First-order Time Research Articles

Frequency Domain Specifications Based Robust Decentralized PI/PID Control Algorithm for Benchmark Variable-Area Coupled Tank Systems.

A decentralized PI/PID controller based on the frequency domain analysis for two input two output (TITO) coupled tank systems is exploited in this paper. The fundamentals of the gain margin and phase margin are used to design the proposed PI/PID controller. The basic objective is to keep the tank at the predetermined level. To satisfy the design specifications, the control algorithm is implemented for decoupled subsystems by employing a decoupler. First-order plus dead time (FOPDT) models are obtained for the decoupled subsystems using the model-reduction technique. In addition, the control law is realized by considering the frequency domain analysis. Further, the robustness of the controller is verified by considering multiplicative input and output uncertainties. The proposed method is briefly contrasted with existing techniques. It is envisaged that the proposed control algorithm exhibits better servo and regulatory responses compared to the existing techniques.

Sign-preserving second-order IMPEC time discretization and its application in compressible miscible displacement with Darcy-Forchheimer models

In this paper, we construct sign-preserving second-order IMplicit Pressure Explicit Concentration (IMPEC) time methods for generalized coupled non-Darcy flow and transport problems in petroleum engineering, extending the algorithm given in [10] which is only applicable to Darcy flows. We use interior penalty discontinuous Galerkin (IPDG) methods for spatial discretization, and develop bound-preserving technique to obtain physically relevant numerical approximations. The sign-preserving second-order IMPEC method is an important innovation. The method is based on the framework of the second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The basic idea is to treat the pressure equation implicitly and the concentration equation explicitly so as to obtain a first-order time integration. Then we introduce a correction stage to compensate the accuracy, maintaining the physical bounds of the numerical cell averages. Unfortunately, the above algorithm is not applicable to non-Darcy problems. There are two main difficulties. Firstly, since the velocity equation is nonlinear and time-independent, all variables in the equation must be calculated at the same time level. However, the treatment in [10] will yield an extremely complicated algorithm and significantly large computational cost, and some iterations whose convergence may not be available if the solutions are not smooth. In our scheme, we linearize the velocity equation and the numerical solutions are not at the same time level, leading to first-order accurate solutions. Therefore, we adopt a completely different approach from [10] to derive the correction stage, keeping the physical bounds of the numerical solutions. Secondly, though with the new correction stage, it is still not easy to solve velocity equations in some stages. In this paper, we construct a direct solver for the velocity equation to save computational cost. Numerical experiments will be given to demonstrate that the improved sign-preserving second-order IMPEC scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. The proposed method also yields much larger CFL number compared with first-order IMPEC schemes. Moreover, the effectiveness of the bound-preserving technique will also be verified.

Simplified two-dimensional model for global atmospheric dynamics

We present a simplified model of the atmosphere of a terrestrial planet as an open two-dimensional system described by an ideal gas with velocity v→, density ρ, and temperature T fields. Starting with the Chern–Simons equations for a free inviscid fluid, the external effects of radiation and the exchange of matter with the strata, as well as diffusion and dissipation, are included. The resulting dynamics is governed by a set of nonlinear differential equations of the first order in time. This defines an initial value problem that can be integrated given the radiation balance of the planet. If the nonlinearities are neglected, the integration can be done in analytic form using standard Green function methods, with small nonlinearities incorporated as perturbative corrections in a consistent way. If the nonlinear approximation is not justified, the problem can be integrated numerically. The analytic expressions as well as the simulations of the linear regime for a continuous range of parameters in the equations are provided, which allows to explore the response of the model to changes of those parameters. In particular, it is observed that a 2.5% reduction in the emissivity of the atmosphere can lead to an increase of 7 °C of the average global temperature.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and 3−α(1<α<2)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

Metric error monitoring for a cleaner record of timing.

A recent line of research has shown that humans can keep track of the direction and magnitude of their timing errors without relying on feedback. But these studies tested temporal error monitoring explicitly by interrogating participants regarding their errors, which might have inadvertently primed the prospective coupling between the first-order timing and second-order metacognitive judgments. The current study utilized an indirect way of testing temporal error awareness while providing a strong objective incentive for maximizing the accuracy of first-order timing performance. In two experiments, participants were asked to maximize the average proximity of their time reproduction to the target by accepting or rejecting their time reproduction depending on the subjective judgment of their proximity to the target time interval. We found that participants more frequently opted out of a trial with a larger distance between their reproductions and the target time interval in both directions, forming a positive quadratic relationship with reproduced time. Resultantly, timing precision was lower in trials that participants opted out of. Our results provide new evidence in support of the temporal error-monitoring performance of human participants. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Information Granulation With Rectangular Information Granules and Its Application in Time-Series Similarity Measurement

Information granules can discover interpretable and meaningful relationships offering a full description for time series. This article presents an information granulation method with rectangular information granules and applies it to time-series similarity measurement. First, the fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$c$</tex-math></inline-formula> -means clustering algorithm transforms the time series and its first-order difference time series to data clusters. With the maximum volume of rectangular information granules viewed as the criterion, the optimal rectangular information granules are formed using the data cluster by the principle of justifiable granularity and the gravitational search algorithm. The time-series similarity is measured by calculating the similarity between the upper and lower bounds of the optimal rectangular information granules built from the time series. Finally, an experiment is performed on a public dataset to verify the feasibility of the proposed method. The result shows that the rectangular information granulation method can capture the change characteristics of time series. The similarity measurement method can effectively evaluate the similarity of the time series belonging to different classes.

A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm.

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

Abstract This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first-order time extrapolation, which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e., the algorithm converges with first order.

Non-Local Seismo-Dynamics: A Fractional Approach

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process.

An efficient hybrid numerical method based on an additive scheme for solving coupled systems of singularly perturbed linear parabolic problems

We construct an efficient hybrid numerical method for solving coupled systems of singularly perturbed linear parabolic problems of reaction‐diffusion type. The discretization of the coupled system is based on the use of an additive or splitting scheme on a uniform mesh in time and a hybrid scheme on a layer‐adapted mesh in space. It is proven that the developed numerical method is uniformly convergent of first order in time and third order in space. The purpose of the additive scheme is to decouple the components of the vector approximate solution at each time step and thus make the computation more efficient. The numerical results confirm the theoretical convergence result and illustrate the efficiency of the proposed strategy.

Математическое моделирование динамики тепловых ударных волн в нелинейных локально-неравновесных средах

We performed a mathematical simulation of heat transfer in a local non-equilibrium medium whose transfer characteristics are functions of the temperature distribution. A homogeneous polynomial of arbitrary degree represents nonlinearities in thermal conductivity and thermal diffusivity. The mathematical model consists of a hyperbolic nonlinear heat transfer wave equation, initial conditions and nonlinear boundary conditions of the second and first kind. To solve this problem, we used a conservative homogeneous finite-difference scheme along the upper time grid line (implicitly). We then used the tridiagonal matrix algorithm of the second order in the spatial variable and of the first order in time to solve the resulting system of linearised algebraic equations. A periodic series of rectangular temperature or heat flux pulses form the boundary conditions of the first and second kind. Computation results reveal ultimate propagation rates of temperature and heat fronts featuring pronounced first-kind discontinuities with attenuating magnitudes. As the process unfolds, the initial pulses heat the region between the boundary and the heat wave front, while the subsequent pulses traverse this region at a higher velocity due to thermal diffusivity being a function of temperature, their fronts "catching up" with the previous fronts, increasing the discontinuity magnitude at the initial pulse front, that is, forming a thermal shock wave front similar to that of a shock wave in gas dynamics. We obtained such thermal shock waves for boundary conditions of both the first and the second kind. We also analysed kinematic and dynamic characteristics of thermal waves

Generalized SAV-Exponential Integrator Schemes for Allen--Cahn Type Gradient Flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.

