Delay Integro-differential Equations Research Articles

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

A Novel Operational Matrix Method for Solving the Fractional Delay Integro-Differential Equations with a Weakly Singular Kernel

Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative

Abstract In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Numerical Solution of Generalized Delay Integro-Differential Equations via Galerkin-Vieta-Lucas Polynomials

In this article, the Galerkin-Vieta-Lucas scheme is presented to find an approximate solution to the generalised delay integro-differential equation using the Vieta-Lucas polynomial as an approximation. The Galerkin approach transforms the delay integro-differential equation into a set of n × (n + m) algebraic equations, which, together with the attached conditions, give (n + m) × (n + m) equations. The effectiveness and accuracy of the proposed technique were tested on some existing examples in the literature, and obviously, the results obtained justify the accuracy of the proposed scheme.

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.

On the qualitative behaviors of stochastic delay integro-differential equations of second order

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE).To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature.

On a second-order delay integro-differential equation in Banach spaces

On a second-order delay integro-differential equation in Banach spaces

Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

Analysis of a class of fractional delay integro-differential equations with Riesz-Caputo derivative

An Actuated Computational Method for Treating Parabolic Partial Delay Integro-Differential Equations Constrained by Infinite Boundary

An Actuated Computational Method for Treating Parabolic Partial Delay Integro-Differential Equations Constrained by Infinite Boundary

Results for Retarded Nonlinear Integral Inequalities with Mixed Powers and Their Applications to Delay Integrodifferential Equations

Results for Retarded Nonlinear Integral Inequalities with Mixed Powers and Their Applications to Delay Integrodifferential Equations

The stability of nonlinear delay integro-differential equations in the sense of Hyers-Ulam

Abstract In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

Stability inequalities and numerical solution for neutral Volterra delay integro-differential equation

The aim of the present paper is to introduce a new numerical method for solving Volterra integro-differential equation involving delay and neutral types. Using the numerical quadrature formula we form a finite difference scheme on a uniform mesh. The presented numerical method obtains a second-order convergence in discrete maximum norm. Furthermore, we illustrate the efficacy of the proposed method by constructing examples.

Stability Properties for the Delay Integro-Differential Equation

In this paper stability inequalities for the linear nonhomogeneous Volterra delay integro-differential equation (VDIDE) is being established. The particular problems are encountered to show the applicability of the method and to confirm the predicted theoretical analysis.

Asymptotic stability of nonlinear neutral delay integro-differential equations

Abstract In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

On a common fixed point theorem in vector-valued $ b $-metric spaces: Its consequences and application

<abstract><p>We introduce a Ćirić type contraction principle in a vector-valued $ b $-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $ W, E:G\rightarrow G $ ($ G $ is a non-empty set), for which these mappings admit a unique common fixed point in $ G $ subject to a nonlinear operator $ {\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m} $. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.</p></abstract>

Global existence and stability results for partial delay integro-differential equations with random impulses

We investigate the global existence, continuous dependence and exponential stability for mild solutions of a class of delay partial integro-differential equation with random impulsive moments. The results are obtained by using the Leray-Schauder alternative fixed point theory and Banach contraction principle.

Approximation of mild solutions of delay integro-differential equations on Banach spaces

Approximation of mild solutions of delay integro-differential equations on 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.

Solution estimates to Caputo proportional fractional derivative delay integro-differential equations

Solution estimates to Caputo proportional fractional derivative delay 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.