Time-dependent Coefficients Research Articles

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

Well-posedness and decay in a system of hyperbolic and biharmonic-wave equations with variable exponents and weak dampings

In this paper, we consider a coupled system of hyperbolic and biharmonic-wave equations with variable exponents in the damping and coupling terms. In each equation, the damping term is modulated by a time-dependent coefficient a(t) (or b(t)). First, we state and prove a well-posedness theorem of global weak solutions, by exploiting Galerkin’s method and some compactness arguments. Then, using the multiplier method, we establish the decay rates of the solution energy, under suitable assumptions on the time-dependent coefficients and the range of the variable exponents. We end our work with some illustrative examples.

An investigation of the variable coefficients modified KdV equation arising in arterial mechanics by using two expansion techniques

This article investigates the exact soliton solutions to the mKdV equation with time-dependent coefficients using the adapted sine-Gordon and modified simple equation approaches. The variations of the achieved solutions are demonstrated by plotting them with the corresponding solutions of the constant coefficients mKdV equation. Solitary waves with discrete wave profiles are developed from the resultant solution functions when specific values are assigned in place of the variable coefficients. The compatible two- and three-dimensional configurations show that the wave profiles are strongly affected by the variable wave velocity and its associated variable parameters. The analysis and interpretation emphasize the importance of the nonlinear evolution equations with time-varying coefficients, and it might open the purview to a notable domain of research.

Investigation of Synechocystis sp. CPCC 534 Motility during Different Stages of the Growth Period in Active Fluids

The motility behavior of suspended microorganisms plays an essential role in the properties of active fluids. Despite the important progress in our understanding of microorganisms’ motility in recent years, there are still several open questions about the dynamics of cell motility in active suspensions. Of special interest is the relationship between cell motility and age. In this study, cyanobacterium Synechocystis sp. CPCC 534 was used as the model microorganism, and the cell trajectories were tracked for 78 days during the cell growth period. Results showed that the length of cell trajectories had substantially increased from the exponential growth phase to the stationary phase and had declined at the end of the stationary phase. Similar trends were observed for the cells’ mean squared displacement (MSD), the time-dependent diffusion coefficient of cell suspensions, and the cell displacement probability density function (PDF). These results suggest that the cellular age of microorganisms has a significant effect on various metrics of cell motility and, therefore, can impact the transport properties of active suspensions.

Two-dimensional Inverse Boundary Value Problem for a Third-order Pseudo-hyperbolic Equation with an Additional Integral Condition

In this paper we study an inverse boundary value problem with an unknown timedependent coefficient for a third-order pseudo-hyperbolic equation with an additional integral condition. The definition of the classical solution of the problem is given. The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is again made, as a result, a conclusion is made about the solvability of the original inverse problem.

Data-driven estimation of time-dependent solar gain coefficient in a two-zone building with synthetic occupants: Two B-splines integrated grey-box modeling approaches

Solar gains are a significant share of the heat input to the buildings’ interior energy balance. Therefore, it is important to estimate the dynamic details of solar gains precisely for predictive works, such as in model predictive control (MPC). Effective solutions to gauge the dynamic details of solar gains are still lacking. To fill this gap, a data-driven B-splines integrated grey-box modelling technique has been presented recently, and its potential to estimate dynamic solar gains based on the limited size of low-frequency on-site data has been demonstrated in a simplified one-zone building case, with a single level on/off heating input signal. Its robustness is examined in a two-zone building with heating inputs of synthetic occupants, in this study. Two thermal zones are established to mimic day and night zones in a test building, and corresponding synthetic occupancy profiles and thermostat systems have been implemented in each zone. This study shows that the B-spline integrated grey-box model is useful in this more realistic case considering the impacts of occupants, but demands more advanced modellings. Specifically, two updated solutions on B-splines integrated grey-box modelling are offered, although challenges in the model selection are noted, mainly attributed to the low statistical significance of solar gain, which had fortunately almost no impact on the effectiveness of the proposed techniques to characterize solar gain dynamics. The reported challenge could be further investigated in the future based on more case studies.

Effectiveness of the Coronavirus Disease 2019 Bivalent Vaccine.

