Time Derivative Research Articles

Data fusion based on short-term memory Kalman filtering using intermittent-displacement and acceleration signal with a time-varying bias

Accurate displacement information plays a vital role in various applications from structural health monitoring to damage detection in physical systems. Due to practical reasons, displacement sensing is often performed using sensors that provide less accuracy or a low sampling rate. This study proposes an algorithm for displacement signal estimation that fuses high sampling rate acceleration data (which consists of unknown time-varying bias) with low sampling rate displacement data. In contrast to the present literature, the state vector in the proposed state-space formulation consists of short-term memory of the past realizations of displacement and velocity, and bias terms which allow (i) online estimation of the displacements by avoiding multi-rate Kalman filtering, and (ii) estimate time-varying bias in the measured acceleration signal. The state equations comprise the finite difference approximations of the time derivative of velocity and displacement which act as a moving average filter to provide smooth estimates, thus preventing the need for offline smoothing procedures. By inclusion of bias terms in the state vector, the drifts in displacement and velocity caused due to time-varying bias in measured acceleration are easily eliminated. The proposed method was found to provide accurate estimates of displacement, velocity, and time-varying bias in measured acceleration signals when tested on different types of bias, including but not limited to constant, linear, and parabolic variations. Additional tests on three oscillators subjected to three earthquake ground motions showed significantly better state estimates for stiff oscillators and improved estimates for flexible oscillators. Experimental validation on a base-excited two-story frame further affirms the capability of the method to estimate the hidden bias in the measured acceleration and provide an accurate estimate of displacement.

Stability analysis of an age structured population model with fractional time derivative

In this paper, we analyse the large time behaviour in a fractional nonlinear model of population daynamics with age dependent. We show the existence and uniqueness of the solution by using the method of seperation of variables, and we studied the Ulam-Hyers stability of the model.

Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences

This study presents the first comprehensive numerical simulation of heat and mass transfer in fractal-like mixed convective nanofluid flows. The flow of non-Newtonian nanofluids over flat and oscillating sheets is modelled mathematically, and a finite difference scheme is used to solve this model. The two-stage scheme can tackle fractal and fractal stochastic mathematical models of partial differential equations. The consistency in the mean square is proved, and Fourier series stability analysis is adopted to find stability conditions for fractal stochastic partial differential equation. The scheme is applied to solve the unsteady Casson nanofluid flow over the flat and oscillatory sheet, which affects thermal radiation, heat source, and chemical reaction. The existence of the solution is also provided for the Navier-Stokes equation of the considered flow model using fractal time derivative. The graph illustrates that the proposed fractal technique achieves faster convergence than the Crank-Nicolson approach. Applications in energy systems, materials science, and environmental engineering are just a few of the domains that could benefit from a better understanding of mixed convective nanofluid flows with fractal features, and that is what this research study hopes to accomplish. Scientists and engineers may better develop efficient and environmentally friendly systems by simulating and analyzing these complicated processes with the suggested finite difference technique.

Reconstruction schemes of scalar field models for the Power Law Entropy Corrected Holographic Dark Energy model with Ricci scalar cut-off

In this work, we examine the cosmological characteristics of the Power Law Entropy Corrected Holographic Dark Energy (PLECHDE) model with infrared (IR) cut-off, which is determined by the curvature parameter k, the time derivative of H, and the average radius of the Ricci scalar curvature R, which varies with the Hubble parameter H squared. We obtain the deceleration parameter q and the Equation of State (EoS) parameter of Dark Energy (DE) ωD. Additionally, we derive the Hubble parameter H and the scale factor a expressions as functions of the cosmic time t. Additionally, we examine the limiting scenario that pertains to a flat Dark Dominated Universe. Furthermore, we establish a correspondence between the DE model considered and some scalar fields, in particular the Generalized Chaplygin Gas, the Modified Chaplygin Gas, the Modified Variable Chaplygin Gas, the New Modified Chaplygin Gas, the Viscous Generalized Chaplygin Gas, the Dirac-Born–Infeld, the Yang–Mills, and the Non Linear Electrodynamics scalar field models.

Dynamics and Stability via Thin‐Shell of Approximated Black Holes in f(Q)$f(\mathbb {Q})$ Gravity

AbstractThe current study is devoted to exploring the geometrical configuration of a thin‐shell in the background of symmetric teleparallel gravity. For this purpose, the well‐known cut‐and‐paste approach by matching the inner flat and outer newly calculated class of approximated black hole (BH) solutions in symmetric teleparallel gravity are considered, i.e., uncharged, charged, and anti‐de‐Sitter BHs. The dynamical analysis of thin‐shell configuration by adopting the massive and massless scalar field via Klein‐Gordon's equation of motion is discussed. The effective potential and proper time derivative of shell radius for both massive and massless scalar shells are used to discuss the collapse, expansion, and oscillatory behavior. The stable configuration of thin‐shell is observed through the linearized radial perturbation approach with a phantomlike equation of state, i.e., quintessence, dark energy, and phantom energy. It is noted that stable/unstable behavior of thin‐shell is found after the expected position of the event horizon of an exterior manifold. It is concluded that the stability of a thin‐shell is greater for BH with cosmological constant as compared to the uncharged and charged  BHs.

