Integral Equation Formulations and Basic Numerical Methods

2002 ◽  
pp. 13-59
Keyword(s):  
Integral Equation ◽  
Numerical Methods

THE RESOLUTION CORRECTION IN THE SCINTILLATION SPECTROMETRY OF CONTINUOUS X RAYS

1958 ◽  
Vol 36 (12) ◽  
pp. 1624-1633 ◽  
Author(s):  
W. R. Dixon ◽  
J. H. Aitken
Keyword(s):  
Integral Equation ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Analytical Procedure ◽  
Pulse Height ◽  
Matrix Inversion ◽  
Height Distribution ◽  
X Rays ◽  
Smooth Solutions ◽  
Pulse Height Distribution

The problem of making resolution corrections in the scintillation spectrometry of continuous X rays is discussed. Analytical solutions are given to the integral equation which describes the effect of the statistical spread in pulse height. The practical necessity of making some kind of numerical analysis is pointed out. Difficulties with numerical methods arise from the fact that the observed pulse-height distribution cannot be defined precisely. As a result it is possible in practice only to find smooth "solutions". Additional difficulties arise if the numerical method is based on an invalid analytical procedure. For example matrix inversion is of doubtful value in making the resolution correction because there does not appear to be an inverse kernel for the integral equation in question.

scholarly journals Numerical methods for a Volterra integral equation with non-smooth solutions

2006 ◽  
Vol 189 (1-2) ◽  
pp. 412-423 ◽  
Author(s):  
Teresa Diogo ◽  
Neville J. Ford ◽  
Pedro Lima ◽  
Svilen Valtchev
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Smooth Solutions

scholarly journals Numerical Methods for Solving Fredholm Integral Equations of Second Kind

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
S. Saha Ray ◽  
P. K. Sahu
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Integral Equations ◽  
Applied Mathematics ◽  
Fredholm Integral Equations ◽  
Nonlinear Fredholm Integral Equations ◽  
Linear And Nonlinear ◽  
Interdisciplinary Effort

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

XVI.—Dual Series Relations. III. Dual Relations Involving Trigonometric Series

1963 ◽  
Vol 66 (3) ◽  
pp. 173-184 ◽  
Author(s):  
R. P. Srivastav
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Integral Equations ◽  
Trigonometric Series ◽  
Analytical Solutions ◽  
Fredholm Integral Equation ◽  
Auxiliary Function

SynopsisThe methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

scholarly journals Two numerical methods for Abel's integral equation with comparison

2012 ◽  
Vol Volume 15, 2012 ◽  
Author(s):  
Abdallah Ali Badr
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Fractional Order ◽  
Quadrature Formula ◽  
Approximate Formula ◽  
Simple Pole ◽  
Numerical Examples ◽  
International Audience ◽  
Modified Formula

International audience Analogy between Abel's integral equation and the integral of fractional order of a given function, j^α f(t), is discussed. Two different numerical methods are presented and an approximate formula for j^α f(t) is obtained. The first approach considers the case when the function, f(t), is smooth and a quadrature formula is obtained. A modified formula is deduced in case the function has one or more simple pole. In the second approach, a procedure is presented to weaken the singularities. Both two approaches could be used to solve numerically Abel's integral equation. Some numerical examples are given to illustrate our results.

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