The purpose of this study was to evaluate whether a bivalent coronavirus disease 2019 (COVID-19) vaccine protects against COVID-19. The study included employees of Cleveland Clinic in employment when the bivalent COVID-19 vaccine first became available. Cumulative incidence of COVID-19 over the following 26 weeks was examined. Protection provided by vaccination (analyzed as a time-dependent covariate) was evaluated using Cox proportional hazards regression, with change in dominant circulating lineages over time accounted for by time-dependent coefficients. The analysis was adjusted for the pandemic phase when the last prior COVID-19 episode occurred and the number of prior vaccine doses. Among 51 017 employees, COVID-19 occurred in 4424 (8.7%) during the study. In multivariable analysis, the bivalent-vaccinated state was associated with lower risk of COVID-19 during the BA.4/5-dominant (hazard ratio, 0.71 [95% confidence interval, .63-79]) and the BQ-dominant (0.80 [.69-.94]) phases, but decreased risk was not found during the XBB-dominant phase (0.96 [.82-.1.12]). The estimated vaccine effectiveness was 29% (95% confidence interval, 21%-37%), 20% (6%-31%), and 4% (-12% to 18%), during the BA.4/5-, BQ-, and XBB-dominant phases, respectively. The risk of COVID-19 also increased with time since the most recent prior COVID-19 episode and with the number of vaccine doses previously received. The bivalent COVID-19 vaccine given to working-aged adults afforded modest protection overall against COVID-19 while the BA.4/5 lineages were the dominant circulating strains, afforded less protection when the BQ lineages were dominant, and effectiveness was not demonstrated when the XBB lineages were dominant.

Inverse Problem of Determining the Unknown Coefficient in the Beam Vibration Equation in an Infinite Domain

For the equation of transverse vibrations of a homogeneous beam, we consider the direct initial value problem in an infinite domain and study the inverse problem of determining the time-dependent beam stiffness coefficient. The solution of the direct problem is given with the help of fundamental solutions, and the existence and uniqueness of this solution are proved. Stability estimates are obtained for the solution of the inverse problem. Existence and uniqueness theorems for the solution of the inverse problem are proved using Banach’s contraction mapping principle.

Exact soliton solutions of a (2+1)-dimensional time-modulated nonlinear Schrödinger equation with cubic-quintic nonlinearity

In the present paper, based on the wave ansatz method, exact bright and dark solitons of a (2+1)-dimensional time variable coefficients nonlinear Schrödinger equation with cubic-quintic nonlinearity term are presented. This model can be used to describe the propagation of optical pulse in inhomogeneous media. These obtained solitons exhibit nonautonomous properties due to the amplitude and velocity of them are time-dependent coefficients. At this time, Tangent and Secant function solutions are also reported.

Modeling of pertinent parameters influence on the time dependent mass transfer coefficient of particulate matter under the sink effect

One of the most crucial evaluation metrics for the performance of particle sink purification technology is the time-dependent mass transfer coefficient (TDMTC). Therefore, it is very helpful for designers and developers to accurately describe the functional relationship between different influence parameters and the TDMTC. In this paper, four influence parameters (the applied voltage (V), interelectrode distance (dc), porosity (P), and shape (n) of the collecting electrode) were considered, and then non-linear multiple regression (NLMR) and multi-gene genetic programming (MGGP) methods were used to establish prediction models of the TDMTC. Results showed that V and n were the most significant factors, followed by dc and P. Both multi-factor models could accurately predict the TDMTC under the sink effect with a maximum prediction error of 20% and 15%, respectively. Moreover, for particulate matter (PM) with different size fractions, MGGP models could improve the prediction accuracy by 5–10% compared to NLMR models.

The slightly compressible Brinkman-Forcheimer equations and its incompressible approximation

This paper investigates the slightly compressible Brinkman–Forchheimer equations (BFEs): ∂tu−Δxu+∇xp+f(u)=g with D−1(t)∂tp+divu=0 in a bounded 3D domain with Dirichlet boundary conditions. The features of this problem is that, formally, this system is partially dissipative, and will recover to incompressible BFEs when the time-dependent coefficient D(t) goes to infinity as t→∞. The well-posedness and dissipation are obtained in the natural energy space. In addition, our result reveals that the velocity field u can be approximated by the solution uin of incompressible BFEs in L2 provided that ∂tpin is uniform bounded in L2, here pin is the pressure of incompressible BFEs.

On the Emergence of Quantum Boltzmann Fluctuation Dynamics near a Bose–Einstein Condensate

In this work, we study the quantum fluctuation dynamics in a Bose gas on a torus $$\Lambda =(L{\mathbb {T}})^3$$ that exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with density $$N\gg 1$$ surrounded by thermal fluctuations with density 1, we assume that the system dynamics is generated by a Hamiltonian with mean-field scaling. We derive a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics, with rigorous error control. Given a quasifree initial state, we determine the time evolution of the centered correlation functions $$\langle a\rangle $$ , $$\langle aa\rangle -\langle a\rangle ^2$$ , $$\langle a^+a\rangle -|\langle a\rangle |^2$$ at mesoscopic time scales $$t\sim \lambda ^{-2}$$ , where $$0<\lambda \ll 1$$ is the coupling constant determining the HFB interaction, and a, $$a^+$$ denote annihilation and creation operators. While the BEC and the HFB fluctuations both evolve at a microscopic time scale $$t\sim 1$$ , the Boltzmann dynamics is much slower, by a factor $$\lambda ^2$$ . For large but finite N, we consider both the case of fixed system size $$L\sim 1$$ , and the case $$L\sim \lambda ^{-2-}$$ . In the case $$L\sim 1$$ , we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in $${\mathbb {Q}}$$ ; this phenomenon is reminiscent of the Talbot effect. For the case $$L\sim \lambda ^{-2-}$$ , we prove that the collision operator is well approximated by the expression predicted in the literature. In either of those cases, we have $$\lambda \sim \Big (\frac{\log \log N}{\log N}\Big )^{\alpha }$$ , for different values of $$\alpha >0$$ .

Information overload: How hot topics distract from news--COVID-19 spread in the US

Information dissemination and the associated change of individual behavior can significantly slow the spread of an epidemic. However, major social events which attract public attention will disturb information spread and affect epidemic transmission in ways that have not been readily quantified. We investigate the interplay between disease spreading and disease-related information dissemination in a two-layer network. We employ the SIR-UAU model with a time dependent coefficient to denote information dissemination. We found that major social events are equivalent to perturbations of information dissemination in certain time intervals and will consequently weaken the effect of information dissemination, and increase prevalence of infection. Our simulation results agree well with the trends of real-world data sets. We find two specific major events explain the trend observed from the coronavirus epidemic in the United States: the online propaganda and international agenda setting of Donald Trump early in 2020 and the 2020 U.S. Presidential Election.

Coefficient identification in a SIS fractional-order modelling of economic losses in the propagation of COVID-19

A fractional-order SIS (Susceptible–Infectious–Susceptible) model with time-dependent coefficients is used to analyse some effects of the novel coronavirus 2019 (COVID-19). This generalized model is suitable for describing the COVID dynamics since it does not presume permanent immunity after contagion. The fractional derivative activates the memory property of the dynamics of the susceptible and infectious population time series. A coefficient identification inverse problem is posed, which consists of reconstructing the time-varying transmission and recovery rates, which are of paramount importance in practice for both medics and politicians.The inverse problem is reduced to a minimization problem, which is solved in a least squares sense. The iterative predictor–corrector algorithm reconstructs the time-dependent parameters in a piecewise-linear fashion. The economic losses emerging from social distancing using the calibrated model are also discussed. A comparison between the results obtained by the classical model and the fractional-order model is included, which is validated by ample tests with synthetic and real data.

The effect of sintering time on the tribological properties of automotive brake pads

In this study, the effect of sintering temperature on the physical, chemical and tribological effects of sintered automotive friction materials was investigated. Using the powder metallurgy method, 3 different samples were produced with the codes A1, A6 and A11. Mixing, pressing and sintering processes were applied respectively as production steps. Mixing time is 90 minutes, speed is 15 rpm. The pressing pressure is 250 MPa and the duration is 1 min. The sintering temperature was 800 °C and the sintering time was 1, 6 and 11 hours, respectively. The sintering process was carried out in the sintering mold using Argon gas. As the sintering time increased, it was observed that there was an increase in the time-dependent friction coefficients of the samples. Increasing the sintering time also increased the density of the samples. As the sintering time increased, the gap in the sample decreased, as it provided better bonding of the dustsers.

