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Stochastic delay-differential equations (SDDEs) are SDEs with dependence on past times. Applications of SDDEs are found in financial modelling, engineering, and various sciences. At present, existing numerical methods that are used for SDDEs are stochastic Taylor approximations. We propose Magnus-type modifications to these schemes in the case of finitely many delays, for improved accuracy with no additional computation cost. These modified schemes are applied step-by-step between delay times. We present error graphs, showing improvements for strong-order 1/2 and 1 schemes, when combined with Magnus methods. We also provide discussion about further applications to equations more general than those considered in these pages. [ABSTRACT FROM AUTHOR]

An attempt to study the stability of first order S-ROCK methods applied to a stochastic differential equation with a delay in the deterministic part. The previous results for stochastic differential equations and delay differential equations are possible to be generalized since the splitting technique used in S-ROCKs for the stochastic part. Thus mean square stability notion is combined with P-stability for delay equations. Some choice of "optimal" damping coefficient of S-ROCKs is done. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior and oscillation criteria of neutral differential equations of Emden-Fowler type of third order are studied. Where some conditions were formulated to ensure oscillation for all solutions of these equations. The obtained conditions can be generalized to higher order delay differential equations of Emden-Fowler type. The obtained results showed that the Emden-Fowler type in the neutral differential equation plays a major role in the presence or absence of the oscillation property, unlike other types of neutral differential equations. Some examples are presented to illustrate and apply the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we used the fractional collocation method based on the B-spline basis to derive the numerical solutions for a special form of fractional delay differential equations (DFDEs). The fractional derivative used is defined in the sense of Caputo. So, we can represent the DFDE under consideration into a matrix form that can be solved using matrix operations and tools from linear algebra. As a result, we get algebraic equations with unknown coefficients that can be solved efficiently using a computer algorithm. To illustrate the validity and efficiency of the method, exact and approximate solutions are compared, and absolute errors are found using an example. The numerical results, which are backed up by simulation, reveal that the absolute error is very small and that the approach is extremely efficient. [ABSTRACT FROM AUTHOR]

In the present work is considered the Cauchy (initial) problem (IP) for a linear delayed system with derivatives in Caputo's sense of incommensurate order, distributed delays and locally Lebesgue integrable initial functions. For this IP is studied the important problem of existence a formula of variation of constants. The obtained results extend the corresponding ones in the particular cases of fractional systems with constant and variable delays. The proposed conditions are almost the same as the conditions which guarantee the same result in the case of linear differential equations with distributed delays with integer order of differentiation. [ABSTRACT FROM AUTHOR]

In this research, some properties of oscillation and nonoscillation of a system of second order half linear delay differential equations were studied, as some conditions were obtained that ensure the oscillation of all solutions of the system, or the convergence of nonoscillating ones to zero. Some illustrative examples of the obtained results are given. [ABSTRACT FROM AUTHOR]

The first order non-linear neutral differential equations with several non-monotonic delays and several variable coefficients with forcing terms were discussed, also the oscillation behavior of all solutions was investigated where some conditions that guarantee the oscillation for each solution were obtained, illustrative examples related to the obtained results were given. [ABSTRACT FROM AUTHOR]

To illustrate the method's validity and effectiveness, this article offers a technique for finding approximation solution to fuzzy delay differential equations (FDDE) using the Implicit Runge – Kutta Method the Seventh Order (7th). To ensure the method's validity and efficiency, the estimated answer is compared to the precise solution. A numerical example is given, and the convergence of the solution is addressed in conclusion. [ABSTRACT FROM AUTHOR]

This paper deals with fractional delay differential equations (FDDE) of single order with fractional Caputo derivatives using the 1-point fractional Adams method. The proposed method is a predictor-corrector approach that can be constructed from the Adams-Bashforth method and Adams-Moulton method to achieve both simplicity and accuracy. The implementation of the fractional Caputo derivatives using the Euler-Gamma function and the approximation of delay term using the Newton divided difference interpolation are discussed. Some of the numerical examples are presented to show the proposed method is reliable and efficient [ABSTRACT FROM AUTHOR]

An efficient semi-analytical approach based on the method of is presented in this work numerical a procedure for solve a Delay Partial Differential Equations(DPDE) with a several constant. A status variable's small delay and retarded. Using High Performance Training Algorithms such as Levenberg-Marquardt with The radial basis function is used to activate every unit's sigmoid activation in a multilayer with one hidden layer with 7 hidden units (neurons) and one linear output unit. Finally, in illustrative instances The outcomes of numerical experiments are checked to the exact answer to ensure that the algorithm is accurate and efficient proposed methodology. [ABSTRACT FROM AUTHOR]

