scholarly journals Deferred correction for the integral equation eigenvalue problem

1981 ◽  
Vol 22 (4) ◽  
pp. 474-487 ◽  
Author(s):  
K. W. Chu ◽  
A. Spence
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Integral Equations ◽  
Convergence Theorem ◽  
Eigenvalues And Eigenfunctions ◽  
Deferred Correction ◽  
Numerical Example

AbstractThis paper considers the improvement of approximate eigenvalues and eigenfunctions of integral equations using the method of deferred correction. A convergence theorem is proved and a numerical example illustrating the theory is given.

THE ADOMIAN DECOMPOSITION METHOD APPLIED TO THE FUZZY SYSTEM OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

2006 ◽  
Vol 14 (01) ◽  
pp. 101-110 ◽  
Author(s):  
S. ABBASBANDY ◽  
T. ALLAHVIRANLOO
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Fuzzy System ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Extension Principle ◽  
Numerical Example ◽  
Adomian Decomposition

In this work, the Adomian decomposition(AD) method is applied to the Fuzzy system of linear Fredholm integral equations of the second kind(FFIE). First the crisp Fredholm integral equation is solved by AD method and then the crisp solution is fuzzified by extension principle. The proposed algorithm is illustrated by solving a numerical example.

scholarly journals Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
K. Maleknejad ◽  
M. Khodabin ◽  
F. Hosseini Shekarabi
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations ◽  
Numerical Example ◽  
Block Pulse Functions ◽  
A New Technique ◽  
Modified Block

We present a new technique for solving numerically stochastic Volterra integral equation based on modified block pulse functions. It declares that the rate of convergence of the presented method is faster than the method based on block pulse functions. Efficiency of this method and good degree of accuracy are confirmed by a numerical example.

scholarly journals Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yabin Shao ◽  
Huanhuan Zhang
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Convergence Theorem ◽  
Existence Theorems ◽  
Fuzzy Integral ◽  
Henstock Integral

By using the strong fuzzy Henstock integral and its controlled convergence theorem, we generalized the existence theorems of solution for initial problems of fuzzy discontinuous integral equation.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

2011 ◽  
Vol 255-260 ◽  
pp. 1830-1835 ◽  
Author(s):  
Gang Cheng ◽  
Quan Cheng ◽  
Wei Dong Wang
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Free Vibration ◽  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
Integral Equation Method ◽  
Free Vibrations ◽  
Equation Method ◽  
Concentrated Mass

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

scholarly journals A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

2020 ◽  
Vol 53 (1) ◽  
pp. 152-166 ◽  
Author(s):  
Getahun B. Wega ◽  
Habtu Zegeye ◽  
Oganeditse A. Boikanyo
Keyword(s):  
Strong Convergence ◽  
Approximation Method ◽  
Convergence Theorem ◽  
Finite Family ◽  
Strong Convergence Theorem ◽  
Monotone Mappings ◽  
Important Class ◽  
Numerical Example ◽  
Minimization Problems ◽  
Nonlinear Mappings

AbstractThe purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

A new extension of Kannan’s fixed point theorem via F-contraction with application to integral equations

2021 ◽  
pp. 2250123
Author(s):  
Pradip Debnath
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Point Theorem ◽  
Extended Version

Our aim is to introduce an updated and real generalization of Kannan’s fixed point theorem with the help of [Formula: see text]-contraction introduced by Wardowski for single-valued mappings. Our result can be useful to ascertain the existence of fixed point for a family of mappings for which neither the Wardowski’s result nor that of Kannan can be applied directly. Our result has been applied to solve a particular type of integral equation. Finally, we establish a Reich-type extended version of the main result.

scholarly journals Integral equations of the first kind of Sonine type

2003 ◽  
Vol 2003 (57) ◽  
pp. 3609-3632 ◽  
Author(s):  
Stefan G. Samko ◽  
Rogério P. Cardoso
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Lebesgue Space ◽  
Inverse Operator ◽  
Orlicz Spaces ◽  
Hypersingular Integral ◽  
The Real ◽  
Integrable Kernel

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

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 Forward

2007 ◽  
Vol 1 (1) ◽  
Author(s):  
Thomas J. Rudolphi
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Equation Approach ◽  
Equation Formulation ◽  
Integral Equation Formulation ◽  
Potential Problems ◽  
Integral Equation Approach

<br /><br /> <table width="530" border="0" cellspacing="0" cellpadding="0"> <tr> <td align="left" valign="top"> <a name="abstract"></a> <span class="subtitle" style="font-weight:bold">Abstract</span><br /> <p><img src="http://ejbe.libraries.rutgers.edu/files/rizzo.gif" align=left HSPACE=20>This is the first of two special issues of the Electronic Journal of Boundary Elements dedicated to Frank Rizzo. To say that Frank Rizzo played an important role in the development of what he referred to as “boundary integral equations� would not give much credit to where much credit is due. While it could be argued that the use of integral equations to formulate and form a computational basis of many of the problems of applied mathematics and engineering would probably have been inevitably developed, it was Frank’s seminal work on using the integral equation approach to classical elastostatics that set a whole new research area into motion. His dissertation (which we thought would be of interest to include in this issue) topic, as suggested by his mentor Marvin Stippes at the University of Illinois, and subsequently so well documented in the oft-cited paper “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics�, Quarterly of Applied Mechanics, 1967, represented the quantum step in the use of integral equations for classical scalar potential problems to the vector potential problems of practical engineering significance. The theoretical basis for this development was Betti’s reciprocal work theorem with the fundamental (response to a point force) solution of the equations of elastostatics, but it was Frank Rizzo who actually breathed the new life into this classical mathematics. A nontrivial contribution of Frank’s original work was to not only to achieve the singular integral equation formulation, but also the systematic methodology of reducing the elegant integral equation formulation to well conditioned, linear algebraic equations by proper analytical integration of the singular terms. Those combined theoretical and practical developments by Frank set into motion a whole new and modern approach to numerically solving partial differential equations, at least of the elliptic type. With Frank’s hard work and the recognition of its elegance and potential by several of his early disciples, the integral equation method blossomed into a powerful and practical computational methodology that would eventually be called “boundary elements�. Amongst the early disciples of the integral equation method, several of which contributed significantly to advancing the methodology to a sophisticated and now mature state, are the authors of this issue and its sequel dedicated to Frank. It is undoubtedly fair to say that most of these authors were, at one time or even continuously, colleagues and personal friends of Frank Rizzo. Frank’s contributions to the boundary integral equation method spanned nearly four decades, from roughly 1964 to 2001. I, too, have been very privileged to become involved with this field in the 1970’s and later to work side by side with Frank, especially in that part of the development of the methodology for what is now referred to as “hypersingular� integral equations. I’m sure that all the present authors can recall numerous occasions and conversations with Frank on a technical point or issue regarding the application of “his� boundary integral method to their own problem of interest. Throughout his productive career, his easy going, collegial, engaging, yet rigorous style earned him respect and admiration that surely befits the “father� of modern boundary integral methods. This commemorative sequence of two issues represents only a small token of tribute and recognition that Frank Rizzo so much deserves for his “singular� contributions to the field that he virtually invented, developed, promoted and nurtured to maturity. Thomas J. Rudolphi Iowa State University <br /><br /><br /> </td> </tr> </table>

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