Volterra-Fredholm Integro-differential Equations Research Articles

A fully discrete scheme on piecewise-equidistant mesh for singularly perturbed delay integro-differential equations

. This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems.

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

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.

Numerical Treatment of Volterra-Fredholm Integro- Differential Equations of Fractional Order and its Convergence Analysis

This work deals with semi-analytical and numerical methods to solve a class of fractional order Volterra-Fredholm integro-differential equations. First, a semi-analytical method is proposed using the Chebyshev and Bernstein polynomials in the Adomian decomposition method. The uniqueness of the solution and convergence of the method are proved. Further, we solve the model using a numerical scheme comparing the L1 scheme for the fractional order derivative in combination with appropriate quadrature rules for the integral parts. Numerical experiments are done by the proposed methods to show their efficiency through a few tabular data and plots. Some comparisons with the existing results show that the proposed methods are highly productive and reliable.

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

This article deals with the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.

On a New Class of Impulsive η-Hilfer Fractional Volterra-Fredholm Integro-Differential Equations

This work addresses the idea of the uniqueness and existence results for a class of boundary value problems (BVPs) for implicit Volterra-Fredholm integro-differential equations (V-FIDEs) with fractional η-Hilfer nonlinear equations and multi-point fractional boundary non-instantaneous conditions. The conclusions are confirmed by the fixed point of Krasnoselskii's theorem and the Banach contraction principle. Finally, a concrete example is given to illustrate our main conclusions.

Some New Results on Hadamard Neutral Fractional Nonlinear Volterra-Fredholm Integro-Differential Equations

Some New Results on Hadamard Neutral Fractional Nonlinear Volterra-Fredholm Integro-Differential Equations

Barycentric Interpolation Collocation Method for Solving Fractional Linear Fredholm-Volterra Integro-Differential Equation

In this article, barycentric interpolation collocation method (BICM) is presented to solve the fractional linear Fredholm-Volterra integro-differential equation (FVIDE). Firstly, the fractional order term of equation is transformed into the Riemann integral with Caputo definition, and this integral term is approximated by the Gauss quadrature formula. Secondly, the barycentric interpolation basis function is used to approximate the unknown function, and the matrix equation of BICM is obtained. Finally, several numerical examples are given to solve one-dimensional differential equation.

On existence and numerical solution of higher order nonlinear integro-differential equations involving variable coefficients

We investigate the approximate solution to a class of fourth-order Volterra–Fredholm integro differential equations (VFIDEs). Additionally, we are able to get some adequate results for the existence of a solution with the use of nonlinear analysis techniques. The basis for the required numerical computation is provided by the Haar wavelet collocations (HWCs) technique, which converts the problems into a system of algebraic equations. Then, in order to obtain the required numerical solution, we solve the resulting system of algebraic equations using Gauss elimination and Broyden’s techniques. This method can be used to show convergence, and the predicted rate of convergence is two. Along with relevant examples, we have included a graphical presentation to demonstrate our suggested approach.

Solving high order of non-linear Volterra-fredholm Integro-differential equation by using bou-baker Polynomials approximation Method

In this paper, the Bou-baker polynomial method is used to evaluate an approximate so1ution initia1 value problem of high-order of non1inear Vo1terra-Fredholm Integro-Differential equation of the second kind. Three different examp1es and their graphics are displayed.

Sixth-Kind Chebyshev Collocation Treatment of Fractional Fredholm-Volterra Integro-Differential Equations

Sixth-Kind Chebyshev Collocation Treatment of Fractional Fredholm-Volterra Integro-Differential Equations

EXISTENCE OF SOLUTIONS FOR NONLINEAR VOLTERRA FREDHOLM INTEGRODIFFERENTIAL EQUATION OF HIGHER ORDER VIA S-ITERATION METHOD

In this paper, we study the existence and other properties of the solution of the nonlinear Volterra Fredholm integrodifferential equation of higher order. The tool employed in the analysis is based on the application of the S-iteration method. Various properties such as dependence on initial data, closeness of solutions and dependence on parameters and functions involved therein are obtained using the S-iteration method. Examples are provided in support of findings.

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Existence, Uniqueness and Stability Results for Nonlinear Neutral Fractional Volterra-Fredholm Integro-Differential Equations

Existence, Uniqueness and Stability Results for Nonlinear Neutral Fractional Volterra-Fredholm Integro-Differential Equations

Existence and controllability of fractional semilinear mixed Volterra-Fredholm integro differential equation

In this paper, the existence of mild solution and controllability of fractional semilinear mixed Volterra–Fredholm integro differential equations of order 1<γ<2 with nonlocal conditions are investigated. First we discuss the existence of mild solution and then controllability of the proposed system by using the concept of Banach fixed point theorem and semigroup theory. For the authenticity and applicability of the presented theory some examples are also provided.

An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation

The scope of this study is to establish an effective approximation method for linear first order singularly perturbed Volterra-Fredholm integro-differential equations. The finite difference scheme is constructed on Shishkin mesh by using appropriate interpolating quadrature rules and exponential basis function. The recommended method is second order convergent in the discrete maximum norm. Numerical results illustrating the preciseness and computationally attractiveness of the proposed method are presented.

Using 2D and 1D block-pulse functions simultaneously for solving the Barbashin integro-differential equations

The goal of this work is to look at how to solve Barbashin equations using a mix of Volterra–Fredholm integro-differential equations and Volterra equations. For the first time, the 1D and 2D block-pulse functions are employed in a hybrid technique to solve this two-dimensional integral problem concurrently. We construct a new matrix called the converter operating matrix for this purpose. The Barbashin equation is reduced to a linear system that can be solved using well-known methods employing direct computing. The proposed method's convergence theorem is demonstrated under appropriate conditions. In addition, some numerical results are presented in the paper, which supports the strong theoretical properties of our approach.

Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.