Nonlinear Volterra-Fredholm Integro-differential Equations Research Articles

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.

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

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.

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.

Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions

<abstract><p>This paper studies the solvability of boundary value problems for a nonlinear integro-differential equation. Converting the problem to an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE) is driven by using a suitable transformation. To investigate the existence and uniqueness of continuous solutions for the NVFIE under certain given conditions, the Krasnoselskii fixed point theorem and Banach contraction principle have been used. Finally, we numerically solve the NVFIE and study the rate of convergence using methods based on applying the modified Adomian decomposition method, and Liao's homotopy analysis method. As applications, some examples are provided to support our work.</p></abstract>

Analytical and numerical approach for a nonlinear Volterra-Fredholm integro-differential equation

An approach for Volterra- Fredholm integro-differential equations using appropriate fixed point theorems of existence, uniqueness is presented. The approximation of the solution is performed using Nystrom method in conjunction with successive approximations algorithm. Finally, we give a numerical example, in order to verify the effectiveness of the proposed method with respect to the analytical study.

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

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

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

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative

<abstract><p>The point of this work was to analyze and investigate the sufficient conditions of the existence and uniqueness of solutions for the nonlinear fuzzy fractional Volterra Fredholm integro-differential equation in the frame of the Atangana-Baleanu-Caputo fractional derivative methodology. To begin with, we give the parametric interval form of the Atangana-Baleanu-Caputo fractional derivative on fuzzy set-valued functions. Then, by employing Schauder's and Banach's fixed point procedures, we examine the existence and uniqueness of solutions for fuzzy fractional Volterra Fredholm integro-differential equation with the Atangana-Baleanu-Caputo fractional operator. It turns out that the last interval model is a combined arrangement of nonlinear equations. In addition, we consider results by applying the Adams Bashforth fractional technique and present two examples that have been numerically solved using graphs.</p></abstract>

On Fractional Integro-Differential Equation with Nonlinear Time Varying Delay

In this manuscript, we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay. Also, we demonstrate the stability analysis for these equations. Our paper provides a convergence of semi-analytical approximate method for these equations. It would be desirable to point out approximate results.

A NEW NUMERICAL METHOD TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.

Galerkin Method for Nonlinear Volterra-Fredholm Integro-Differential Equations Based on Chebyshev Polynomials

We aim in this paper to develop a new algorithm for approximating the analytic solution for the integrodifferential equations using the Galerkin method. The bases of the solution obtained by the proposed algorithm are Chebyshev polynomials. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton's method is used to solve the resulting nonlinear algebraic system. Examples are introduced to prove the effectiveness of this algorithm in comparison with some other methods.

Uniqueness and Stability Results for Caputo Fractional Volterra-Fredholm Integro-Differential Equations

In this paper, we established some new results concerning the uniqueness and Ulam’s stability results of the solutions of iterative nonlinear Volterra-Fredholm integro-differential equations subject to the boundary conditions. The fractional derivatives are considered in the Caputo sense. These new results are obtained by applying the Gronwall–Bellman’s inequality and the Banach contraction fixed point theorem. An illustrative example is included to demonstrate the efficiency and reliability of our results

Analysis of a nonlinear Volterra-Fredholm integro-differential equation

The objective is to study the existence and uniqueness questions for a nonlinear Volterra-Fredholm integro-differential equation, then, develop a numerical approach using the Nyström method coupled with successive approximations algorithm. A numerical example is presented to examine the efficiency of the proposed method according to the existence and uniqueness conditions.

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations

Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations

On a Nonlinear Volterra-Fredholm Integrodifferential Equation on Time Scales

The main aim in this work is to obtain an integral inequality with a clear estimate on time scales. The obtained inequality is used as a tool to investigate some basic qualitative properties of solutions to certain nonlinear Volterra-Fredholm integrodifferential equations on time scales.

Stability of nonlinear Volterra-Fredholm integro differential equation: A fixed point approach

The objective of the present work is to analyze stability in the sense of Hyers-Ulam and Hyers-Ulam-Rassias for nonlinear Volterra Fredholm integro-differential equation by using fixed point approach.

An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations

We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique.