Integer Time Research Articles

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

Enhancing Integer Time Series Model Estimations through Neural Network-Based Fuzzy Time Series Analysis

This investigation explores the effects of applying fuzzy time series (FTSs) based on neural network models for estimating a variety of spectral functions in integer time series models. The focus is particularly on the skew integer autoregressive of order one (NSINAR(1)) model. To support this estimation, a dataset consisting of NSINAR(1) realizations with a sample size of n = 1000 is created. These input values are then subjected to fuzzification via fuzzy logic. The prowess of artificial neural networks in pinpointing fuzzy relationships is harnessed to improve prediction accuracy by generating output values. The study meticulously analyzes the enhancement in smoothing of spectral function estimators for NSINAR(1) by utilizing both input and output values. The effectiveness of the output value estimates is evaluated by comparing them to input value estimates using a mean-squared error (MSE) analysis, which shows how much better the output value estimates perform.

A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator

In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)∈(1,2) and space-fractional order s∈(0,1), where v=u′(t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O(τ2+h2), different from the common L1,2−α schemes with convergence rate O(τ2,3−α(x,t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method.

A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model

This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.

Unconditionally Stable System Incorporated Factorization-Splitting Algorithm for Blackout Re-Entry Vehicle

A high-temperature plasma sheath is generated on the surface of the re-entry vehicle through the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. Such a condition results in the forming of a blackout which significantly affects the communication components. To analyze the re-entry vehicle at the hypersonic speed, an unconditionally stable system incorporated factorization-splitting (SIFS) algorithm is proposed in finite-difference time-domain (FDTD) grids. The proposed algorithm shows advantages in the entire performance by simplifying the update implementation in multi-scale problems. The plasma sheath is analyzed based on the modified auxiliary difference equation (ADE) method according to the integer time step scheme in the unconditionally stable algorithm. Higher order perfectly matched layer (PML) formulation is modified to simulate open region problems. Numerical examples are carried out to demonstrate the performance of the algorithm from the aspects of target characteristics and antenna model. From resultants, it can be concluded that the proposed algorithm shows considerable accuracy, efficiency, and absorption during the simulation. Meanwhile, plasma sheath significantly affects the communication and detection of the re-entry vehicle.

Model for Reinvestment Policy in Risk-Free Assets with Various Maturities

Abstract Logistic tasks are aimed at the optimal distribution of material, energy, financial and human resources. This research has a narrow field aimed at optimal management of financial resources and their redistribution. Specifically, a reinvestment policy model is derived by maximizing the profit of a business entity. Reinvestment is done with risk-free assets, but they have different maturity periods. This makes it difficult to assess the optimal investment strategy, as reinvestment can be done at the end of the maturity period. This study develops a model for a dynamic control process, which leads to the formalization of a discrete integer time optimization problem. Its solution gives a sequence of investments and a total optimal return. The solution to the problem is illustrated in an EXCEL environment. The added value of this study stems from the formalization and quantification of the model for the reinvestment strategy in the optimization problem.

Energy-conserving interface dynamics with asynchronous direct time integration employing arbitrary time steps

The derivation and implementation of an asynchronous direct time integration scheme for domain-decomposed finite element models is presented. To maximize clarity in the description of the proposed asynchronous integration, the scheme is restricted to the linear-elastic stress wave propagation case. The proposed method allows the integration of individual subdomains with independent time steps. There is no requirement for an integer time steps ratio of the interacting domains while maintaining zero interface energy. The subdomains are connected by the condition of the continuity of the acceleration field at the interface. In addition, the a posteriori technique is applied to satisfy the continuity of the displacement and velocity fields. Another important contribution of this paper lies in the description of the implementation — we offer the reader a general description of the implementation of the case of any number of subdomains with any number of constraints between them, while the basics of the algorithm are explained on a single domain pair. The functionality of the asynchronous integrator is verified by solving selected problems and comparing with analytical solutions and experimental measurements obtained using a Split Hopkinson pressure bar setup.

Using Thermal Imaging to Track Cellulitis.

Cellulitis is a common soft tissue infection and a major cause of morbidity. The diagnosis is based almost exclusively on clinical history and physical exam. To improve the diagnosis of cellulitis, we used a thermal camera to track how skin temperature of the affected area changed during a hospital stay for patients with cellulitis. We recruited 120 patients admitted with a diagnosis of cellulitis. Daily thermal images of the affected limb were taken. Temperature intensity and area were analyzed from the images. Highest daily body temperature and antibiotics administered were also collected.We estimated a longitudinal linear mixed-effects model with a random intercept for the affected body area. All observations on a given day were included, and we used an integer time indicator indexed to the initial day (ie, t = 1 for the first day the patient was observed, etc.). We then analyzed the effect of this time trend on both severity (ie, normalized temperature) and scale (ie, area of skin with elevated temperature). We analyzed thermal images from the 41 patients with a confirmed case of cellulitis who had at least 3 days of photos. For each day that the patient was observed, the severity decreased by 1.63 (95% CI, -13.45 to 10.32) units on average, and the scale decreased by 0.63 (95% CI, -1.08 to -0.17) points on average. Also, patients' body temperatures decreased by 0.28°F each day (95% CI, -0.40 to -0.17). Thermal imaging could be used to help diagnose cellulitis and track clinical progress.

Networks with correlated edge processes

Abstract This article proposes methods to model non-stationary temporal graph processes motivated by a hospital interaction data set. This corresponds to modelling the observation of edge variables indicating interactions between pairs of nodes exhibiting dependence and evolution in time over interactions. This article thus blends (integer) time series models with flexible static network models to produce models of temporal graph data, and statistical fitting procedures for time-varying interaction data. We illustrate the power of our proposed fitting method by analysing a hospital contact network, and this shows the challenge in modelling and inferring correlation between a large number of variables.

Efficient monolithic immersed boundary projection method for incompressible flows with heat transfer

Efficient monolithic immersed boundary projection methods (MIBPMs) with staggered time discretization have been proposed for incompressible viscous flows with heat transfer. The main idea is to use a two-step approximate lower-upper decomposition technique to decouple the momentum and energy equations, including immersed boundary forcing. The momentum and energy forcing are treated as Lagrangian multipliers to impose divergence-free constraints and no-slip conditions at the immersed boundary surfaces. A staggered time discretization is applied with the Crank-Nicolson scheme to decouple the temperature and velocities, which means that the velocity fields are described at integer time levels (n+1), while the temperature fields are described at half-integer time levels (n+1/2). To investigate the effect of forcing schemes in monolithic formulation, several MIBPM variants based on forcing schemes are formulated and evaluated numerically. The proposed MIBPM presents an accurate imposition of no-slip conditions on the immersed boundary surface and exhibits good stability for two-dimensional forced and natural convection problems. Further, simulations with the proposed MIBPM are implemented for the three-dimensional natural convection problem. Numerical simulation results for single- and multi-particle sedimentation demonstrate the robustness of the proposed method for complex heat transfer flows over moving objects.

Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

<p style='text-indent:20px;'>For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis.</p>

Numerical Simulations of the Oscillating Second-Grade Fluid through a Rectangular Cross Duct with Fractional Constitution Relationship

An oscillating second-grade fluid through a rectangular cross duct is studied. A traditional integer time derivative in the kinematic tensors is substituted by a fractional operator that considers the memory characteristics. To treat the fractional governing equation, an analytical method was obtained. To analyze the impact of the parameters more intuitively, the difference method was applied to determine the numerical expression and draw with the help of computer simulation. To reduce the cost of the amount of computation and storage, a fast scheme was proposed, one which can greatly improve the calculation speed. To verify the correctness of the difference scheme, the contrast between the numerical expression and the exact expression—constructed by introducing a source term—was given and the superiority of the fast scheme is discussed. Furthermore, the influences of the involved parameters, including the parameter of retardation time, fractional parameter, magnetic parameter, and oscillatory frequency parameter, on the distributions of velocity and shear force at the wall surface with oscillatory flow are analyzed in detail.