Error analysis for discontinuous Galerkin method for time‐fractional Burgers' equation

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy.

Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetime

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.

Novel Robust Estimation-Based Control of One-Sided Lipschitz Nonlinear Systems Subject to Output and Input Delays

This paper highlights the design of a controller established on estimated states for one-sided Lipschitz (OSL) nonlinear systems subject to output and input delays. The controller has been devised by involving Luenberger-like estimated states. The stability of the time-delayed nonlinear system is reckoned by assuming a Lyapunov functional for delayed dynamics and for which a delay-range dependent criterion is posed with a delay ranging between known upper and lower bounds. The time derivative of the functional is further exploited with linear matrix inequality (LMI) procedures, and employing Wirtinger’s inequality for the integral terms instead of the traditional and more conservative Jensen’s condition. Moreover, a sufficient and necessary solution is derived for the proposed design by involving the tedious decoupling technique to attain controller and observer gain simultaneously. The proposed methodology validates the observer error stability between observers and states asymptotically. The solution of matrix inequalities was obtained by employing cone-complementary linearization techniques to solve the tiresome constraints through simulation tools by convex optimization. Additionally, a novel scheme of an observer-based controller for the linear counterpart is also derived for one-sided Lipschitz nonlinear systems with multiple delays. Finally, the effectualness of the presented observer-based controller under input and output delays for one-sided Lipschitz nonlinear systems is validated by considering a numerical simulation of mobile systems in Cartesian coordinates.

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

How the spherical modes of gravitational waves can be detected despite only seeing one ray

Abstract The spherical modes of gravitational waves (GWs) have become a major focus of recent detection campaigns due to the additional information they can provide about different properties of the source. However, GW detection is restricted to only detecting one ray and hence it is not obvious how we can extract information about angular properties. In this note, we introduce a new gauge that makes visible GW detection does not only contain information on the second time derivative but also on the angular derivatives of the GW. In particular, we show that the angular derivatives are of the same order as the time derivatives of the wave thus allowing us to constrain the spherical modes. To further illustrate the detection of the spherical modes, we discuss how the evolution of the orbit of the source and thus the phase of the wave depends on them.

The solution of the nonlinear Poisson-Boltzmann equation using lattice-Boltzmann

In this paper we present the solution of the Poisson-Boltzmann equation using the Lattice-Boltzmann method. In order to obtain the solution, we use a redefinition of tensor n°, which is declared as a symmetric tensor whose diagonal components are chosen as the second derivative in time of the first moment of the distribution function, and the components outside of the diagonal give account of the nonlinear terms. The results are presented in two dimensions employing the D2Q9 lattice velocity scheme. We obtain results for the scalar field and its gradient for several kinds of initial conditions.

Dynamical dark energy in light of cosmic distance measurements I: a demonstration using simulated datasets

Abstract We develop methods to extract key dark energy information from cosmic distance measurements including the BAO scales and supernovae luminosity distances. Demonstrated using simulated datasets of the complete DESI, LSST and Roman surveys designed for BAO and SNe distance measurements, we show that using our method, the dynamical behaviour of the energy, pressure, equation of state (with its time derivative) of dark energy and the cosmic deceleration function can all be accurately recovered from high-quality data, which allows for robust diagnostic tests for dark energy models.

Convenient and efficient N-methylation of secondary amines under solvent-free ball milling conditions

In the present work, we report the development of a rapid, efficient, and solvent-free procedure for the N-methylation of secondary amines under mechanochemical conditions. After optimization of the milling parameters, a vibrational ball mill was used to synthesize 26 tertiary N-methylated amine derivatives in a short time of 20 min (30 Hz frequency) and high yields ranging from 78 to 95%. An exception was compounds having a hydroxyl group in their structure, for which a decrease in reaction efficiency was observed. During our research, we investigated alternate reaction selectivity occurring in compounds able to form ring closure products that are 3,4-dihydro-2H-1,3-benzoxazine derivatives instead of N-methylated products. The liquid-assisted grinding technique has been applied using formalin as a methylating agent and sodium triacetoxyborohydride as a reducing agent in a reductive amination reaction.

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.