In the paper, we present a numerical method for solving fractional delay differential equations with variable coefficients based on the fractional Laplace transform and series Adomian decomposition. We consider the fractional derivative in the Caputo sense. Using the proposed method we get easily solvable iterative algebraic equations for uknown coefficients of power series solutions. Numerical example is included to demonstrate the validity and applicability of this technique. [ABSTRACT FROM AUTHOR]

The paper considers a delayed system of discrete equations x (k + 1) = A x (k) + B x (k − m) + f (k) , k = 0 , 1 ... , an initial problem x(k) = φ(k), k = −m, ..., 0 and prescribed impulses x(k + 1) = Cx(k + 1 − 0) + Jk + 1, k = 0, 1, ..., where A, B and C are regular square matrices satisfying ACB = BCA, f and Jk + 1 are column vectors, m > 0 is an integer and x is an unknown vector. A formula is derived for a solution satisfying the initial problem and the prescribed impulses. [ABSTRACT FROM AUTHOR]

Second order stabilized Runge–Kutta–Chebyshev methods (RKCs) are considered. Variants of their application to dis-crete delay differential equations (DDEs) are presented, namely reuse of stages rom previous steps, linear and quadratic interpola-tions. Linear stability analysis with real coefficients is made for all three variants. P-stability regions are plotted. Linear interpolation shows better stabiity properites than any quadratic (though the second being more accurate). [ABSTRACT FROM AUTHOR]

In this paper, we are going to deal with the pricing formulae and an exchange option for the asset pricing process, which is represented by stochastic delay differential equations; we are into introducing a recently come into existence model in which Poisson jumps are occurring as an increase in stochastic delay differential equation using the Black Scholes Model. This method would help the investors to speculate on the market trends, control their level of risk and grow their market value most efficiently. [ABSTRACT FROM AUTHOR]

Systems of equations with delay are widely used in various fields of modern mathematical modeling. The development and implementation of effective algorithms for the numerical solution of various classes of equations with lagging arguments is an urgent task. In this article, we considered the modeling of the learning process, which are described by classes of differential equations with a lagging argument. When developing the structure of a mathematical model, it is necessary to apply the modified Runge-Kutta method, which allows for integration with a variable step and localize the points of discontinuity of derivatives. The application of sequential integration methods (step-by-step method) for solving a differential equation with a lagging argument and comparing the resulting solution with numerical methods using the Matlab software package is considered. [ABSTRACT FROM AUTHOR]

In this paper we have taken different types of system of fractional delay differential equations with coefficient matrices considering constant as well as variable entries and discussed their oscillatory and non oscillatory behaviour. By giving suitable examples, theoretical results have been justified. [ABSTRACT FROM AUTHOR]

The characteristics of transportation problems may not be known precisely because of unpredictable circumstances in real life. Herein, fuzzy transportation problem with cost, supply and demand as triangular intuitionistic fuzzy numbers is considered and a method to solve that fully Intuitionistic Fuzzy Transportation Problems(FIFTP) is proposed, where the weights of the trader are provided in the form of Singularly Perturbed Delay Differential Equations (SPDDEs). The Trader weights are computed using the numerical solutions with a finite difference scheme on Shish kin mesh of SPDDEs. Finally, the new approach is illustrated by a numerical example and the solution is compared with the solution of existing methods. [ABSTRACT FROM AUTHOR]

In this paper, an initial value problem for a linear singularly perturbed delay differential equation with Robin initial condition is considered. A Shishkin piecewise uniform mesh is implemented and combined with a classical finite difference discretization to create a numerical method for solving this problem. The numerical approximations obtained are essentially first order convergence uniformly with respect to the singular perturbation parameter. Numerical experiment is givento support the theory. [ABSTRACT FROM AUTHOR]

