Time-delay Differential Equation Research Articles

MODELING STOCHASTIC ADVERSE EFFECTS OF CBN 2023 REDESIGNED NAIRA NOTES POLICY ON RURAL FARMERS IN NIGERIA

The recent Central Bank of Nigeria (CBN) 2023 redesigned naira notes is of good benefits to strengthen the economy of the country by checking counterfeiting and hoarding of large volume of banknotes by the public. Despite all the efforts made by the CBN for citizens to enjoy the benefits of this implementation, most rural farmers were faced with adverse effects of uncertainties in the production and marketing of their agricultural produce due to lack of redesigned new naira notes in circulation. The adverse effects of these uncertainties are modeled as Advanced Stochastic Time-Delay Differential Equation (ASTDDE). The modeled equation is solved using Extended Second Derivative Block Backward Differentiation Formulae Method (ESDBBDFM) without the use of interpolation techniques in the evaluations of the delay term and noise term. In comparing the numerical results of this method with other existing methods in literature, the newly developed mathematical expressions for the evaluations of the delay term and the noise term in solving ASDDEs with the discrete schemes of ESDBBDFM gives better results for step number than step numbers and 3 by producing Least Minimum Absolute Random Error (LMARE) in a Lower Computational Processing Unit Time (LCPUT) faster than other existing methods that applied interpolation techniques in evaluations of the delay term and the noise term.

Nonlinear chatter and reliability analysis of milling Ti-6Al-4V with slender ball-end milling cutter

Slender ball-end milling cutters with high length/diameter ratio can offer increased accessibility in the deep cavity machining of parts in the aviation industry. In this paper, a comprehensive nonlinear machining dynamics model for milling Ti-6Al-4V with the slender ball-end milling cutter is proposed to predict and understand on cutting stability, vibration characteristics and dynamic reliability. The tool tip dynamics model of the slender ball-end milling cutter is established including the structural nonlinearity of bearing joints, the contact flexibility of joint interfaces and the influence of tool geometry. Generated nonlinear dynamic cutting force is calculated considering the multiple regenerative effect, the nonlinearity of tool away from cutting and the run-out of tool. The effect of process damping is included in the nonlinear time-delayed differential equation of the machining dynamics model. Furthermore, the reliability evaluation method for the dynamic accuracy of milling Ti-6Al-4V workpieces is established with the application of Monte Carlo method and active learning Kriging model. Some milling experiments on Ti-6Al-4V workpiece are designed to validate the proposed model by comparing the predicted and measured results. The milling stability, vibration characteristics and dynamic accuracy reliability of milling Ti-6Al-4V with the slender ball-end milling cutter are discussed. The increase in the fractal dimension of joint interfaces can lead to the improvement of milling stability, and increased fractal roughness results in reduced milling stability. The reliability in the dynamic accuracy of milling Ti-6Al-4V workpiece is 99.26 %, and the distribution parameters in the fractal dimension of joint interfaces have the greatest influence on the reliability of dynamic accuracy.

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Stability Analysis of Milling Based on the Barycentric Rational Interpolation Differential Quadrature Method

Chatter causes great damage to the machining process, and the selection of appropriate process parameters through chatter stability analysis is of great significance for achieving chatter-free machining. This article proposes a milling stability analysis method based on the barycentric rational interpolation differential quadrature method (DQM). The dynamics of the milling process considering the regeneration effect is first modelled as a time-delay differential equation (DDE). When adjacent pitch angles of the milling cutter are symmetric, the milling dynamic equation contains a single time delay. Otherwise, when adjacent pitch angles are asymmetric, the dynamic equation contains multiple time delays. The barycentric rational interpolation DQM is then used to approximate the differential and delay terms of the milling dynamics equation, and to construct a state transition matrix between adjacent milling periods. Finally, the chatter stability lobe diagram (SLD) is obtained based on the Floquet theory. According to the SLD, the appropriate spindle speed can be selected to obtain the maximum stable axial depth of cutting, thereby effectively improving the material removal rate. The accuracy and efficiency of the proposed method have been validated by two widely used milling models, and the results show that the proposed method has good accuracy and computational efficiency.