A Priori Estimates for the Solution of an Initial Boundary Value Problem of Fluid Flow through Fractured Porous Media

The paper studies a model of fluid flow in a fractured porous medium in which fractures are distributed uniformly over the volume. This model includes a nonlinear equation containing several terms with fractional derivatives in the sense of Caputo of order belonging to the interval 1,2. The relevance of studying this problem is determined by its practical significance in the oil industry, since most of the world’s oil reserves are in these types of reservoirs. The uniqueness of the solution to the problem in a differential form and its dependence on the initial data and the right-hand side of the equation is proved. A numerical method is proposed based on the use of the finite difference approximation for integer and fractional time derivatives and the finite element method in the spatial direction. A change of variables is introduced to reduce the order of the fractional derivatives. Furthermore, the fractional derivative is approximated by using the L1-method. The stability and convergence of the proposed numerical method are rigorously proved. The theoretical order of convergence is confirmed by the results of numerical tests for a problem of fluid flow in fractured porous media with a known exact solution.

Stationary states of the one-dimensional discrete-time facilitated symmetric exclusion process

We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on Z. In this model, all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site, we choose one randomly to succeed. The ETIS states are qualitatively different for densities ρ < 1/2, ρ = 1/2, and 1/2 < ρ < 1, but in each density region, we find states that may be grouped into families, each of which is in natural correspondence with the set of all ergodic measures on {0,1}Z. For ρ < 1/2, there is one such family, containing all the ergodic states in which the probability of two adjacent occupied sites is zero. For ρ = 1/2, there are two families, in which configurations translate to the left and right, respectively, with constant speed 2. For the high density case, there is a continuum of families. We show that all ETIS states at densities ρ ≤ 1/2 belong to these families and conjecture that also at high density there are no other ETIS states. We also study the possible ETIS states that might occur if the conjecture fails.

Exponential and gamma form for tail expansions of first-passage distributions in semi-markov processes

AbstractWe consider residue expansions for survival and density/mass functions of first-passage distributions in finite-state semi-Markov processes (SMPs) in continuous and integer time. Conditions are given which guarantee that the residue expansions for these functions have a dominant exponential/geometric term. The key condition assumes that the relevant states for first passage contain an irreducible class, thus ensuring the same sort of dominant exponential/geometric terms as one gets for phase-type distributions in Markov processes. Essentially, the presence of an irreducible class along with some other conditions ensures that the boundary singularity b for the moment generating function (MGF) of the first-passage-time distribution is a simple pole. In the continuous-time setting we prove that b is a dominant pole, in that the MGF has no other pole on the vertical line $\{\text{Re}(s)=b\}.$ In integer time we prove that b is dominant if all holding-time mass functions for the SMP are aperiodic and non-degenerate. The expansions and pole characterisations address first passage to a single new state or a subset of new states, and first return to the starting state. Numerical examples demonstrate that the residue expansions are considerably more accurate than saddlepoint approximations and can provide a substitute for exact computation above the 75th percentile.

Optimizing reachability sets in temporal graphs by delaying

A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. We introduce control mechanisms for reachability sets that are based on two natural operations of delaying. The first operation, termed merging, is global and batches together consecutive time labels into a single time label in the whole network simultaneously. The second, imposes independent delays on the time labels of every edge of the graph. We provide a thorough investigation of the computational complexity of different objectives related to reachability sets when these operations are used.

