Volterra Integro-differential Equations Research Articles

On the second-order neutral Volterra integro-differential equation and its numerical solution

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme.

The Way to Construct Innovative Methods for Solving Initial-Value Problem of the Volterra Integro-Differential Equation

Mathematical models for many problems in the natural sciences are often simplified to solving initial-value problems (IVPs) for the Volterra integro-differential equations (VIDE). Numerical methods of a multistep type are typically used to solve these problems. It is known that in some cases, the multi-step method (MSM) is applied to solving the IVPs of both ordinary differential equations (ODEs) and VIDE encountered in solving some problems in mathematical biology. Here, to solve such problems by combining different methods, some modifications of established methods were developed, and it was demonstrated that these methods outperform the existing ones. As is known, one of the main issues in solving the aforementioned problems is determining the reliability of calculating values using the known mathematicalstatistical models (MSMs). In this regard, some experts utilize the predictor-corrector method. Having highlighted the disadvantages of this method, the proposal is to develop an innovative approach and assess the errors that may arise when applying this method to solve various problems. Here, the IVPs for the VIDE of the first order are primarily investigated. To illustrate the benefits of the innovative methods proposed here, we discuss the use of simple numerical methods to solve some common examples.

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

Numerical treatment of linear Volterra integro differential equations using variational iteration algorithm with collocation

To numerically solve the linear Volterra integro-differential equation, this study employs fourth-kind Chebyshev polynomials and the variational iteration algorithm with collocation, which is a combination of the variational iteration strategy and collocation technique. By applying fourth-kind Chebyshev polynomials to the variational iteration method with collocation for solving Volterra integro-differential equations, mathematical problems with a broad range of multidisciplinary applications are addressed, and numerical techniques that produce more accurate and efficient results are developed. The recommended method is then used, and the fourth-kind Chebyshev polynomials generated for the given integro-differential equation serve as the trial functions for the approximation. As a result, the suggested method's significance probably goes beyond a particular equation or application, as it contributes to the larger field of mathematical modeling and numerical analysis. Research methods employing a variational iteration algorithm with collocation aim to provide general techniques that can be applied to a wide range of problems. Additionally, numerical examples were provided to highlight the applicability and dependability of the proposed methodology. The mathematical computations were carried out using the Maple 18 software.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Numerical Solution on Neutral Delay Volterra Integro-Differential Equation

Numerical Solution on Neutral Delay Volterra Integro-Differential Equation

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Well-Posed Solvability of Volterra Integro-Differential Equations Arising in Viscoelasticity Theory

Well-Posed Solvability of Volterra Integro-Differential Equations Arising in Viscoelasticity Theory

METRICAL BOCHNER CRITERION AND METRICAL STEPANOV ALMOST PERIODICITY

We analyze the metrical Bochner criterion and a new class of multi-dimensional metrically Stepanov almost periodic type functions. We clarify the main structural properties for the introduced classes of functions, including the Bochner criterion, and provide certain applications to Doss-p-almost periodic functions. We also study the extensions of almost periodic sequences and briefly explain how we can apply the established theoretical results to the abstract Volterra integro-differential equations.

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations

Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

Two-step Runge–Kutta methods for Volterra integro-differential equations

In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses.

Generalized Jacobi Spectral Galerkin Method for Fractional-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Generalized Jacobi Spectral Galerkin Method for Fractional-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

NUMERICAL METHOD FOR SOLUTION OF FOURTH-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY GREEN’S FUNCTION

In this paper, we generalize Picard-Green’s Embedded method for solving fourth-order Volterra integro-differential equations. We prove the existence and uniqueness theorems. Moreover, we illustrate some numerical examples to present the better approximation with a minimum error. We use MATLAB for numerical solutions.