An Optimal Dead Time Compensator Design for Nonsquare Process with Disturbance Rejection

This work aims to establish an optimal design of the Smith dead-time compensator for a nonsquare multivariable system with a transfer function matrix containing first-order dead time elements. The scheme has a set point tracking controller along with disturbance rejection using an optimal estimator. An inverse signal from the estimator eliminates the disturbance. The proposed method significantly improves the disturbance rejection and performance index compared to the other established methods. The common evolutionary algorithms such as the grey wolf optimization technique and teaching learning based algorithm have been used to tune the controllers.

A multi-splitting method to solve 2D parabolic reaction–diffusion singularly perturbed systems

In this paper we design and analyze a numerical method to solve a type of reaction–diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction–diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal.

A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems

A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of L^2-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the ell ^infty (L^2)- and ell ^2(H^1_0)-norms. For the time increment Delta t, the mesh size h and a conforming finite element space of polynomial degree k in {mathbb {N}}, the convergence order is of O(Delta t^2 + h^k) in the ell ^infty (L^2)cap ell ^2(H^1_0)-norm and of O(Delta t^2 + h^{k+1}) in the ell ^infty (L^2)-norm if the duality argument can be employed. Error estimates of O(Delta t^{3/2}+h^k) in discrete versions of the L^infty (H^1_0)- and H^1(L^2)-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.

Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved.

Invariance analysis and some new exact analytic solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations

In this work, the fractional Lie symmetry method is used to find the exact solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations with the Riemann-Liouville fractional derivative. Time-fractional coupled Drinfeld-Sokolov-Wilson equations are obtained by replacing the first-order time derivative to the fractional derivatives (FD) of order $\alpha$ in the classical Drinfeld-Sokolov-Wilson (DSW) model. Using the fractional Lie symmetry method, the Lie symmetry generators are obtained. With the help of symmetry generators, FCDSW equations are reduced into fractional ordinary differential equations (FODEs) with Erd$\acute{e}$lyi-Kober fractional differential operator. Also, we have obtained the exact solution of FCDSW equations and shown the effects of non-integer order derivative value on the solutions graphically. The effect of fractional order $\alpha$ on the behavior of solutions is studied graphically. Finally, new conservation laws are constructed along with the formal Lagrangian and fractional generalization of Noether operators. It is quite interesting the exact analytic solutions are obtained in explicit form.

Method of System Identification for Air Conditioning Systems in Operation

This study aims to optimize the operation of an air conditioning (AC) system by tracking situational changes due to outside temperature, number of people and computers, and other factors. Therefore, we studied the accurate estimation of system parameters of an AC unit during operation. We modeled the AC system using the first-order plus dead time model and discretized it using the autoregressive with exogenous input model. We developed a technique to estimate the system parameters using Bayesian optimization. Here, the system parameters are values that determine the physical characteristics of the combined air conditioning system and room. Therefore, we determined that there are cases where the characteristics deteriorate after repeated estimation. By solving this problem, we were able to establish a practical system. Keywords—ARX time-series model, Bayesian optimization method, First-order plus dead time, Gaussian process, PID control

Analysis and Prediction of the Trend Features for Teaching Development Based on Knowledge Discovery

The existing research on teaching development of teachers fails to effectively quantify the teaching development trend. This paper deeply mines the evaluation data on the teaching quality of college teachers, before analyzing and predicting the trend features for teaching development of college teachers based on knowledge discovery. Firstly, the knowledge features of the teaching development trend of college teachers were examined. Next, the fluctuation features of the time series on the teaching quality development of college teachers were described based on chaotic time series. In addition, a prediction model for teaching development of college teachers was established for weighted first-order chaotic time series, and used to simulate the nonlinear features of the time series on the teaching quality development of college teachers. The prediction model was proved effective through experiments.