Numerical Solution of Volterra-Fredholm Integral Equations with Hosoya Polynomials

In this paper, we present a numerical solution of nonlinear Volterra-Fredholm integral equations using Haar wavelet collocation method. Properties of Haar wavelet and its operational matrices are utilized to convert the integral equation into a system of algebraic equations, solving these equations using MATLAB to compute the Haar coefficients. The numerical results are compared with exact and existing method through error analysis, which shows the efficiency of the technique.

Download Full-text

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

Problems of applied mathematics and mathematic modeling ◽

10.15421/321906 ◽

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.

Download Full-text

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Journal of Scientific Research ◽

10.3329/jsr.v2i2.4483 ◽

2010 ◽

Vol 2 (2) ◽

pp. 264-272 ◽

Cited By ~ 7

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)

Download Full-text

A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

Applied Mathematics and Nonlinear Sciences ◽

10.21042/amns.2018.2.00035 ◽

2018 ◽

Vol 3 (2) ◽

pp. 447-458 ◽

Cited By ~ 3

Author(s):

S.C. Shiralashetti ◽

H. S. Ramane ◽

R.A. Mundewadi ◽

R.B. Jummannaver

Keyword(s):

Error Analysis ◽

Comparative Study ◽

Numerical Solution ◽

Integral Equations ◽

Haar Wavelet ◽

Fredholm Integral Equations ◽

Polynomial Method ◽

Wavelet Method ◽

Haar Wavelet Method ◽

Hosoya Polynomial

AbstractIn this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

Download Full-text

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691316500041 ◽

2016 ◽

Vol 14 (01) ◽

pp. 1650004 ◽

Cited By ~ 3

Author(s):

M. Tahami ◽

A. Askari Hemmat ◽

S. A. Yousefi

Keyword(s):

Integral Equation ◽

Tensor Product ◽

Numerical Solution ◽

Fredholm Integral Equation ◽

Fredholm Integral Equations ◽

Wavelet Basis ◽

Two Dimensional ◽

Legendre Wavelets ◽

Numerical Examples ◽

One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

Download Full-text

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

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488506003868 ◽

2006 ◽

Vol 14 (01) ◽

pp. 101-110 ◽

Cited By ~ 29

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.

Download Full-text

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

Modern Innovations, Systems and Technologies ◽

10.47813/2782-2818-2021-1-4-1-7 ◽

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

Download Full-text

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

Mathematical Problems in Engineering ◽

10.1155/2018/3194907 ◽

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.

Download Full-text

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

Journal of the Indian Mathematical Society ◽

10.18311/jims/2018/16260 ◽

2018 ◽

Vol 85 (1-2) ◽

pp. 53 ◽

Cited By ~ 4

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.

Download Full-text

Chebyshev collocation treatment of Volterra–Fredholm integral equation with error analysis

Arabian Journal of Mathematics ◽

10.1007/s40065-019-0243-y ◽

2019 ◽

Vol 9 (2) ◽

pp. 471-480 ◽

Cited By ~ 6

Author(s):

Y. H. Youssri ◽

R. M. Hafez

Keyword(s):

Integral Equation ◽

Error Analysis ◽

Numerical Solution ◽

Fredholm Integral Equation ◽

Chebyshev Polynomials ◽

Matrix Equation ◽

Chebyshev Collocation ◽

Chebyshev Nodes ◽

Potential Applicability ◽

The Given

Abstract This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.

Download Full-text