An optimal algorithm for 2-bounded delay buffer management with lookahead

The bounded delay buffer management problem, which was proposed by Kesselman et al. (STOC 2001 and SIAM Journal on Computing 33(3), 2004), is an online problem focusing on buffer management of a switch supporting Quality of Service (QoS). The problem definition is as follows: Packets arrive to a buffer over time and each packet is specified by the release time, deadline and value. An algorithm can transmit at most one packet from the buffer at each integer time and can gain its value as the profit if transmitting the packet by its deadline after its release time. The objective of this problem is to maximize the gained profit. We say that an instance of the problem is s-bounded if for any packet, an algorithm has at most s chances to transmit it. For any s≥2, Hajek (CISS 2001) showed that the competitive ratio of any deterministic algorithm is at least (1+5)/2≥1.618. Recently, Veselý et al. (SODA 2019) designed an online algorithm matching the lower bound.Böhm et al. (ISAAC 2016 and Theoretical Computer Science, 2019) introduced the lookahead ability to an online algorithm. At a time t, the algorithm obtains information about packets arriving at time t+1, and showed that for s=2, there is an algorithm which achieves the competitive ratio of (−1+13)/2≤1.303. Also, they showed that the competitive ratio of any deterministic algorithm is at least (1+17)/4≥1.280. In this paper, for the 2-bounded model with lookahead, we design an algorithm with a matching competitive ratio of (1+17)/4.

Implicit Approximate Crank–Nicolson Theory for Anisotropic Ferrite Structure Simulation with Enhanced Absorption

AbstractTo simulate ferrite structures with anisotropic characteristics efficiently, an unconditionally stable implementation is proposed by incorporating the approximate Crank–Nicolson (CN) theory and a complex‐frequency‐shifted perfectly matched layer (CFS‐PML) formulation in the finite‐difference time‐domain lattice. More specifically, the proposed implementation combines the CN approximate factorization splitting procedure with bilinear transformation in the z‐domain within the CFS‐PML regions. To simulate the unique anisotropic characteristic of the ferrite structure, the auxiliary differential equation (ADE) method, which is modified at the integer time step, is introduced during the formulation. Numerical examples are presented to demonstrate the effectiveness and efficiency theoretically and experimentally. According to the results, it can be concluded that the proposed scheme takes full advantage of the approximate CN theory, CFS‐PML formulation, and modified ADE method in terms of the considerable computational accuracy, outstanding absorbing performance, and remarkable efficiency. Moreover, it can be demonstrated that the proposed algorithm can maintain its stability with the dissatisfaction of the Courant–Friedrichs–Lewy condition.

Dynamical distributed controller for the synchronization problem of integer and fractional order partial differential equation systems

In this work we present a methodology to design dynamical distributed controllers for the synchronization problem of systems governed by partial differential equations of integer and fractional order. We consider fractional systems whose space derivatives are of integer order and time derivatives are of fractional commensurate order. The methodology is based on finding canonical forms by means of a change of variable, such that a distributed controller can be designed in a natural way in the form of a chain of integrators. To study the stability of the integer order closed loop system, we propose to use the spectral and semi-group theory for infinite dimensional Hilbert spaces. On the other hand, the stability criterion previously established by Matignon is used for the stability analysis of the fractional order case. Additionally, we tackle the synchronization problem of multiple systems, which is reduced to a problem of generalized multi-synchronization. To illustrate the effectiveness of the proposed methodology, examples and numerical results are given.

Variational Discrete Action Theory.

Here we propose the variational discrete action theory (VDAT) to study the ground state properties of quantum many-body Hamiltonians. VDAT is a variational theory based on the sequential product density matrix (SPD) ansatz, characterized by an integer N, which monotonically approaches the exact solution with increasing N. To evaluate the SPD, we introduce a discrete action and a corresponding integer time Green's function. We use VDAT to exactly evaluate the SPD in two canonical models of interacting electrons: the Anderson impurity model and the d=∞ Hubbard model. For the latter, we evaluate N=2-4, where N=2 recovers the Gutzwiller approximation (GA), and we show that N=3, which exactly evaluates the Gutzwiller-Baeriswyl wave function, provides a truly minimal yet precise description of Mott physics with a cost similar to that of the GA. VDAT is a flexible theory for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework.