scholarly journals Solution of system of the Fredholm integral equation of the first kind

2021 ◽  
Vol 1 (4) ◽  
pp. 1-7
Author(s):  
Vladimir Uskov
Keyword(s):  
Integral Equation ◽  
Unique Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Production Costs ◽  
Fredholm Integral Equations ◽  
System Of Equations ◽  
Blurred Image ◽  
Basic Functions ◽  
Image Production

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered

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) 

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 A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Meilan Sun ◽  
Chuanqing Gu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Recursive Algorithm ◽  
High Order ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Initial Value ◽  
Numerical Examples ◽  
Type Approximation

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation

2018 ◽  
Vol 85 (1-2) ◽  
pp. 53 ◽  
Author(s):  
Ahmed A. Hamoud ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution ◽  
Fuzzy Solution

In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

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)

scholarly journals Comparative analysis of algorithms for solving the second-order Fredholm integral equation in Matlab environment

2019 ◽  
Author(s):  
L. L. Hart ◽  
M. O. Vasenyn ◽  
N. V. Baleyko
Keyword(s):  
Integral Equation ◽  
Comparative Analysis ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Analysis Of Algorithms ◽  
Qualitative Comparative Analysis ◽  
Fredholm Integral Equations ◽  
Approximate Methods ◽  
Aircraft Wings ◽  
Matlab Programming

The most common approximate methods for solving the linear Fredholm integral equation of the second kind are investigated, corresponding computational schemes are developed, and the order of their accuracy is estimated. For experiments, a software implementation of the selected methods was executed in the Matlab programming language. A qualitative comparative analysis of the results of the implemented algorithms was carried out on the example of solving specific problems. The problems of modeling complex physical processes are one of the most advanced and important ones throughout human history and today. One of the tools that helps to create a model of a process or phenomenon is integral equations. It is a very large class of problems and equations, consisting of many varieties. One of the types of equations of this class is the Fredholm integral equations of the second kind, because these equations help to solve problems such as the analysis of dynamic machines and mechanisms in mechanics, the problem of self-oscillations of aircraft wings in aerodynamics, the problem of forced vibrations of a string, the problem of determining the critical criticality shaft rotation and a huge range of tasks in the fields of electrical engineering, physics, auto-regulation, astronomy, acoustics and more. However often these processes are quite complex, and it is very difficult to solve the integral equation explicitly. Therefore, it is advisable to make a comparative analysis of approximate methods for solving Fredholm second kind equations and to conclude in which case one or the other method produces the best results. The results of the studies can be applied to the modeling of physical oscillation or regulation processes that require the solution of a linear Fredholm equation of the second kind with a complex kernel and a free term, which makes it impossible to find the exact solution of the equation. 

scholarly journals Bezier Curves for Solving Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
F. Ghomanjani ◽  
M. H. Farahi ◽  
A. Kılıçman
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Direct Algorithm ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

The Bezier curves are presented to estimate the solution of the linear Fredholm integral equation of the second kind. A direct algorithm for solving this problem is given. We have chosen the Bezier curves as piecewise polynomials of degreenand determine Bezier curves on [0, 1] byn+1control points. Numerical examples illustrate that the algorithm is applicable and very easy to use.

scholarly journals Numerical Solution of Volterra-Fredholm Integral Equations with Hosoya Polynomials

2021 ◽  
Author(s):  
Ercan Çelik ◽  
Merve Geçmen
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Programming Language ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Matlab Programming

In this study, Volterra-Fredholm integral equation is solved by Hosoya Polynomials. The solutions obtained with these methods were compared on the figure and table. And error analysis was done. Matlab programming language has been used to obtain conclusitions, tables and error analysis within a certain algorithm.

scholarly journals A Study of Some Iterative Methods for Solving Fuzzy Volterra-Fredholm Integral Equations

2018 ◽  
Vol 11 (3) ◽  
pp. 1228 ◽  
Author(s):  
Ahmed A. Hamoud ◽  
Ali Dhurgham Azeez ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fredholm Integral Equations ◽  
Analysis Method ◽  
Adomian Decomposition ◽  
Homotopy Analysis ◽  
Fuzzy Solution

<div>This paper mainly focuses on the recent advances in the some approximated methods for solving fuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iteration method and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to a system of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find the approximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzy Volterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6 according is used. Also, some numerical examples are included to demonstrate the validity and applicability</div><div>of the proposed techniques.</div>This paper mainly focuses on the recent advances in the some approximated methods for solvingfuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iterationmethod and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to asystem of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find theapproximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzyVolterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6according is used. Also, some numerical examples are included to demonstrate the validity and applicabilityof the proposed techniques.

scholarly journals Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Iqra Shamas ◽  
Saif Ur Rehman ◽  
Hassen Aydi ◽  
Tayyab Mahmood ◽  
Eskandar Ameer
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Fredholm Integral Equations ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Contractive Type

This paper aims at proving some unique fixed-point results for different contractive-type self-mappings in fuzzy metric spaces by using the “triangular property of the fuzzy metric”. Some illustrative examples are presented to support our results. Moreover, we present an application by resolving a particular case of a Fredholm integral equation of the second kind.

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