
MA Matin
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Quadrature Method ◽  
Adomian Decomposition ◽  
Non Linear ◽  
Nonlinear Fredholm Integral Equations

In this paper, we present a numerical method to solve a non-linear Fredholm integral equations. We intend to approximate the solution of this equation by Newton-Kantorovich-quadrature method and Adomian Decomposition method compare both the methods accurately for solving the non-linear Fredholm integral equation. Dhaka Univ. J. Sci. 65(1): 61-66, 2017 (January)

AN ACOUSTIC INVERSE PROBLEM: NUMERICAL EXPERIMENT

1995 ◽  
Vol 03 (03) ◽  
pp. 229-240 ◽  
Author(s):  
R. P. GILBERT ◽  
ZHONGYAN LIN
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Numerical Experiment ◽  
Fredholm Integral Equation ◽  
Numerical Experiments ◽  
Goursat Problem ◽  
Index Of Refraction ◽  
Numerical Treatment ◽  
Moments Problem

As a sequel to Refs. 1 and 2, this paper gives a numerical treatment of the inverse problem associated with the determination of the index of refraction. We show that the problem can be solved in two steps. First we must recover a function from its moments, problem (IM), which we may reformulate as a Fredholm integral equation of the first kind, problem (IE). Second we solve an inverse Goursat problem, (IG). Numerical schemes for both steps are given along with the results of some numerical experiments.

scholarly journals ON THE DISCRETE LINEAR ILL‐POSED PROBLEMS

1999 ◽  
Vol 4 (1) ◽  
pp. 147-152
Author(s):  
A. A. Stepanov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Regularization Methods ◽  
Acoustic Spectroscopy ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Photo Acoustic Spectroscopy

An inverse problem of photo‐acoustic spectroscopy of semiconductors is investigated. The main problem is formulated as the integral equation of the first kind. Two different regularization methods are applied, the algorithms for defining regularization parameters are given.

scholarly journals Some Probability Characteristics Functions of the Solution of a Stochastic Non-Linear Fredholm Integral Equation of the Second Kind

2011 ◽  
Vol 8 (2) ◽  
pp. 394-399
Author(s):  
Baghdad Science Journal
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Spectral Density ◽  
Probability Density ◽  
Fredholm Integral Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Non Linear

In this research, some probability characteristics functions (probability density, characteristic, correlation and spectral density) are derived depending upon the smallest variance of the exact solution of supposing stochastic non-linear Fredholm integral equation of the second kind found by Adomian decomposition method (A.D.M)

An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part I: Theoretical Approach

1993 ◽  
Vol 60 (3) ◽  
pp. 595-600 ◽  
Author(s):  
Weichung Yeih ◽  
Tatsuhito Koya ◽  
Toshio Mura
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Linear Elasticity ◽  
Fredholm Integral Equation ◽  
Theoretical Approach ◽  
Regularization Method ◽  
Numerical Examples

A Cauchy problem in linear elasticity is considered. This problem is governed by a Fredholm integral equation of the first kind and cannot be solved directly. The regularization method, which has been originally employed by Gao and Mura (1989), is formulated from a different perspective in order to address some of the difficulties experienced in their formulation. The theoretical details are discussed in this paper. Numerical examples are treated to Part II.

scholarly journals An inverse problem of synthesis of nanooptical security elements for visual and automated authenticity verification

2020 ◽  
pp. 56-63
Author(s):  
А.А. Гончарский ◽  
С.Р. Дурлевич
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
White Light ◽  
Fredholm Integral Equation ◽  
Phase Function ◽  
Focal Plane ◽  
Optical Element ◽  
Coherent Radiation ◽  
Nonlinear Fredholm Integral Equation ◽  
Elementary Region

Статья посвящена решению обратных задач синтеза нанооптических защитных элементов. Синтез нанооптического элемента включает в себя как решение обратной задачи расчета его фазовой функции, так и прецизионное формирование микрорельефа. При освещении микрорельефа в любой точке нанооптического элемента когерентным излучением в фокальной плоскости, параллельной плоскости оптического элемента, формируется изображение, используемое для автоматизированного контроля. Область оптического элемента разбивается на элементарные области. Изображение в элементарных областях формируется с помощью бинарных киноформов, фазовая функция которых рассчитывается с помощью решения нелинейного интегрального уравнения Фредгольма первого рода. Глубина микрорельефа в каждой элементарной области постоянна и определяет цвет элементарной области при освещении оптического элемента белым светом. Разработанные элементы могут быть использованы для защиты документов, акцизных марок, брендов и др. This paper is concerned with solving inverse problems of the synthesis of nanooptical security elements. The synthesis of a nanooptical element involves calculating its phase function via solving an inverse problem and fabricating the microrelief with high precision. The microrelief of the nanooptical element illuminated at any point with coherent radiation produces an image in the focal plane parallel to the plane of the optical element. This image is used for the automated authenticity verification. The area of the optical element is divided into elementary regions. In each elementary region, the image is formed using binary kinoforms whose phase function is calculated via solving a nonlinear Fredholm integral equation of the first kind. The depth of the microrelief is constant in each elementary region and determines the color of that region when the optical element is illuminated with white light. The developed elements can be used to protect documents, excise stamps, and brands.

scholarly journals An Inverse Problem Solution Scheme for Solving the Optimization Problem of Drug-Controlled Release from Multilaminated Devices

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Xinming Zhang
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Controlled Release ◽  
Fredholm Integral Equation ◽  
Regularization Method ◽  
Optimization Problem ◽  
Problem Solution ◽  
Drug Controlled Release ◽  
Inverse Problem Solution ◽  
Solution Scheme

The optimization problem of drug release based on the multilaminated drug-controlled release devices has been solved in this paper under the inverse problem solution scheme. From the viewpoint of inverse problem, the solution of optimization problem can be regarded as the solution problem of a Fredholm integral equation of first kind. The solution of the Fredholm integral equation of first kind is a well-known ill-posed problem. In order to solve the severe ill-posedness, a modified regularization method is presented based on the Tikhonov regularization method and the truncated singular value decomposition method. The convergence analysis of the modified regularization method is also given. The optimization results of the initial drug concentration distribution obtained by the modified regularization method demonstrate that the inverse problem solution scheme proposed in this paper has the advantages of the numerical accuracy and antinoise property.

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