We examine at boundary value problems for a system of 'n' singularly perturbed reaction-diffusion delay differential equations with discontinuous source terms. Using a finite element technique on an appropriate Shishkin mesh, we constructed a numerical method that resolves not only the normal boundary layers but also the interior layers generated by the delay terms. Assume that the point d ∈ (0,2) arises as a source term discontinuity. This approach is illustrated to be second order convergent uniformly with respect to the perturbation parameters in the energy norm. The article draws to a close with numerical evidence of the proposed technique. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to propose an efficient technique for solution of nonlinear differential equation involving constant and proportional delay terms. The presented method is combination of method of steps (MOS), differential transform and Bell polynomials. The delay differential equation (DDE) is transformed into proportional delay equation with the help of method of steps. After that differential transform is applied on DDE to provide recursive relation. Finally, Faà di Bruno formula is used to handle nonlinearity in the problem. Two numerical test problems are discussed to show accuracy and reliability of the method. The comparison of results with exact solution and MATLAB function DDENSD are presented. [ABSTRACT FROM AUTHOR]

The principle of mining machines digital control system simulation is considered. It is shown, that shaft-boring machine is the complex multi-contour object, which include actuators with significant nonlinearities in every control contour. When control the machines with Von Neumann type controllers it is necessary to take into account controller's real performance linked with step-by-step interpretation of a managing algorithm. The shaft-boring machine control system model proposed includes transfer functions matrix equations, describing object under control and semantics of data processing in controller, two delay matrices, describing Von Neumann type controller as a physical device, and description of actuator with nonlinearities by non-linear differential equation in Cauchy form. For estimation of delay matrices parameters semi-Markov process model of control algorithm is used. Initially semi-Markov matrix describes ergodic process, but after proposed transformation it permits to estimate time intervals when wandering between states. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy analysis method have been utilized for solving non-linear delay differential equations of multi-term fractional order, and the fractional derivative is expressed in the Caputo sense. In this approach, the solutions are found in the form of convergent power series components are easily computed. Some illustrative examples are simulated using computer programs written in Mathcad 15 computer software and then a comparison between the exact solution and the approximate results are presented to show the reliability and the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The goal of this paper is to look into some criteria for the oscillatory behavior of second order neutral delay difference equations. Starting with a general overview of difference equations, we present some recent findings on the use of neutral delay difference equations based on a comparison principle and summation averaging techniques. The acquired results can be utilised to build and offer theoretical support for the given equations. [ABSTRACT FROM AUTHOR]

In the present paper, the initial value problem for the third order partial delay differential equation with involution and nonlocal boundary condition is studied. The first order of accuracy difference schemes for the numerical solution of the third order partial delay differential equation with involution and nonlocal boundary condition is presented. The illustrative numerical results is provided. [ABSTRACT FROM AUTHOR]

It this paper we study a system of nonlinear delay impulsive fractional differential equations with Riemann-Liouville fractional derivatives taken from both, the unknown function at the present time and the unknown function at the variable time delay. We consider the case when the lower limit of the fractional derivative is changed at each impulsive time point. In the case studied the solution has a singularity at the initial time and any impulsive time point of the impulses. This leads to an appropriate definition of both the initial condition and the impulsive conditions. It is obtained the integral presentation of the solutions on a finite interval. Also, it is definite a special type finite time Lipschitz stability and some sufficient conditions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation criteria for the solutions of half-linear delay difference equations with order three and non-canonical operators. To give contradiction for existing results, oscillation of the studied equation is established under two different conditions. These results improve, extend and significantly simplify the existing results and the importance of the results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

In this work,we proposed a novel technique for solving fuzzy integro-differential equations with fractional derivative (FFIDEs) and fuzzy delay differential equations with fractional derivative (FFDDEs) involving gH Caputo differentiability. In this approach, 0-cut and 1-cut solutions of the differential equation are combined using the convex combination. The proposed method is elucidated in detail through illustrations. [ABSTRACT FROM AUTHOR]

We studied the possibility of describing both long-periodic solutions and short-periodic solutions for the Goodwin business cycle model with a continuously distributed investment lag using differential equations. It is shown that such equations can be formulated on the basis of the Padé approximants of the exponential function. The resulting differential equations of the 4th-7th order qualitatively describe the temporal dynamics of the national income. [ABSTRACT FROM AUTHOR]

The third order non linear difference equations of the form Δ (c n Δ (d n Δ x n)) + p n Δ x n + 1 + q n f (x n − σ) = 0 for n ≥ n0 are considered. Here {cn}, {dn}, {pn} and {qn} are sequences of positive real numbers for n0 ∈ Nf is a continuous function such that f (u) u ≥ K > 0 for u ≠ 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results. Where n0 ∈ N is a fixed integer, Δ denotes to forward difference operator Δxn = xn+1xn and σ is a nonnegative integer. [ABSTRACT FROM AUTHOR]