Existence and stability of solution for time-delayed nonlinear fractional differential equations

ABSTRACT The objective of this study is to analyze the existence and stability of solutions for a mathematical model of nonlinear systems of time-delayed fractional differential equations involving Atangana-Baleanu-Caputo ( ABC ) fractional derivatives with vaccination. The epidemiological states of the total population are classified as susceptible individuals S ( t ) , infected individuals I ( t ) , quarantined individuals Q ( t ) , recovered individuals R ( t ) , vaccinated individuals V ( t ) and insusceptible/protected persons P ( t ) . By considering coordinate transformation, we linearize our system and obtain a matrix coefficient for the proposed time-delayed fractional differential equation. We establish sufficient requirements and inequalities using the generalized Gronwall inequality and Krasnoselskii's fixed-point theorem to prove the existence of solution for our problem. Additionally, we construct sufficient requirements for stability analysis by employing the Laplace transform and matrix measure theory. We also investigate criteria for the local stability of the epidemic and disease-free equilibrium points. Finally, we provide a numerical illustration to validate the effectiveness of our results.

A study on inventory control strategies of fresh food supply chain considering advertising delay effect

PurposeThe existence of the advertising delay effect and its impact on supply chain operations have been demonstrated in the current study. Therefore, this study develops a timely inventory control strategy for the fresh produce supply chain to address the advertising delay effect in the fresh produce supply chain.Design/methodology/approachThis study proposes a game model based on the Nerlove-Arrow time delay differential equation and Pontryagin's maximum principle. Through comparative analyses of the optimal equilibrium strategies, the authors compare the optimal equilibrium strategies, product goodwill and optimal inventory trajectories for suppliers and retailers under secondary replenishment decisions and decentralized decisions.FindingsThe authors find that (1) Only when the sales cycle meets certain conditions can the overall profit of the supply chain under the secondary replenishment decision be greater than that under the decentralized decision. As the price markup coefficient increases, the total profit of the supply chain first increases and then decreases. (2) With the increase in the delay time, the replenishment quantity during the initial period gradually decreases. After the delay time elapses, the inventory depletion rate under secondary replenishment decisions is faster than that under decentralized decision-making. (3) Although there is a continuously increasing maximum value of product goodwill with the increase in delay time, it becomes difficult to achieve this value for longer delays.Practical implicationsThe authors’ findings provide a theoretical basis for supply chain members of fresh agricultural products to select replenishment and inventory control strategies when adopting different levels of delay in advertising marketing.Originality/valueFirstly, this paper explains the impact of advertising delay effect on fresh produce supply chain from a dynamic perspective, and secondly, it provides guidance on advertising formulation and inventory replenishment for fresh produce retailers under the influence of advertising delay effect.

Cluster Synchronization of Multiple Fractional-Order Recurrent Neural Networks With Time-Varying Delays.

This article focuses on the cluster synchronization of multiple fractional-order recurrent neural networks (FNNs) with time-varying delays. Sufficient criteria are deduced for realizing cluster synchronization of multiple FNNs via a pinning control by applying an extended Halanay inequality applicable for time-delayed fractional-order differential equations. Moreover, an adaptive control applicable for the synchronization of fractional-order systems with time-varying delays is proposed, under which sufficient criteria are derived for realizing cluster synchronization of multiple FNNs with time-varying delays. Finally, two examples are presented to illustrate the effectiveness of the theoretical results.

Stability prediction via parameter estimation from milling time series

Machine tool vibrations impose severe limitations on industry. Recent progress in solving for the stability behavior of delay differential equations and in modeling milling operations with time delay differential equations has provided the potential to significantly reduce the aforementioned limitations. However, industry has yet to widely adopt the current academic knowledge due to the cost barriers in implementing this knowledge. Some of these cost prohibitive tasks include time-consuming experimental cutting tests used to calibrate model force parameters and experimental modal tests for every combination of tool, tool holder, tool length, spindle, and machine. This paper introduces an alternative approach whereby the vibration behavior of a milling tool during cutting is used to obtain the necessary model parameters for the common delay differential equation models of milling.

Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

Passively mode-locked high-frequency dual-VCSEL system.

Two VCSELs placed facing each other with one biased chip while the second chip is unbiased is shown as a promising alternative to the popularly used conventional SESAM mode-locked VECSEL to generate mode-locked pulses. We propose a theoretical model using time-delay differential rate equations and numerically show that the proposed dual-laser configuration functions as a typical gain-absorber system. Parameter space defined by laser facet reflectivities and current are used to show general trends in the exhibited nonlinear dynamics and pulsed solutions.

