Linear Differential Model Research Articles

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

Dance Rehearsal System Based on Linear Partial Differential Mathematical Equation

Abstract In this paper, a linear partial differential mathematical choreography model is proposed for the accuracy of action recognition in dance rehearsal systems. This paper more accurately represents the action change features. The article uses the pyramid algorithm (LK) to calculate pixels’ optical flow vector change in dt time. Then this paper adopts a multi-feature fusion module to fuse multi-channel features. Finally, the system completes the accurate identification of dance choreography movements. The research results show that the algorithm can perform dance movement recognition. The dance rehearsal system realizes the movement correction of dancers.

PECLOS path-following control of underactuated AUV with multiple disturbances and input constraints

This paper presents a new position error-constrained line-of-sight (PECLOS) path-following (PF) control strategy for an underactuated AUV under multiple disturbances and input constraints. Firstly, considering that many previous studies ignored the influence of stochastic noises, which may degrade the designed controller performance or even lead to instability in practical applications, the multiple disturbances are divided into two categories based on their features, i.e., the slowly time-varying deterministic disturbances and the stochastic noises. On this basis, the time-varying deterministic disturbances are further treated as a single equivalent disturbance and a linear stochastic differential model of underactuated AUV is established by augmenting the equivalent disturbance as an extended state. Secondly, a novel PF control strategy is proposed based on an extended state based Kalman-Bucy filter (ESKBF) and auxiliary dynamic compensators. The PECLOS guidance law is established to maintain the PF errors within the specified time-varying bounds, the ESKBF technology is utilized to estimate and compensate for the equivalent disturbances under the stochastic noises, and the auxiliary dynamic compensators are adopted to overcome the input constraints. Eventually, the stability of the overall closed-loop system is verified. Simulation and comparative analysis are presented to demonstrate the enhanced performance of the presented controller.

On the Solution of the Variable Order Time Fractional Schrödinger Equation

In the present paper, we used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional Schrödinger equations. Twelve illustrative applications have been presented. When the fractional order is unity, the results are the same as those obtained by the standard integer order derivative. Some of the obtained results are discussed graphically. Some of the obtained results 1st appear in the literature. This technique can be applied to other linear or nonlinear fractional differential models.
 HIGHLIGHTS
 
 We study a class of time variable-order fractional nonlinear Schrödinger equations (tVOFNLSEs)
 We introduce a new procedure for solving the tVOFNLSE and tVOFSE
 We used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional equations
 The calculations are performed in the frame of Caputo sense
 
 GRAPHICAL ABSTRACT

Laplace decomposition method for series solutions of systems of linear partial differential models

Most of the real-life problems experienced in the industry these days cannot be expressed as a single differential equation but as a system of differential equations. Such structures of linear partial differential models are considered in this work with the aid of the Laplace Adomian decomposition method (LADM) in terms of solutions. Various examples are taken into consideration. The results are easily obtained, are in good agreement when compared to their exact forms. In addition, the proposed method seems effective and efficient; the solutions are graphically presented.

Iterative Method for Approximate-Analytical Solutions of Linear Schrödinger Equation

In this study, the modified Picard Iterative Method (MPIM) is used to provide analytic and numerical solutions to linear Schrödinger Equations. These approximate-analytical solutions for the examples under consideration are easily computed. The suggested method is employed without any transformation, discretization, linearization, or limiting assumptions. The obtained results are similar to their exact forms. As a result, the approach is highly suggested for both linear and non-linear time-space fractional partial differential models with applications in various applied disciplines.

Discrete choice experiment to determine preferences of decision-makers in healthcare for different formats of rapid reviews

BackgroundTime-saving formats of evidence syntheses have been developed to fulfill healthcare policymakers’ demands for timely evidence-based information. A discrete choice experiment (DCE) with decision-makers and people involved in the preparation of evidence syntheses was undertaken to elicit preferences for methodological shortcuts in the conduct of abbreviated reviews.MethodsD-efficient scenarios, each containing 14 pairwise comparisons, were designed for the DCE: the development of an evidence synthesis in 20 working days (scenario 1) and 12 months (scenario 2), respectively. Six attributes (number of databases, number of reviewers during screening, publication period, number of reviewers during data extraction, full-text analysis, types of HTA domains) with 2 to 3 levels each were defined. These were presented to the target population in an online survey. The relative importance of the individual attributes was determined using logistic regression models.ResultsScenario 1 was completed by 36 participants and scenario 2 by 26 participants. The linearity assumption was confirmed by the full model. In both scenarios, the linear difference model showed a preference for higher levels for “number of reviewers during data extraction”, followed by “number of reviewers during screening” and “full-text analysis”. Subgroup analyses showed that preferences were influenced by participation in the preparation of evidence syntheses.ConclusionThe surveyed persons expressed preferences for quality standards in the process of literature screening and data extraction.

