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In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature. [ABSTRACT FROM AUTHOR]

This paper aims to establish the finite difference scheme on Bakhvalov-type mesh for the singularly perturbed pseudo-parabolic problems with time-delay. Using the energy estimates, the bounds of the solution are investigated. An error analysis is derived in the discrete energy norm and the almost second order convergence is obtained. The reliability of the suggested method is displayed by means of a numerical example. [ABSTRACT FROM AUTHOR]

Abstract.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. [ABSTRACT FROM AUTHOR]

In this paper, the stability of an equilibrium of a feedback nonlinear system with time delay and structural switching is studied. The critical case of a zero root of the characteristic equations of the linearised systems is treated by applying a Malkin-type theorem using a complete Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

This paper studies a class of fractional stochastic delay differential equations driven by a Wiener process. The sufficient conditions for the existence and uniqueness of the solution of fractional stochastic delay differential equations are obtained by means of the Picard iteration method. Moreover, the upper bound of the error using the Picard approximation is estimated. Numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies the numerical solution of fractional stochastic delay differential equations driven by Brownian motion. The proposed algorithm is based on linear B-spline interpolation. The convergence and the numerical performance of the method are analyzed. The technique is adopted for determining the statistical indicators of stochastic responses of fractional Langevin and Mackey-Glass models with stochastic excitations. [ABSTRACT FROM AUTHOR]

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε. [ABSTRACT FROM AUTHOR]

A class of singularly perturbed parabolic delay differential equation with discontinuous data is solved numerically. Numerical scheme based on the upwind finite difference method is presented on a specially generated mesh. The proposed numerical method has been analysed for consistency, stability, and convergence. It is found that the method proposed is unconditionally stable, and the convergence obtained is parameter uniform. Numerical illustrations are presented to demonstrate the effectiveness of the technique. [ABSTRACT FROM AUTHOR]

In this manuscript, a class of fractional delay differential equation is considered under multi-point boundary conditions. Two important aspects including existence theory and stability results are developed. For the concerned results prior estimate method and some results of nonlinear analysis are used. By giving a pertinent example the main results are justified. [ABSTRACT FROM AUTHOR]

We develop a delay differential equation model for the interactive wild and sterile mosquitoes. Different from the existing modelling studies, we assume that only those sexually active sterile mosquitoes play a role for the interactive dynamics. We consider the cases where the release amount is either constant or described by a given function of time. For the constant releases, we establish a threshold of releases to determine whether the wild mosquito suppression succeeds or fails. We study the existence and stability of the model equilibria. When the releases are described by given functions, the trivial equilibrium is no longer globally but locally uniformly asymptotically stable if the amount of releases is below the threshold whereas it is still globally uniformly asymptotically stable if the release amount is above the threshold. Numerical examples demonstrating the model dynamical features and brief discussions of our findings are also provided. [ABSTRACT FROM AUTHOR]

In this article, we propose a parameter uniform numerical method for singularly perturbed delay differential equations with turning point. Using interpolation, a new algorithm is developed to tackle the retarded term. A fitted operator finite difference scheme using Il'in Allen Southwell fitting factor is used for numerical discretization. Bounds on the analytical solution and its derivatives are obtained. The efficiency of the proposed numerical method is illustrated via applying the proposed method on some test problems. The solution is proved to be stable and ε-uniform error estimates are derived. It is shown that the delay argument has significant effect on the solution behavior. [ABSTRACT FROM AUTHOR]

In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method. [ABSTRACT FROM PUBLISHER]

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs. [ABSTRACT FROM AUTHOR]