New process to approach linear Fredholm integral equations defined on large interval

2019 ◽  
Vol 12 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Samir Lemita ◽  
Hamza Guebbai
Keyword(s):  
Algebraic System ◽  
Iterative Scheme ◽  
Jacobi Method ◽  
Fredholm Integral Equations ◽  
Bounded Operators ◽  
Large Interval ◽  
Numerical Tests ◽  
Diagonal Part ◽  
New Process ◽  
Numerical Process

To tackle a linear Fredholm integral equation on great interval, two numerical processes are involved: discretization and iterative scheme. The conventional numerical process is discretize first then use an iterative scheme as Jacobi’s method to approach the solutions of the huge algebraic system. In this paper, we propose an alternative numerical process, we apply an iterative scheme based on construction of a generalization of the iterative scheme for Jacobi method which is adapted to the system of linear bounded operators, then we use Nyström method to discretize only the diagonal part of the system. The convergence analysis of this new method is proved and numerical tests developed show its effectiveness.

New method for the numerical solution of the Fredholm linear integral equation on a large interval

2020 ◽  
pp. 387-400
Author(s):  
Samir Lemita ◽  
Hamza Guebbai ◽  
Ilyes Sedka ◽  
Mohamed Zine Aissaoui
Keyword(s):  
Integral Equation ◽  
Iterative Scheme ◽  
New Method ◽  
Linear Integral Equation ◽  
Bounded Operators ◽  
Large Interval ◽  
Diagonal Part ◽  
Product Integration Method ◽  
Numerical Process ◽  
Linear Integral

The traditional numerical process to tackle a linear Fredholm integral equation on a large interval is divided into two parts, the first is discretization, and the second is the use of the iterative scheme to approach the solutions of the huge algebraic system. In this paper we propose a new method based on constructing a generalization of the iterative scheme, which is adapted to the system of linear bounded operators. Then we don’t discretize the whole system, but only the diagonal part of the system. This system is built by transforming our integral equation. As discretization we consider the product integration method and the Gauss–Seidel iterative method as iterative scheme. We also prove the convergence of this new method. The numerical tests developed show its effectiveness.

scholarly journals Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

2020 ◽  
Vol 146 (4) ◽  
pp. 699-728
Author(s):  
Patricia Díaz de Alba ◽  
Luisa Fermo ◽  
Giuseppe Rodriguez
Keyword(s):  
Integral Equations ◽  
Numerical Approximation ◽  
Gauss Quadrature ◽  
Upper And Lower Bounds ◽  
Stability And Convergence ◽  
Quadrature Rules ◽  
Fredholm Integral Equations ◽  
Numerical Tests ◽  
Gauss Quadrature Formula ◽  
Gauss Quadrature Rules

AbstractThis paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.

A Regularizing Parameter for Some Fredholm Integral Equations

2010 ◽  
Vol 10 (2) ◽  
pp. 177-194
Author(s):  
L. Fermo
Keyword(s):  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Numerical Tests

AbstractThe regularizing parameter appearing in some Fredholm integral equations of the second kind is discussed. Theoretical estimates and the results of numerical tests confirming the theoretical expectations are given.

scholarly journals A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 83
Author(s):  
José M. Gutiérrez ◽  
Miguel Á. Hernández-Verón
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Integral Equations ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Iterative Scheme ◽  
Fredholm Integral Equations

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

scholarly journals Solving second kind linear Fredholm integral equations via Quarter-Sweep SOR iterative method

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Mohana Sundaram Muthuvalu ◽  
Jumat Sulaiman
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Integral Equations ◽  
Numerical Solutions ◽  
Fredholm Integral Equations ◽  
Quadrature Method ◽  
Numerical Tests ◽  
Successive Over Relaxation

In this paper, we consider the numerical solutions of linear Fredholm integral equations of the second kind. The Quarter-Sweep Successive Over-Relaxation (QSSOR) iterative method is applied to solve linear systems generated from discretization of the second kind linear Fredholm integral equations using quadrature method. In addition, the formulation and implementation of the proposed method to solve the problem are also presented. Numerical tests and comparisons with other existing methods are given to illustrate the effectiveness of the proposed method.

scholarly journals General viscosity implicit midpoint rule for nonexpansive mapping

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 225-237
Author(s):  
Shuja Rizvi
Keyword(s):  
Nonexpansive Mapping ◽  
Evolution Equations ◽  
Variational Inequality Problem ◽  
Nonlinear Evolution ◽  
Iterative Scheme ◽  
Nonlinear Evolution Equations ◽  
Strong Convergence Theorem ◽  
Fredholm Integral Equations ◽  
Implicit Midpoint Rule ◽  
Midpoint Rule

In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mapping in the framework of Hilbert space. Further, under the certain conditions imposed on the sequence of parameters, strong convergence theorem is proved by the sequence generated by the proposed iterative scheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore, we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evolution equations and give a numerical example to justify the main result. The results presented in this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.

scholarly journals Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1747
Author(s):  
José Manuel Gutiérrez ◽  
Miguel Ángel Hernández-Verón ◽  
Eulalia Martínez
Keyword(s):  
Integral Equations ◽  
Theoretical Study ◽  
Iterative Scheme ◽  
Semilocal Convergence ◽  
Iterative Solution ◽  
Fredholm Integral Equations ◽  
Existence Of A Solution ◽  
Separable Kernel ◽  
Starting Point ◽  
Domain Of Existence

This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.

scholarly journals A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2994
Author(s):  
Malik Zaka Ullah
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Iterative Scheme ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Numerical Tests ◽  
Maximal Solutions ◽  
Minimal And Maximal Solutions ◽  
Nonlinear Matrix ◽  
Convergence Of The Scheme

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

scholarly journals On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2078
Author(s):  
Ajmal Ali ◽  
Thabet Abdeljawad ◽  
Azhar Iqbal ◽  
Tayyaba Akram ◽  
Muhammad Abbas
Keyword(s):  
Dirichlet Boundary ◽  
Iterative Scheme ◽  
Relaxation Method ◽  
Approximate Solutions ◽  
Dirichlet Boundary Conditions ◽  
Group Method ◽  
Numerical Tests ◽  
The Matrix ◽  
Fractional Group ◽  
The Stability

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions.

scholarly journals An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1957
Author(s):  
José M. Gutiérrez ◽  
Miguel Á. Hernández-Verón
Keyword(s):  
Integral Equations ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Semilocal Convergence ◽  
Fredholm Integral Equations ◽  
Linear Integral ◽  
Ulm’S Method ◽  
Theoretical Results ◽  
Quadratic Order ◽  
Linear Integral Equations

In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels.

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