Fredholm Integro-differential Equations Research Articles

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

Boundary value problem B1x = f with an abstract linear operator B1, corresponding to an Fredholm integro-differential equation with ordinary or partial differential operator is researched. An exact solution to B1x = f in the case when a bijective operator B1 has a factorization of the form B1 =BB0 where B and B0 are two linear more simple than B1 second degree abstract operators, received. Conditions for factorization of the operator B1 and a criterion for bijectivity of B1 are found.

Numerical scheme for singularly perturbed Fredholm integro-differential equations with non-local boundary conditions

Numerical scheme for singularly perturbed Fredholm integro-differential equations with non-local boundary conditions

A numerical solution of singularly perturbed Fredholm integro-differential equation with discontinuous source term

In this paper, we investigate a singularly perturbed Fredholm integro-differential problem with a discontinuous source term, leading to the formation of interior layers in the solution at the point of discontinuity. We apply the exponentially fitted mesh method to solve the problem. Our analysis demonstrates that the method exhibits almost first-order convergence in the maximum norm, independent of the diffusion parameter. Numerical experiments are conducted to verify the validity of our theoretical results and they support the accuracy of our estimates.

On solvability of some systems of Fredholm integro-differential equations with mixed diffusion in a square

On solvability of some systems of Fredholm integro-differential equations with mixed diffusion in a square

Integrating Bernstein and Improved Block-Pulse Functions for Solving Linear Fredholm Integro-Differential Equations

Integrating Bernstein and Improved Block-Pulse Functions for Solving Linear Fredholm Integro-Differential Equations

A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations

This study deals with two numerical schemes for solving a class of singularly perturbed nonlinear Fredholm integro-differential equations. The nonlinear terms are linearized using the quasi-linearization technique. On the layer adapted Shishkin mesh, the numerical solution is initially calculated using the finite difference scheme for the differential part and quadrature rule for the integral part. The method proves to be first-order convergent in the discrete maximum norm. Then, using a post-processing technique we significantly enhance the accuracy from first order to second order. Further, a hybrid scheme on the nonuniform mesh is also constructed and analyzed whose solution converges uniformly, independent of the perturbation parameter and directly gives second order accuracy. Parameter uniform error estimates are demonstrated and the theoretical results are validated through some numerical tests.

A fitted approximate method for solving singularly perturbed Volterra–Fredholm integro-differential equations with an integral boundary condition

UDC 517.9 We consider a novel numerical approach for solving boundary-value problems for the second-order Volterra–Fredholm integro-differential equation with layer behavior and an integral boundary condition. A finite-difference scheme is proposed on suitable Shishkin-type mesh to obtain the approximate solution of the presented problem. It is proven that the method is first-order convergent in the discrete maximum norm. Two numerical examples are included to show the efficiency of the method.

Boundary-Value Problems with Control for Fredholm Integrodifferential Equations with Degenerate Kernel in Banach Spaces

Boundary-Value Problems with Control for Fredholm Integrodifferential Equations with Degenerate Kernel in Banach Spaces

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term

A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

ObjectivesIn this paper, a uniformly convergent numerical scheme is proposed for solving a singularly perturbed Fredholm integro-differential equation with an integral initial condition. The equation involves a left boundary layer which makes it difficult to solve it using the standard numerical methods. A fitted operator finite difference method is used to approximate the differential part of the equation and the composite Simpson 13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{3}$$\\end{document} rule is used for the integral parts of the equation and the initial condition.ResultThe stability bound and error estimation of the approximated solution are performed, to show the uniform convergence of the scheme with order one in the maximum norm. Numerical test examples are provided to calculate the maximum absolute errors, thrgence, and the uniform error for a couple of examples to support theoretical analysis.

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

<abstract><p>The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.</p></abstract>

Solvability of some Fredholm integro-differential equations with mixed diffusion in a square

Solvability of some Fredholm integro-differential equations with mixed diffusion in a square

A stable numerical method for singularly perturbed Fredholm integro differential equation using exponentially fitted difference method

This paper implemented a stable numerical scheme for solving singularly perturbed linear second-order Fredholm integro-differential equation. A parameter-uniform numerical method was constructed using an exponentially fitted finite difference method to approximate the differential part and the composite Simpson 1/3 rule for the integral part of the equation. The scheme’s stability and convergence analysis has been carried out. The maximum absolute errors and the rate of convergence are tabulated for different values of the perturbation parameter ɛ and mesh sizes using different numerical test examples.

An equidistributed grid-based second-order scheme for a singularly perturbed Fredholm integro-differential equation with an interior layer

In this study, we introduce and evaluate the effectiveness of the central finite difference coupled with a composite trapezoidal rule on adapted meshes in solving a second-order singularly perturbed Fredholm integro-differential equation with a discontinuous source term. A grid equidistribution technique was employed to address the challenge posed by the perturbation parameter in classical numerical methods. Our approach demonstrated superior precision compared to existing schemes in the literature and achieved second-order uniform convergence.

Rational approximation for solving Fredholm integro-differential equations by new algorithm

Abstract In this article, we used a novel semi-analytical approach, named the optimal auxiliary function method (OAFM), to solve integro-differential equations (IDEs). The OAFM includes an auxiliary function and convergence control parameters, which expedite the convergence of the method. To apply the proposed method, some assumptions regarding small or large parameters in the problem are necessary. We present numerical outcomes acquired via the OAFM alongside those obtained from other numerical techniques in tables. Furthermore, we demonstrate the efficacy and ease of implementing the proposed method for various IDEs using 2D graphs.

Superconvergence of Legendre spectral projection methods for [formula omitted]th order integro-differential equations with weakly singular kernels

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.

New General Solution to a Quasilinear Fredholm Integro-Differential Equation and Its Application

New General Solution to a Quasilinear Fredholm Integro-Differential Equation and Its Application

Nonlocal Direct and Inverse Problems for a Second Order Nonhomogeneous Fredholm Integro-Differential Equation with Two Redefinition Data

Nonlocal Direct and Inverse Problems for a Second Order Nonhomogeneous Fredholm Integro-Differential Equation with Two Redefinition Data

NUMERICAL ACCURACY OF FREDHOLM LINEAR INTEGRO-DIFFERENTIAL EQUATIONS BY USING ADOMIAN DECOMPOSITION METHOD, MODIFIED ADOMIAN DECOMPOSITION METHOD AND VARIATIONAL ITERATION METHOD

In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4.