scholarly journals Invertible linear relations generated by integral equations with operator measures

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1589-1607
Author(s):  
Vladislav Bruk
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Linear Relations ◽  
Minimal Relation

We define a minimal relation L0 generated by an integral equation with operators measures and give a description of the relations L0- ?E, L*0- ?E, where L*0 is adjoint for L0, ?? C. The obtained results are applied to a description of relations T(?) such that L0- ?E ? T(?) ? L*0- ?E and T-1(?) are bounded everywhere defined operators.

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>

scholarly journals Solution of Variavle Delay integral eqiations using Variational approach

2006 ◽  
Vol 3 (3) ◽  
pp. 481-487
Author(s):  
Baghdad Science Journal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Variational Approach

The main objective of this research is to use the methods of calculus ???????? solving integral equations Altbataah When McCann slowdown is a function of time as the integral equation used in this research is a kind of Volterra

scholarly journals Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 180
Author(s):  
Oleg Avsyankin
Keyword(s):  
Integral Equation ◽  
Asymptotic Behavior ◽  
Integral Equations ◽  
Spherical Harmonics ◽  
Special Class ◽  
Continuous Functions ◽  
Asymptotic Behavior Of Solutions ◽  
Free Term ◽  
Research Technique ◽  
Multidimensional Integral

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

scholarly journals New Explicit Bounds on Gamidov Type Integral Inequalities for Functions in Two Variables and Their Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Kelong Cheng ◽  
Chunxiang Guo
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Integral Inequalities ◽  
Fredholm Integral Equations ◽  
Uniqueness Of Solutions ◽  
Fredholm Type ◽  
Type Integral ◽  
Explicit Bounds ◽  
Linear And Nonlinear

Some linear and nonlinear Gamidov type integral inequalities in two variables are established, which can give the explicit bounds on the solutions to a class of Volterra-Fredholm integral equations. Some examples of application are presented to show boundedness and uniqueness of solutions of a Volterra-Fredholm type integral equation.

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

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 From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

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