Global stability analysis of a COVID-19 epidemic model with incubation delay

<abstract><p>In this paper, we propose, analyze and simulate a time delay differential equation to investigate the transmission and spread of Coronavirus disease (COVID-19). The basic reproduction number of the model is determined and qualitatively used to investigate the global stability of the model's steady states. We use numerical simulations to support the analytical results in the study. From the simulation results, we note that whenever the basic reproduction number is greater than unity, the model solutions will be associated with periodic oscillations for a considerable time scale from the start before attaining stability. This suggests that the inclusion of the time delay factor destabilizes the endemic equilibrium point leading to periodic solutions that arise due to Hopf bifurcations for a certain time frame.</p></abstract>

Nonlinear Langevin time-delay differential equations with generalized Caputo fractional derivatives

Nonlinear Langevin time-delay differential equations with generalized Caputo fractional derivatives

Computational solution of advanced stochastic time-delay differential equations using hybrid block extended Adams Moulton methods for customer’s satisfaction when using electronic payment systems in Nigerian banks

This study examined the implementation of Hybrid Block Extended Adams Moulton Methods (HBEAMM) for the computational solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) through the use of various electronic payment systems such as ATM, POS and internet banking offered by the Nigerian banks. This work attempts to proffer solution to customer challenges in using the ATM of most banks in Nigeria. The challenges are considered in this work as capable of resulting to advanced time-delay and volatility noise. It is therefore modeled as advanced stochastic time-delay differential equation (ASTDDE) and solved using Hybrid Extended Block Adams Moulton Methods (HEBAMMs). The discrete schemes of the proposed method (HBEAMM) were obtained by continuous formulations of multistep collocation method by matrix inversion approach. The necessary and sufficient conditions for convergence and stability of the method were analyzed and proved satisfactory. The discrete schemes obtained were applied in solving some advanced stochastic time-delay differential equations and the results obtained revealed the degree of customers’ satisfaction in the use of e-payment systems. The accuracy of the method is compared with other existing methods and presented graphically to prove its superiority.

High efficiency and precision approach to milling stability prediction based on predictor-corrector linear multi-step method

Regenerative chatter is the most important factor affecting the stability of the milling process. It is core for suppressing chatter and improving production efficiency to accurately and efficiently identify the stable region of milling chatter. Therefore, according to the theory of predictor-corrector, three predictor–corrector methods (PCM) are, respectively, proposed for the milling stability region by applying the fourth-order Adams-Bashforth-Moulton formula, Simpson formula, and Hamming formula. Firstly, the regenerative chatter milling process is described as a second-order time-delay differential equation (DDE) with periodic coefficients. Thus, the forced vibration time can uniformly be discretized as a time node set. Secondly, the fourth-order Adams-Bashforth formula is used to predict the displacement at every time node, whereas the fourth-order Adams-Moulton formula can be employed to correct this predicted value. In addition, the fourth-order Simpson formula and Hamming formula can also correct the predicted value. Thus, a higher precision discrete prediction-correction expansion is constructed for the transformation of DDE into the state transition express. The Floquet theory can be depended on to present the judgment criterion of milling stability. Moreover, finally, under the same milling process parameters, comparisons of both the stability lobe curve and the local discrete error curve show that the PCM has a faster convergence rate than the 1st-SDM (first-order semi-discretization method) and 2nd-FDM (second-order full-discretization method). This shows that the PCM can obtain better computational accuracy under the same discrete number, whereas the PCM is significantly higher computational efficiency over 1st-SDM and 2nd-FDM. Meanwhile, considering the actual machining environment, helix angle effect and multiple modes effect of the tool are analyzed; experimental verification considering multiple modes with helix angle further indicates the applicability of the PCM.

Multiplicity–Induced–Dominancy property for second–order neutral differential equations with application in oscillation damping

This paper addresses the exponential stability of linear time-delay systems of neutral type. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be necessarily dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). The aim of this paper is to explore the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay. The ensuing control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. The main ingredient of the proposed method is the dominancy proof for multiple spectral values based on frequency bounds established via integral equations. As an illustration, the stabilization of the classical oscillator benefits from the obtained results.

Optimal disturbances for periodic solutions of time-delay differential equations

Abstract A concept of optimal disturbances of periodic solutions for a system of time-delay differential equations is defined. An algorithm for computing the optimal disturbances is proposed and justified. This algorithm is tested on the known system of four nonlinear time-delay differential equations modelling the dynamics of the experimental infection caused by the lymphocytic choriomeningitis virus. The results of numerical experiments are discussed.

Uniqueness and Ulam–Hyers–Mittag–Leffler stability results for the delayed fractional multiterm differential equation involving the Φ-Caputo fractional derivative

The principal aim of this paper is to establish the uniqueness and Ulam–Hyers–Mittag–Leffler (UHML) stability of solutions for a new class of multiterm fractional time-delay differential equations in the context of the Φ-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag–Leffler functions (M–LFs) are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying the well-known Banach contraction principle coupled with the Φ-fractional Bielecki-type norm, while the Φ-fractional Gronwall type inequality and the Picard operator (PO) technique, combined with the abstract Gronwall lemma, are used to prove the UHML stability results for the proposed problem. Lastly, an example is offered to assure the validity of the obtained theoretical results.

Predicting Milling Stability Based on Composite Cotes-Based and Simpson's 3/8-Based Methods.

Avoiding chatter in milling processes is critical for obtaining machined parts with high surface quality. In this paper, we propose two methods for predicting the milling stability based on the composite Cotes and Simpson’s 3/8 formulas. First, a time-delay differential equation is established, wherein the regenerative effects are considered. Subsequently, it is discretized into a series of integral equations. Based on these integral equations, a transition matrix is determined using the composite Cotes formula. Finally, the system stability is analyzed according to the Floquet theory to obtain the milling stability lobe diagrams. The simulation results demonstrate that for the single degree of freedom (single-DOF) model, the convergence speed of the composite Cotes-based method is higher than that of the semi-discrete method and the Simpson’s equation method. In addition, the composite Cotes-based method demonstrates high computational efficiency. Moreover, to further improve the convergence speed, a second method based on the Simpson’s 3/8 formula is proposed. The simulation results show that the Simpson’s 3/8-based method has the fastest convergence speed when the radial immersion ratio is large; for the two degrees of freedom (two-DOF) model, it performs better in terms of calculation accuracy and efficiency.

Modeling pulsativity in the hypothalamic–pituitary–adrenal hormonal axis

A new mathematical model for biological rhythms in the hypothalamic–pituitary–adrenal (HPA) axis is proposed. This model takes the form of a system of impulsive time-delay differential equations which include pulsatile release of adrenocorticotropin (ACTH) by the pituitary gland and a time delay for the release of glucocorticoid hormones by the adrenal gland. Numerical simulations demonstrate that the model’s response to periodic and circadian inputs from the hypothalamus are consistent with those generated by recent models which do not include a pulsatile pituitary. In contrast the oscillatory phenomena generated by the impulsive delay equation mode occur even if the time delay is zero. The observation that the time delay merely introduces a small phase shift suggesting that the effects of the adrenal gland are “downstream” to the origin of pulsativity. In addition, the model accounts for the occurrence of ultradian oscillations in an isolated pituitary gland. These observations suggest that principles of pulse modulated control, familiar to control engineers, may have an increasing role to play in understanding the HPA axis.

A novel dynamic time-delay grey model of energy prices and its application in crude oil price forecasting

The correct forecasting of energy prices can better regulate energy market allocation, reduce the unit energy allocation cost, and maximize the economic benefits of the energy allocation cost. First, this paper aims to study the time-delay in the energy supply and demand relationship, which directly affects changes and fluctuations in the energy price. Starting from the market economic model, this paper analyses the dynamic time-delay process of energy prices by using the relationship between energy supply, demand and prices, and it establishes the time-delay differential equation of energy prices. Second, a novel dynamic grey time-delay forecasting model of energy prices is selected based on the differential information of the differential equation and difference equation and the principle of data reduction. The parameters and optimization methods of the model are studied to obtain the modeling process of the model. Finally, the novel model is applied to predict the Novel York Mercantile Exchange (NYMEX) West Texas Intermediate (WTI) crude oil futures price, the Intercontinental Exchange (ICE) Brent crude oil futures price, and the Organization of the Petroleum Exporting Countries (OPEC) basket crude oil spot price. The forecasting results are better than those of the other four grey prediction models, which indicates that the novel model can effectively predict the oil price and that the novel model can describe the dynamic change law of the energy price system with a time-delay.