Investigated some new results on the oscillatory behavior of second order nonlinear neutral delay difference equation of the form Δ (α n (Δ (x n − p n x n − r)) α) + q n f (x n − σ) = 0 for all n ∈ N(n0) N (n 0) = { n 0 , n 0 + 1 , ... } and n0 is a nonnegative integer. Where 0 ≤ pn ≤ p < 1, qn > 0 and α is a ration of odd positive integers. [ABSTRACT FROM AUTHOR]

We study the approximate controllability results for nonlocal impulsive fractional neutral stochastic integrodifferential equations with state-dependent delay conditions in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The results are obtained by using fractional calculus, semigroup theory, stochastic analysis and fixed point theorem. An example is provided to show the application of our result. [ABSTRACT FROM AUTHOR]

It this paper we study a system of nonlinear neutral fractional differential equations with Riemann-Liouville fractional derivatives and suppremum deay taken from both, the unknown function at the present time and the unknown function at the time delay and non-instantaneous impulses. We consider the case when the lower limit of the fractional derivative is changed at each stop point of the action of the impulses. In the case studied the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. It is obtained the integral presentation of the solutions on a finite interval. Finite time Lipschitz stability is defined for the studied system and some sufficient conditions are obtained. [ABSTRACT FROM AUTHOR]

A system of nonlinear neutral delay non-instantaneous impulsive differential equations is studied. It is obtained the integral presentation of the solutions on a finite interval. Also, it is studied the closeness of solutions over a finite time interval. A finite time Lipschitz stability system is defined and sufficient conditions are obtained. This type of stability gives us an information about of the closeness between two solutions of the studied initial value problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend an oscillation criterion for the half-linear second order differential equations with delay to a more general form and to discuss relationship between these two variants and the link to the perturbation principle. The proving technique employs the Riccati transformation. [ABSTRACT FROM AUTHOR]

In this note, we present some criteria for the oscillation of second-order neutral delay differential equations. Precisely, we state a result related to the oscillation of unbounded solutions of the equation under consideration and a result related to the oscillation and the convergence for every solution of the same equation. [ABSTRACT FROM AUTHOR]

In the paper we consider first order stabilized Runge–Kutta–Chebyshev methods (RKCs) application to discrete delay differential equations (DDEs) and perform a linear stability analysis studying the standard linear test equation with real coefficients. We try two variants of RKCs extension for DDEs: the first, suitable for constant delays and constant time-steps; the second, with linear interpolation between the time-mesh points. It is shown that delay-independent stability regions are larger if using interpolation. As for ordinary differential equations RKCs have points of stability vanishing along the real values of the coefficient of the non-delayed term. We use damped RKCs to improve the stability regions and find an "optimal" damping factor to maximize the numerical stabiity region coverage of the exact stability domain. All the results are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

Recently, spectral methods in time for the numerical solution of many kinds of differential equations have been considered and analysed. Here, the use of spectral methods in time is extended to cope with Delay Differential Equations (DDEs), where they also proved to be very effective. In particular, we show an application to the mrSI2R2 model used to obtain forecast of the COVID-19 epidemic in Italy. [ABSTRACT FROM AUTHOR]

A system of nonlinear neutral delay impulsive differential equations is studied. It is obtained the integral presentation of the solutions on a finite interval. A finite time Lipschitz stability is defined and sufficient conditions for the initial value problem of the studied equation are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an IS-LM macroeconomic model with delay-dependent coefficients. The model assumes that the tax revenues have two complementary components, one based on the current income and the other on a past income where the fixed discrete time delay arises. Moreover, we assume that the tax share on each component depends on the time delay, and thus the resulting model is a system of delay differential equations with delay-dependent coefficients. Previously, De Cesare and Sportelli (2005) studied a similar model but the tax shares that they used were fixed constants. They showed that the equilibrium point may undergo a sequence of alternated stability switches. This can be observed for example in the case with low tax rate and low share of delayed tax revenue where loci of Hopf bifurcation points form closed curves with some of these curves overlap, and the equilibrium is unstable inside these loci. Our proposed model introduces a new parameter μ in the tax share terms, and the case where μ = 0 captures the previously studied model with constant tax shares. We showed that increasing this new parameter results to decoupling and shrinking of the 'islands of instability', and eventually results to the absolute stability of the equilibrium. [ABSTRACT FROM AUTHOR]

