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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 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]

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 this paper, we first prove two existence and uniqueness results for fractional-order delay differential equation with respect to Chebyshev and Bielecki norms. Secondly, we prove the above equation is Ulam–Hyers–Mittag-Leffler stable on a compact interval. Finally, two examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical study of singularly perturbed boundary value problems for delay differential equations with a turning point. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighbourhood of the boundary layers. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments are carried out to illustrate, in practice, the result of convergence proved theoretically and demonstrate the effect of the delay argument and the coefficient of the delay term on the layer behaviour of the solution. [ABSTRACT FROM AUTHOR]

This paper considers the Runge-Kutta methods used on one-dimensional delay differential equations (DDEs), which undergo Hopf bifurcation from the origin. It not only shows the numerical approximations undergo Neimark-Sacker bifurcation, but also these Neimark-Sacker bifurcation points close the Hopf bifurcation point of the DDEs with the order of methods. Meanwhile, compared with another interpolation method, we conclude that the given interpolation method is better in the convergence of Neimark-Sacker bifurcation points. Finally, by using 2-stage Gauss method to delay Logistic equation, the theoretical analysis is supported. [ABSTRACT FROM AUTHOR]

In this paper a class of delay differential equations with a perturbation parameter ε is examined. A hybrid finite difference scheme on an appropriate piecewise uniform mesh of Shishkin-type is derived. We show that the scheme is almost second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results. [ABSTRACT FROM AUTHOR]

The paper discusses the notion of a delay dynamic equation on time scales and describes some asymptotic properties of its solutions. The application of the derived results to continuous and discrete time scales presents new qualitative results for delay differential and difference equations. In particular, our approach faciliates the joint investigation of stability properties of the exact equations and their numerical discretizations. [ABSTRACT FROM AUTHOR]

The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one-dimensional parametrized stochastic differential equation with point delay; the basic properties of which we check. With a view to capturing the effective dynamics and, in particular, the resonance-like behavior of the reference model, we construct a simplified or reduced model, the two-state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance. Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two-state model and the reference model. The resonance characteristics developed for the reduced model can thus be applied to the original model. [ABSTRACT FROM AUTHOR]

A numerical method based on the Adomian decomposition method which has been developed by Adomian [Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method . Kluwer Academic Publishers, Boston, MA] is introduced in this paper for the approximate solution of delay differential equations. The algorithm is illustrated by studying an initial value problem. The results obtained are presented and show that only a few terms are required to obtain an approximate solution which is found to be accurate and efficient. [ABSTRACT FROM AUTHOR]

Marchuk's model of a general immune reaction is presented in the paper. The results of investigation of the model are summarized. The qualitative behaviour of solutions to the model and its simplification is described. Many illustrations of recovery process, oscillations or lethal outcomes of a disease are shown. The model with time-dependent immune reactivity is also considered. Periodic dynamics caused by different reasons are compared. In the last section adaptation of the model to tumour-immune interactions is proposed. The role of interleukins in immune processes is taken into account to adapt Marchuk's model to tumour growth dynamics. Possible immunotherapy is also considered in the proposed model. Some preliminary analysis of the model is presented. [ABSTRACT FROM AUTHOR]