On solvability of the inverse problem for the one-dimensional parabolic&#x0D; equation with unknown time-dependent coefficient under integral observation

In this article, we study the inverse problem of determination of time-dependent coefficient in the parabolic equation. We prove existence and uniqueness theorem for the solution of the inverse problem with nonlocal boundary conditions and integral observation. The proof is based on a priori estimates obtained in this article and the results on solvability of corresponding direct problem for the equarion under consideration.

A Time-Nonlocal Inverse Problem for the Beam Vibration Equation with an Integral Condition

We study the direct problem for transverse vibrations of a homogeneous beam of finite length with time-nonlocal conditions and obtain necessary and sufficient conditions for the existence of its solution. For the direct problem, the inverse problem of determining the time-dependent coefficients of the lower derivative and the right-hand side in the equation is studied. The existence and uniqueness of the solution of the inverse problem are proved. The solution is based on separation of variables, which is used to reduce the problems to an integral equation and a system of integral equations.

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω⊂Rn in the unknown u=u(x,t), Dtν0(ϱ0u)−Dtν1(ϱ1u)−L1u−∫0tK(t−s)L2u(x,s)ds=f(x,t)+g(u),0&lt;ν1&lt;ν0&lt;1, where Dtνi are the Caputo fractional derivatives, ϱi=ϱi(x,t) with ϱ0≥μ0&gt;0, and Li are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator Dtν0(ϱ0u)−Dtν1(ϱ1u), which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν0 and ν1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ0 and ϱ1 and the nonlinearity g(u), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

Analysis of SIRVI model with time dependent coefficients and the effect of vaccination on the transmission rate and COVID-19 epidemic waves.

COVID-19 epidemic models with constant transmission rate cannot capture the patterns of the infection data in the presence of pharmaceutical and non-pharmaceutical interventions during a pandemic. Because of this, a new modification of SIR model that contain the vaccination compartment with time dependent coefficients and weak/loss-immunity is explored. Literature review confirms that the effect of vaccination on the time dependent transmission rate is still an open problem. This study answers this open problem. In this study, we first prove the well-posedness and investigate the model dynamics to show their continuous dependence on the model parameters. We then provide an algorithm to derive the time-dependent transmission function for the epidemiologic model and the data of the infected cases. The derived coupled nonlinear differential equations show the effect of vaccination on the transmission rate. Unlike previous studies, we first filter the published data and solve the nonlinear coupled differential equations using the finite difference technique, where the coefficient of the coupled nonlinear differential equations is a function of given data. We then show that time-dependent transmission function can be represented by linear combinations of Gaussian radial base function. We then validate the prediction of our models using numerical simulations, where we used the published data of COVID-19 confirmed cases by the Ministries of Health in Saudi Arabia and Poland. Finally, the numerical solutions of a SIRVI model with time dependent transmission rate show that the waves for currently active cases are in good agreement with the data of Saudi Arabia and Poland.

Inference of main ion particle transport coefficients with experimentally constrained neutral ionization during edge localized mode recovery on DIII-D

The plasma and neutral density dynamics after an edge localized mode are investigated and utilized to infer the plasma transport coefficients for the density pedestal. The Lyman-Alpha Measurement Apparatus (LLAMA) diagnostic provides sub-millisecond profile measurements of the ionization and neutral density and shows significant poloidal asymmetries in both. Exploiting the absolute calibration of the LLAMA diagnostic allows quantitative comparison to the electron and main ion density profiles determined by charge-exchange recombination, Thomson scattering and interferometry. Separation of diffusion and convection contributions to the density pedestal transport are investigated through flux gradient methods and time-dependent forward modeling with Bayesian inference by adaptation of the Aurora transport code and IMPRAD framework to main ion particle transport. Both methods suggest time-dependent transport coefficients and are consistent with an inward particle pinch on the order of 1 m s−1 and diffusion coefficient of 0.05 m2 s−1 in the steep density gradient region of the pedestal. While it is possible to recreate the experimentally observed phenomena with no pinch in the pedestal, low diffusion in the core and high outward convection in the near scrape-off layer are required without an inward pedestal pinch.