New decomposition method for the solutions of linear Schrödinger equation

The Schrödinger equation serves as the fundamental equation for quantum mechanics. In this article, we consider the theoretical and numerical solutions of Schrödinger’s linear equations. This is achieved using the new semi-analytical approach as an alternative to the classical Adomian Decomposition Method (ADM). The new method is referred to as Telescoping Decomposition Method. Some cases are considered; the results obtained display high level of convergence to their exact forms. The improved version is very useful and reliable; it requires less analytical effort, even without giving up precision. Thus, it is also highly recommended for the solution of related linear and nonlinear differential models in the fields of applied research.

Construction of a method for representing an approximation model of an object as a set of linear differential models

This paper has demonstrated the need to use models not only at the stage of theoretical research and design operations but also when studying existing objects. The techniques to build them on the basis of identification methods have been analyzed. The identification methods have been shown when determining the parameters of processes and objects. The difficulty of defining the models' structures has been emphasized.A method has been proposed to determine the structure of an arbitrary object's model as the approximating set of linear differential models. The data on the object's response to external impact have been used as source data. Demonstrating the method's feasibility employed a set of standard links and a standard external influence in the form of a stepped function as a model. This approach helps assess the adequacy of the obtained approximation results based on the precise solutions available. In a general case, there are no specific requirements for the form of an external influence and an object's reaction.The data that reflect the object's response should allow their approximation using a polynomial. That makes it possible to represent them following a Laplace transform in the form of a truncated power series in the image domain. The transfer function is written in a general form as a rational fraction. It underlies a Padé approximant of the truncated power series.The comparison of the available accurate calculation results and those derived on the basis of the built model has shown good agreement. In the cases under consideration, the computation error did not exceed the 5 % value permissible for engineering calculations. This is also the case when using the approximation of original data over a limited period.The response of the resulting model to the external influence that simulates a real pulse was investigated. The comparison with precise results showed a discrepancy not exceeding the value permissible for engineering calculations (<5 %)

Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical linear ordinary differential equations linear fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of ±20 dB per frequency decade and, respectively, deviate from integer multiples of ±π/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.

Karhunen-Loéve expansion of Brownian motion for approximate solutions of linear stochastic differential models using Picard iteration

This work presents an application of the Picard Iterative Method (PIM) to a class of Stochastic Differential Equations where the randomness in the equation is considered in terms of the Karhunen-Loeve Expansion finite series. Two applicable numerical examples are considered to illustrate the convergence of the approximate solutions to the exact solutions and also to check the efficiency of the method. The results obtained show clearly that accuracy will be more visible by increasing the number of terms in the iteration. Thus, it is recommended for nonlinear financial models of different classes of Stochastic Differential Equations.

Centralized and decentralized distributed control of longitudinal vehicular platoons with non‐uniform communication topology

AbstractIn this paper, the distributed control of a longitudinal platoon of vehicles with non‐uniform communication topology is studied. In the case of non‐uniform communication topology, some eigenvalues of the network's matrix may be complex which complicates the stability analysis of the platoon. Most previous studies on vehicular platooning focus mainly on uniform topologies such as uni‐directional, bi‐directional, and multi predecessors following. Since all eigenvalues of these topologies are real, the stability analysis can be performed in a straightforward manner. A third‐order linear differential model is employed to describe the upper‐level dynamics of each vehicle. The 3 N‐order closed‐loop dynamics of the platoon are decoupled to individual third‐order dynamics by presenting a new approach. Two new centralized and decentralized control protocols are introduced to perform the stability analysis of the closed‐loop dynamics. A constant time headway strategy is employed to adjust the inter‐vehicle spacing. Simulation results with different scenarios are presented to illustrate the effectiveness of the proposed approaches.

ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНЫЕ И ДИФФЕРЕНЦИАЛЬНО-РАЗНОСТНЫЕ МОДЕЛИ СИСТЕМ С АКУСТИЧЕСКОЙ ОБРАТНОЙ СВЯЗЬЮ

The paper reports the investigations of the functional differential and differential models of system with acoustic feedback. The aim of the work is a for-mation, analysis and identification of linear and nonlinear functional differential models of echo, reverberation and acoustic feedback; the analysis of model sensitivity, the analysis of stability of telecommunication systems with the acoustic feedback. There is presented a mechanism of an acoustic feedback mechanism formation. 
 At the analysis of sound propagation in closed rooms with the use of the geometric acoustics method there is obtained a nucleus equation which is approximated by nonnegative functions taking into account a model of reverberation and a model of multiple reflections. The considered procedure of echo formation and reverberation is an element of the feedback model common for warning systems and technological communication. There is investigated a model of acoustic systems with delayed feedback. A diagram of the kernel of a functional differential equation describing a system with feedback is shown. 
 The models of systems with acoustic feedback presented in this work ensure the approximation to reality at that, an account of possibilities of the emergence in channels of non-linear distortion sound propagation are considered to be the next problem.

Adjacent-Agent Dynamic Linearization-Based Iterative Learning Formation Control.

The dynamical relationship of the multiple agents' behavior in a networked system is explored and utilized to enhance the control performance of the multiagent formation in this paper. An adjacent-agent dynamic linearization is first presented for nonlinear and nonaffine multiagent systems (MASs) and a virtual linear difference model is built between two adjacent agents communicating with each other. Considering causality, the agents are assigned as parent and child, respectively. Communication is from parent to child. Taking the advantage of the repetitive characteristics of a large class of MASs, an adjacent-agent dynamic linearization-based iterative learning formation control (ADL-ILFC) is proposed for the child agent using 3-D control knowledge from iterations, time instants, and the parent agent. The ADL-ILFC is a data-driven method and does not depend on a first-principle physical model but the virtual linear difference model. The validity of the proposed approach is demonstrated through rigorous analysis and extensive simulations.

Exact and nonstandard numerical schemes for linear delay differential models

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.

Control design and stability analysis of homogeneous traffic flow under time delay: A new spacing policy

This paper details control design and stability analysis of homogeneous traffic flow by considering it as the interaction between inter-connected vehicular platoons. A third-order linear differential model is used to describe the longitudinal motion of each vehicle. As the lead vehicle of the whole traffic flow may be not available, the inter-platoon communication structure is assumed to be a bidirectional virtual leader following (BDVLF) topology. Both communication and parasitic delays are considered in the system modeling and control design. By employing an appropriate state transformation, the 3 N-order closed-loop dynamics are decoupled to N third-order dynamical equations. The cluster treatment characteristic root (CTCR) method is employed to perform the inter-platoon stability analysis. The intra-platoon communication structure is assumed to be general. Therefore, the eigenvalues of matrix H maybe complex which makes the stability analysis more intricate. By introducing a new decoupling approach and applying a centralized control for each vehicle in the platoon, the necessary conditions on control parameters satisfying intra-platoon stability are presented. The most important merit of this method compared to previous works, is that control parameters are independent of eigenvalues of matrix H. Several simulation studies are provided to show the effectiveness of the proposed approaches.

Dynamic analysis of nonlinear variable frequency water supply system with time delay

In this paper, we study the mathematical model about variable frequency water supply system in the process of applying glue to particleboard. Based on the original linear ordinary differential model, the effects of time delay and the nonlinear factor are considered. Then, we obtain the delayed nonlinear differential equation associated with variable frequency water supply system. Further, we consider the existence and stability of the equilibria and the existence of several types of bifurcations in this functional differential equation. Next, we derive the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation by using the multiple time scales method and the center manifold reduction method, respectively, and analyze the classifications of local dynamics. Finally, by using Matlab software, we obtain numerical simulations with estimated values of parameters and show the existence of stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and a complex chaotic attractor from a sequence of period-doubling bifurcations.

Posterior Estimates of Dynamic Constants in HIV Transmission Modeling.

In this paper, we construct a linear differential system in both continuous time and discrete time to model HIV transmission on the population level. The main question is the determination of parameters based on the posterior information obtained from statistical analysis of the HIV population. We call these parameters dynamic constants in the sense that these constants determine the behavior of the system in various models. There is a long history of using linear or nonlinear dynamic systems to study the HIV population dynamics or other infectious diseases. Nevertheless, the question of determining the dynamic constants in the system has not received much attention. In this paper, we take some initial steps to bridge such a gap. We study the dynamic constants that appear in the linear differential system model in both continuous and discrete time. Our computations are mostly carried out in Matlab.

Profile error evaluation of free-form surface using sequential quadratic programming algorithm

Profile error of free-form surface is evaluated in this paper based on sequential quadratic programming (SQP) algorithm. The optimal localization model is established with the minimum zone criterion firstly. Subsequently, the surface subdivision method or STL (STeror Lithography) model is used to compute the point-to-surface distance and the approximate linear differential movement model of signed distance is deduced to simplify the updating process of alignment parameters. Finally, the optimization model on profile error evaluation of free-form surface is solved with SQP algorithm. Simulation examples indicate that the results acquired by SQP method are closer to the ideal results than the other algorithms in the problem of solving transformation parameters. In addition, real part experiments show that the maximum distance between the measurement points and their corresponding closest points on the design model is shorter by using SQP-based algorithm. Lastly, the results obtained in the experiment of the workpiece with S form illustrate that the SQP-based profile error evaluation algorithm can dramatically reduce the iterations and keep the precision of result simultaneously. Furthermore, a simulation is conducted to test the robustness of the proposed method. In a word, this study purposes a new algorithm which is of high accuracy and less time-consuming.

Methods of Pseudo-Inverse Algebra in State Identification Problems for Thick Elastic Plates

The authors solve identification problems for the three-dimensional function of transverse displacements of points of a thick elastic plate of finite dimensions, whose dynamics can be described by a linear differential model with discrete and continuous observations of the initial---boundary-value external dynamic perturbations of the plate. The working criterion is the RMS criterion of fitting the obtained solution and observations of the plate and methods of linear pseudo-inverse algebra. The conditions of accuracy and uniqueness of the identification results are formulated.