In this paper the semi-analytical solution is investigated for the delayed diffusive neural network model. Delay partial differential equations are approximated to the delay ordinary differential equation systems, by using Galerkin technique method. The main aim of this work is to determine the effect of diffusion and delay parameters and exploring the full map of stability analysis of the system. Stability analysis and bifurcation maps are discussed, as well, and in order to further explain certain concepts. The effect of diffusion parameter and delay values is comprehensively studied and as a result both variables can destabilize or stabilize the system. Illustrated examples of the unstable and stable limit cycles, and the Hopf bifurcation points are shown to prove the formerly revealed outcomes in the Hopf bifurcation map. In addition, comparisons between the numerical results and semi-analytical outcome provide comparison for all figures shown in the work. [ABSTRACT FROM AUTHOR]

We propose an approach in solving a parameter estimation problem in delay differential equations (DDE). By taking the interpolation of the data set, the problem is posed as a minimization problem using the L2–norm. We use a heuristic algorithm called Genetic Algorithm with multi-parent crossover to numerically solve the optimization problem. We consider four DDE models of biological systems with discrete and continuous delay to test our method. [ABSTRACT FROM AUTHOR]

A scheme combining the classical method of steps with expansion of the state space (MSSSE) was earlier proposed for an analysis of systems of stochastic ordinary differential equations with one constant time delay (SODDEs). This two-stage scheme is adapted for the analysis of new models described by stochastic partial differential equations (SPDEs) with delays (SPDDEs). The modified scheme together with a usage of the generalized Fokker–Planck– Kolmogorov equation (FPK Eq.) for the probability density functional makes it possible to construct a chain of SPDEs without delays. We exploit this chain to obtain new sequence of PDEs for calculating the first moment functions (fields) of the solution on successive time intervals. Some results of symbolic and numeric calculations for parabolic and hyperbolic equations that carried out in the environment of the mathematical package Mathematica, are presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for existence and uniqueness (EU) of solution for ABC-fractional order neutral differential equations with impulses in Banach space. The results are obtained by using fixed point theorems. [ABSTRACT FROM AUTHOR]

The aim of the paper is to obtain numerical solution of some delay differential equations (DDEs) using the Banach contraction method (BCM). The BCM is developed and applied to some problems involving proportional delays to show the efficiency of the method. Convergence analysis and error estimation are presented. It is showed that error becomes small with a large number of iterations. The BCM reduces complex calculations, avoids discretization, linearization, perturbation, and easy to implement. [ABSTRACT FROM AUTHOR]

Linear Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied. Both cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. The initial conditions as well as impulsive conditions are defined in an appropriate way for both cases. The explicit solutions are obtained and applied to study finite time stability. [ABSTRACT FROM AUTHOR]

In this paper, the direct Lyapunov method is extended to fractional case of Volterra integro-Caputo fractional differential equations with delays. Two types of homogeneous scalar Voterra integro-Caputo fractional differential equations with delays are studied. Three types of fractional derivatives of Lyapunov functions known in the literature are given. Uniform stability of trivial solution of that retarded IFrDEs is investigated using the Razumikhin method and different types of fractional derivatives of Lyapunov functions. Examples to show the application of the theorems are provided. [ABSTRACT FROM AUTHOR]

A system of nonlinear Riemann-Liouville fractional differential equations with a constant delay is studied. The initial condition is set up similarly to the case of ordinary derivative. Sufficient conditions for finite time stability are obtained. [ABSTRACT FROM AUTHOR]

Malware is a type of malicious program that can damage information systems. Malware can be move randomly, so it is inconvenient to detect. In this paper, we will analyze the propagation of malware on computer networks with two-time delays. The computer network is divided into three types of nodes. In the next step, the propagation of malware is modeling in the system of Differential Delay Equation (DDEs). Furthermore, it is analyzed dynamically from the model especially the local stability of malware. In addition, a simulation is performed to determine the effect of two-time delays. Based on this analysis, it was concluded that the model was effective to be applied so that it could be used to control the propagation of malware on computer networks. [ABSTRACT FROM AUTHOR]

In this paper several problems arising when implementing methods to numerically solve delay-differential equations, such as overlapping, difficulties with error estimation, discontinuities of solution derivatives and the detection of such discontinuity points, are considered. Several approaches to each issue are discussed. The best solutions for explicit and diagonally implicit Runge–Kutta methods for DDEs are reported. A numerical comparison between a method implementing the handling of such problems, and the Matlab code ddesd, demonstrates that our method is more efficient. [ABSTRACT FROM AUTHOR]