Volterra Integro-differential Equations Research Articles

Multi-dimensional Weyl almost periodic type functions and applications

In this paper, we analyze multi-dimensional Weyl almost periodic type functions in Lebesgue spaces with variable exponents. The introduced classes seem to be new and not considered elsewhere even in the constant coefficient case. We provide certain applications to the abstract Volterra integrodifferential equations in Banach spaces.

(ω,c)-almost periodic type functions and applications

In this paper, we introduce several various classes of (?, c)-almost periodic type functions and their Stepanov generalizations. We also consider the corresponding classes of (?, c)-almost periodic type functions depending on two variables and related composition principles. We provide several illustrative examples and applications to the abstract Volterra integro-differential equations in Banach spaces.

Numerical Approach for Delay Volterra Integro-Differential Equation

The delay integro-differential equation for the Volterra type has been solved by using the two-point multistep block (2PBM) method with constant step-size. The proposed block method of order three is formulated using Taylor expansion and will simultaneously approximate the numerical solution at two points. The 2PBM method is developed by combining the predictor and corrector formulae in the PECE mode. The predictor formulae are explicit, while the corrector formulae are implicit. The algorithm for the approximate solutions were constructed and analyzed using the 2PBM method with Newton-Cotes quadrature rules. This paper focused on constant and pantograph delay types, and the previous values are used to interpolate the delay solutions. Moreover, the studies also carried out on the stability analysis of the proposed method. Some numerical results are tested to validate the competency of the multistep block method with quadrature rule approach.

Solving Fuzzy Volterra Integro-Differential Equations By Using Fuzzy Kamal Transform

In this paper, the fuzzy Kamal transform is used to solve fuzzy Volterra integral-differential equations, which is based on Kamal transform. Kamal transform takes very little computation and time. Numerical examples are given to prove the effectiveness of Kamal transform in solving fuzzy Volterra integro-differential equations

New results on the qualitative analysis of solutions of VIDEs by the Lyapunov–Razumikhin technique

UDC 517.9 A new mathematical model described by a Volterra integro-differential equation (VIDE) with constant delay is examined. New agreeable conditions on the uniformly asymptotic stability, boundedness, and square integrability of solutions of the VIDE are obtained by using the Lyapunov–Razumikhin technique. The established conditions improve some former results and they are also nonlinear generalizations of these results. Moreover, they are weaker than some available results cited in the bibliography of this paper. Two examples are presented to demonstrate applications of these results and the introduced concepts. The use of the Lyapunov–Razumikhin technique leads to a significant difference and gives an advantage over the related methods used in the books and papers cited in the bibliography.

Approximate Solution of Singularly Perturbed Problems with Numerical Integration Method

Bu çalışmada, singüler pertürbe Volterra integro-diferansiyel denklemlerin yaklaşık çözümü için nümerik integrasyon yöntemi uygulanır. İlk olarak düzgün bir şebeke üzerinde sonlu fark metodu ile başlanır daha sonra integraller için trapez metodu kullanılır. Buradan elde edilen denklem sistemi Thomas algoritması ile çözülür. Önerilen yöntemin doğruluğunu ve ekonomikliğini ortaya koyan bir örnek sunulur.

Asymptotically ρ-almost periodic type functions in general metric

In this paper, we analyze various classes of multi-dimensional asymptotically ρ-almost periodic type functions in general metric. We clarify the main structural properties for the introduced classes of asymptotically ρ-almost periodic type functions, and provide some applications of our results to the abstract Volterra integro-differential equations.

A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

Output regulation for general heterodirectional linear hyperbolic PDEs coupled with nonlinear ODEs

This paper considers general heterodirectional linear hyperbolic PDEs with boundary actuation and collocated measurement, that are bidirectionally coupled with nonlinear ODEs at the unactuated boundary. An output feedback regulator is designed to have the control output track a reference in the presence of disturbances, with a nonlinear signal model generating the reference and the disturbances. This leads to new challenges for both the observer and controller design. The regulator design makes use of results from output regulation theory for nonlinear lumped-parameter and distributed-parameter systems. The derivation of the state feedback regulator requires solving a new type of regulator equations, which consist of a Cauchy problem for linear and semilinear hyperbolic PDEs. A key component of the novel observer design is a nonlinear retarded observer for a finite-dimensional subsystem. Both the controller and the observer design are systematic in nature and derived in several successive steps, that include backstepping transformations and, most notably, predictions of PDE and ODE states. The latter are shown to be the unique solution of general nonlinear Volterra integro-differential equations. Combining the observer with the state feedback regulator is proven to achieve output regulation for the closed-loop system. A numerical example illustrates and confirms the theoretical results.

The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis

The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis

Existence of Atangana-Baleanu fractional neutral Volterra integro-differential equations with non-instantaneous impulses

The main focus of this manuscript is centered around Atangana-Baleanu semilinear fractional integrodifferential equations accompanied by non-instantaneous impulses. The outcomes are demonstrated using the Darbo fixed point theorem along with its results when the measure of non-compactness collaborates. Eventually, a demonstration example is proposed.

Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

Fractional Linear Volterra Integro-Differential Equations in Banach Spaces

The paper presents the foundations of the theory of linear fractional Volterra integro-differential equations of convolution type in Banach spaces. It is established that the existence of a fractional resolvent operator for such equations is equivalent to the well-posedness of the formulation of the initial problem for them. Within the framework of this approach, a theorem of the Hille–Yosida type is proved.

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

Collocation Method for Solving Two-Dimensional Fractional Volterra Integro-Differential Equations

Collocation Method for Solving Two-Dimensional Fractional Volterra Integro-Differential Equations

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

Piecewise Fractional Jacobi Polynomial Approximations for Volterra Integro-Differential Equations with Weakly Singular Kernels

This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the existence, uniqueness and convergence properties of Galerkin solutions by exploiting the decaying rate of the coefficients of the transformed fractional Jacobi series. Finally, numerical experiments are carried out to illustrate the performance of the piecewise Galerkin solution.

The operational matrix of Chebyshev polynomials for solving pantograph-type Volterra integro-differential equations

In this work, the Chebyshev collocation scheme is extended for the Volterra integro-differential equations of pantograph type. First, we construct the operational matrices of pantograph and derivative based on Chebyshev polynomials. Also, the obtained operational matrices are utilized to approximate the derivatives of unknown functions. Furthermore, a detailed analysis of convergence is discussed in the weighted square norm. We conduct some numerical experiments to verify the high performance of the suggested numerical approach. The results show that the computational scheme is accurate.

Well-Posed Solvability of Volterra Integro-Differential Equations in Hilbert Spaces

Well-Posed Solvability of Volterra Integro-Differential Equations in Hilbert Spaces