Deployment of a space tether in a centrifugal force field with alignment to the local vertical

This study is concerned with a small orbital tether of two bodies to be deployed from a spacecraft so that upon completion of the deployment it turns out to be aligned along the local vertical. The bodies of the tether have equal masses, and the thread connecting the bodies is supposed to be massless. The objective of the study is to build a program law of tether length control taking into account the variation of the angular momentum of the tether under the action of the gravitational torque from the central Newtonian field of forces. The deployment mode of the space tether in a centrifugal force field with its alignment at the conclusion of the deployment along the local vertical is studied. To produce centrifugal forces, the tether is pre-spinned about the orbit binormal. The study consists of two steps. The first step involves the construction of a tether length control law that would provide the planned deployment. At this step, use is made of the tether motion equations written in spherical coordinates for the special case of the tether motion in the orbital plane. A numerical simulation of the tether deployment dynamics is carried out at the second step using the constructed program law of tether length control. Hill-Clohessy-Wiltshire’s equations are used as a mathematical model of the tether. They describe the spatial motion of the tether bodies. These equations do not contain the tether length as a variable in explicit form. Therefore, these equations are modified. The tether tension force appearing in these equations is expressed in terms of the program law of tether length change and its two first time derivatives. The novelty of the study consists in the construction of a program control law that allows the tether to be deployed along the local vertical in a single stage. The study used methods of analytical mechanics, numerical methods, and methods developed by the authors. The obtained results make it possible to find the ranges of values of the deployment system parameters allowing a deployment of this type. The errors of the numerical simulation are estimated. The practical significance of the obtained results consists in the possibility of deploying small tethers in orbit with their alignment at the conclusion of the deployment along the local vertical in a single stage with controlling the tether length without the need for further dumping of libratory oscillations.

Effects of base frequency, duty cycle, and waveform repetition on TEM responses: Insights from models of a deep-buried conductor

In transient electromagnetic (TEM) methods, the recorded data for both the magnetic field and its time derivative are influenced by the waveforms of the transmitting current. These waveforms are characterized by parameters such as the durations of the turn-on, steady, and turn-off stages, as well as the base frequency, duty cycle, and waveform repetition. The full-waveform effects encompass all counteracting effects observed in TEM responses that arise from using a waveform other than a step-off or step-on waveform. To address these complexities, a 3D forward-modeling solver was developed for TEM methods, capable of calculating TEM responses while considering realistic waveforms with considerations for base frequency, duty cycle, and waveform repetition. The effects of these parameters on the magnetic field and its time derivative were investigated through numerical experiments using models involving deep-buried horizontal and vertical conductors. The results indicated that as the waveform period increases, there is a significant improvement in the detection and discrimination capability of the magnetic field for perfect conductors with a conductivity of 1000 S/m or higher in the models presented in this study. However, the improvement in the time derivative of the magnetic field is minimal. Interestingly, the improvement in both the magnetic field and its time derivative does not show a consistent trend as the conductor conductivity increases. In addition, a novel equivalence phenomenon was uncovered in both the magnetic field and its time derivative data. The results also suggested that as the number of waveform repetitions increases or the duty cycle decreases, the discrepancies between the recorded TEM responses at the two off-time stages within a single period gradually diminish. However, the influence of the full waveform on the TEM responses becomes more pronounced with an increasing number of waveform repetitions or a decrease in the duty cycle, especially for the magnetic field.

A machine learning approach for entropy due to natural convection flow of a nanofluid under the uniform inclined magnetic field

In this study, entropy generation of natural convection flow of Cu-water nanofluid under the influence of a uniform inclined magnetic field (MF) is firstly modeled by a machine learning approach using the data obtained from a numerical process. The two dimensional, time dependent dimensionless equations are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second order backward differentiation formula (BDF2) in time derivatives. In a set of combined dimensionless problem parameters, numerical simulations are performed. In the obtained data, the inputs are a set of combined dimnesionless problem parameters and the outputs are entropy indicators. Then, global and local entropy indicators are modeled by Neural Network (NN). In view of smaller mean squared error metric results, small number of layers or layer sizes in NN seems enough in smaller data while trilayer NN (TNN) is a better modeler in case of larger data.

Friction tensor for an ellipsoid enclosed in a rarefied gas

This research renders calculations of rotranslational friction and diffusion tensors in a rarefied gas, considering both specular and diffuse reflection off ellipsoidal particles. Calculations were conducted in a spherical polar coordinate system, revealing the dynamic nature of these off-diagonal tensors. The tensor elements were determined analytically, intending to compute the surface integrals. All diagrams, and pictures of two-dimensional surfaces were obtained by the numerical integration process in MATLAB. Additionally, we performed calculations without using the time parameter and time derivatives in the relevant relations. Deviations from the state of spherical symmetry were observed in the results for an ellipsoid and show that the tensors are dynamic quantities. Therefore, it is necessary to provide average values. We established that the non-zero elements of the friction tensor for an ellipsoid represent the directions of linear momentum transfer. Also, the non-zero components of the diffusion tensor signify the extent of fluctuations in relevant momentum transfers in corresponding directions.

Mathematical frameworks for investigating fractional nonlinear coupled Korteweg-de Vries and Burger’s equations

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger’s equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient.