Time-dependent Solutions Research Articles

Quantum oscillons may be long-lived

Hertzberg has constructed a quantum oscillon that decays into pairs of relativistic mesons with a power much greater than the radiation from classical oscillon decay. This result is often construed as a proof that quantum oscillons decay quickly, and so are inconsequential. We apply a construction similar to Hertzberg’s to the quantum kink. Again it leads to a rapid decay via the emission of relativistic mesons. However, we find that this is the decay of a squeezed kink state to a stable kink state, and so it does not imply that the quantum kink is unstable. We then consider a time-dependent solution, which may be an oscillon, and we see that the argument proceeds identically.

Electrodynamic suspension of particles with finite electrical conductivity

The electrodynamic suspension of micrometric inertialess particles with finite electrical conductivity in the gap between two parallel horizontal electrodes subjected to a DC potential difference is analyzed in a collision-dominated regime in which the mean free path between particle collisions is small compared with the interelectrode distance. Operation with a constant number of suspended particles and with an excess of particles deposited on the lower electrode are modelled. Stationary solutions in a laterally unbounded gap are computed and their linear stability is analyzed. In both cases, these solutions become unstable when the number of suspended particles increases, and they transition to time dependent solutions. Numerical computations show that particle and charge accumulations develop at the lower electrode and eventually become electrohydrodynamic plumes rising across the gap. The electric current carried by the suspended particles from the lower to the upper electrode is a rapidly increasing function of the particle conductivity.

Laplace-domain BEM for dynamic analysis of size-dependent couple stress elasticity problems

The size-dependent elastodynamic plane problems are described by a consistent couple-stress theory (C-CST) in this paper. The time-domain basic equations are then transformed to frequency domain by the Laplace transform method. The Laplace-domain boundary integral equations (BIEs) together with the fundamental solutions are derived. Then, these BIEs are numerically solved by a collocation method in conjunction with the numerical treatment of singular integrals. The time-dependent solutions are obtained by the Durbin's method for the inverse Laplace-transform. Some typical size-dependent elastodynamic problems are numerically studied by the developed boundary element method (BEM). Good agreements are observed between the present results and the corresponding analytical or numerical solutions, which validates the high accuracy of the present BEM.

Dynamics of fixed-volume pinned films – dealing with a non-self-adjoint thin-film problem

The use of thin liquid films has expanded beyond lubrication and coatings, and into applications in actuators and adaptive optical elements. In contrast to their predecessors, whose dynamics can be typically captured by modelling infinite or periodic films, these applications are characterized by a finite amount of liquid in an impermeable domain. The global mass conservation constraint, together with common boundary conditions (e.g. pinning), create quantitatively and qualitatively different dynamics than those of infinite films. Mathematically, this manifests itself as a non-self-adjoint problem. This work presents a combined theoretical and experimental study for this problem. We provide a time-dependent closed-form analytical solution for the linearized non-self-adjoint system that arises from these boundary conditions. We highlight that, in contrast to self-adjoint problems, here, special care should be given to deriving the adjoint problem to reconstruct the solution based on the eigenfunctions properly. We compare these solutions with those obtained for permeable and periodic boundary conditions, representing common models for self-adjoint thin-film problems. We show that, while the initial dynamics is nearly identical, the boundary conditions eventually affect the film deformation as well as its response time. To experimentally illustrate the dynamics and to validate the theoretical model, we fabricated an experimental set-up that subjects a thin liquid film to a prescribed normal force distribution through dielectrophoresis, and used high-frame-rate digital holography to measure the film deformation in real time. The experiments agree well with the model and confirm that confined films exhibit a different behaviour which could not be predicted by existing models.

Photothermal enhanced salinity-gradient osmotic energy conversion for sustainable power generation with phase change material

Salinity-gradient osmotic energy conversion is one of the key energy-effective strategies to alleviate global power scarcity and is highly dependent on the saline solution temperature. In this study, a solar thermal boosted salinity-gradient osmotic energy conversion with phase change thermal storage is proposed. Phase change materials (PCMs) are integrated into a solar-driven salinity-gradient osmotic energy conversion system to overcome intermittent solar illumination during the entire diurnal cycle. A graphene oxide membrane (GOM)-based experiment is conducted to study the time-dependent solution temperature, current, and power density with three thermal management materials: Cu/PCM composite, pure PCM, and a reference case without PCM in one diurnal cycle, as well as their periodic stability. The solution temperature can be effectively improved by the solar thermal effect. The utilized PCMs, which absorb solar heat during the daytime and release latent heat at nighttime, can effectively maintain the solution temperature to suppress the power decline at night, thereby improving the power generation capacity of the osmotic system during the entire cycle. The combined interaction mechanism of temperature and solution evaporation on the solution concentration, ion diffusivity, ion conductance, ion selectivity, and power density is revealed. The contributions of ion diffusivity, ion conductance, ion selectivity and solution concentration on the power density are obtained. In an illumination cycle, the power density undergoes three processes of growth stage (0 < t < 5 h), high power output stage (5 < t < 13h), and decline stage (13 < t < 23 h). In addition, the present design also demonstrates a consistent solution temperature and short-current stability during successive cycles.

Approximate analytical solution of a (V,m,h) reduced system for backpropagating action potentials in sparsely excitable dendrites

We derive an approximate analytical solution of a nonlinear cable equation describing the backpropagation of action potentials in sparsely excitable dendrites with clusters of transiently activating, TTX-sensitive Na+ channels of low density, discretely distributed as point sources of transmembrane current along a continuous (non-segmented) passive cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na+ ion channels, included known cumulative inactivation kinetics observed at the microscopic level. In such a reduced third-order system, the ‘recovery’ variable is an electrogenic sodium-pump and/or a Na+-Ca2+ exchanger imbedded in the passive membrane, and a high leakage conductance stabilizes the system. A nonlinear cable equation was used to investigate back-propagation and repetitive activity of action potentials, exhibiting characteristics of the modified Hodgkin-Huxley kinetics (in the presence of suprathreshold input). In particular, a time-dependent analytical solution was obtained through a perturbation expansion of the non-dimensional membrane potential (Φ) for all voltage dependent terms including the voltage dependent Na+ activation μ) and state-dependent inactivation (η) gating variables and then solving the resulting system of integral equations. It was shown that back-propagating action potentials attenuate in amplitude with the frequency following experimental findings and that the discrete and low-density distributions of transient Na+ channels along the cable structure contribute significantly to their discharge patterns. A major significance of integrative modelling is the provision of a continuous description of the non-dimensional membrane potential (Φ) as a function of position.

The impact of time aggregation and travel time models on time-dependent routing solutions

Traffic and congestion have a big impact on the performance of transportation systems. Travel time models are required to calculate trip durations and arrival times when traffic information is available. These models rely on the availability of detailed information about the traffic state. With the growing availability of onboard devices, we can now capture very precise data with a very high frequency. The challenge is now to efficiently use these data to solve routing problems and evaluate routing solutions. A key question that emerges is how to determine the best compromise between a huge amount of very precise data and a smaller volume of aggregated data. In this article, we analyze the impact of time aggregation on the performance of the main travel time models, namely the link travel model (LTM), the flow speed model (FSM) and the smoothed travel time model (STTM). We also analyze the impact of different time aggregation levels on these models. Our results show that all models share similar performance, particularly with large intervals. Finally, we show that the LTM largely respects the FIFO property, which is an important hypothesis for routing algorithms.

Instability analysis of double diffusive convection under time dependent solute boundary conditions in the presence of internal heat generator

The present study provides heat and mass transport of double-diffusive convective flow of Newtonian fluid with the presence of internal heat source. Here, we have considered two infinite horizontal plates apart from a constant distance under the solutal modulated boundary condition. To obtain the weakly nonlinear instability analysis, we drive Ginzburg–Landau equation. The effects of physical parameters obtained during the study on the Nusselt number and Sherwood number have been examined graphically. The Prandtl number and Richardson number enhance the rate of heat and mass transport, and Lewis number suppressed it. The major finding of the current study is that the internal heat source has a negligible effect on mass transfer, but a significant effect on the heat transfer.

Counting statistics based on the analytic solutions of the differential-difference equation for birth-death processes

Birth-death processes take place ubiquitously throughout the universe. In general, birth and death rates depend on the system size (corresponding to the number of products or customers undergoing the birth-death process) and thus vary every time birth or death occurs, which makes fluctuations in the rates inevitable. The differential-difference equation governing the time evolution of such a birth-death process is well established, but it resists solving for a non-asymptotic solution. In this work, we present the analytic solution of the differential-difference equation for birth-death processes without approximation. The time-dependent solution we obtain leads to an analytical expression for counting statistics of products (or customers). We further examine the relationship between the system size fluctuations and the birth and death rates, and find that statistical properties (variance subtracted by mean) of the system size are determined by the mean death rate as well as the covariance of the system size and the net growth rate (i.e., the birth rate minus the death rate). This work suggests a promising new direction for quantitative investigations into birth-death processes.

Study of Dynamic Solutions for Human–Machine System with Human Error and Common-Cause Failure

This work investigates a dynamic solution of human–machine systems with human error and common-cause failure. By means of functional analysis, it is proved that the semigroup generated by the underlying operator converges exponentially to a projection operator by analyzing the spectral property of the underlying operator, and the asymptotic expressions of the system’s time-dependent solutions are presented. We also provide numerical examples to illustrate the effects of different parameters on the system and the theoretical analysis’s validity.

High-Order Perturbation Method for Updating Response Functions for Time-Dependent Transport Calculations

The hybrid stochastic deterministic continuous-energy coarse mesh transport method (COMET) has been recently extended for high-fidelity efficient kinetics calculations in highly heterogeneous reactor cores. The method discretizes the time variable as a series of time grids and solves the resulting set of steady-state neutron transport equations. In this work, a high-order perturbation method is developed to update the COMET unperturbed response function library on the fly for changes in the discretized time step size Δ t h . The unperturbed response functions are precomputed with 1 / v Δ t h = 0 . The perturbation expansion coefficients are also generated during the unperturbed response function library precomputation. The adjoint solution needed by the perturbation method is calculated using the reciprocity relation without solving the corresponding adjoint problems. As a result, the method can be easily implemented into any stochastic code to generate the perturbation expansion coefficients together with the unperturbed response functions. The high-order perturbation method is benchmarked by comparing both the response functions and the time-dependent COMET core solution (fission density) with the corresponding reference solutions. It is found that the response functions generated by the perturbation method at second order are in excellent agreement with those directly computed by the Monte Carlo method. When Δ t h changes from 1.0E-4 s to infinity, the average and maximum relative differences in the various response functions were found to be in the range of 0.0000106% to 0.00116% and 0.000011% to 0.00121%, respectively. The fission density as a function of time calculated by COMET using the perturbation method is in excellent agreement with the reference solutions, with an average relative difference of 0.0065% to 0.075%. These comparisons indicate that the perturbation method at second order is highly accurate.

Principal spectral theory for nonlocal systems and applications to stem cell regeneration models

We first study the eigenvalue problem for a system of coupled integral operators and investigate existence and monotonicity of the principal eigenvalue and asymptotic behavior of the principal eigenpair. As applications of the spectral theory, we investigate a single genotype stem cell regeneration model with epigenetic transitions, and multiple genotype stem cell regeneration models with epigenetic transitions, with or without gene mutations. These models give rise to three classes of nonlinear integro-differential evolution equations. We consider existence, uniqueness and multiplicity of positive steady states to these equations, as well as the long time behavior of time-dependent solutions. Various explicit formulae for threshold values for tissue development, degeneration and abnormal growth are obtained.

Existence, uniqueness, and asymptotic stability results for the 3-D steady and unsteady Navier–Stokes equations on multi-connected domains with inhomogeneous boundary conditions

We consider both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω ⊂ R 3 with inhomogeneous boundary values on ∂ Ω = Γ; here Γ is a union of disjoint surfaces Γ 0 , Γ 1 , … , Γ l . Our starting point is Leray’s classic problem, which is to find a weak solution u ∈ H 1 ( Ω ) of the stationary problem assuming that on the boundary u = β ∈ H 1 / 2 ( Γ ). The general flux condition ∑ j = 0 l ∫ Γ j β · n d S = 0 must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl. 12 (1933) 1–82) assumed the more restrictive flux condition ∫ Γ j β · n d S = 0 for each j = 1 , … , l. More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II 1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations 2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray’s problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense.

Dual Solutions of Hybrid Nanofluid Flow Over a Cone with the Influence of Thermal Radiation and Chemical Reaction and Its Stability Analysis

The main intention of this study is to differentiate the stable and realisable solutions between the dual solutions of the water-based hybrid nanofluid flow driven by a solid cone along with energy transfer in the form of heat and mass by employing a new approach called stability analysis. The deviation of thermal radiation, chemical responses and heat absorption/generation are reserved into account. The leading equations which support the mathematical representation of this study are renovated by utilizing a set of similarity variables and solved by the MATLAB built-in bvp4c solver scheme. The outcomes of this study are presented both graphically and numerically. From this study, two kind of flow solutions have been achieved where one of them is related to the time-independent solutions and stable in nature. Also, the speed of the hybrid nanofluid can be controlled by applying magnetic field, but we should keep in mind that excessive amount of magnetic parameter may damage the system by burning.

Application of q-HATM over Fractional in Time Model for Advection-Dispersion Equation with Variable Migration Parameters

In this article, an extended and more generalized method of homotopy analysis method (HAM), known as q-homotopy analysis transform method (q-HATM) is employed to develop solutions for highly nonlinear form of various time-fractional advection-dispersion models (TFADE). The proposed methodology of q-HATM is a conjunction of two different strong methodologies, that are HAM and Laplace transform technique (LTT) which makes the scheme much more capable to develop the convergent series solutions for differential equations with high nonlinearity. In this work, two different examples are constructed for the nonlinear time-fractional form of dispersion models corresponding to distance dependent dispersion and velocity components and also for the time dependent decay rate system and solutions are constructed by using q-HATM in generalized form. The graphical comparison is made for different types of variable functions and also for the different fractional order indexes. Prominent impact of fractional derivative indexes is significantly observed over both the examples of dispersion models. Hence, proposed examples shows that the q-HATM is much convenient to handle the nonlinear systems of fractional models.

Analytical Solutions for Scale and Time Dependent Solute Transport in Heterogeneous Porous Medium

Contaminated groundwater has been a serious problem across the world for many years as it has a bad impact on the quality of groundwater as well as on the environment. This study considers the solute transport problem in a heterogeneous porous medium with scale and time-dependent dispersion. The heterogeneity of porous media at the microscopic level facilitates dispersion, which affects groundwater flow patterns and solute distribution. For this work, the porous formation is assumed to be of semi-infinite length and of adsorbing nature. The key parameters such as dispersion coefficient and groundwater velocity are considered to be spatially and temporally dependent functions in degenerated forms. In addition, the first-order decay and zero-order production terms are also considered as time-dependent functions. Initially, it is assumed that the aquifer is uniformly polluted. Two different types of input sources namely uniform and varying nature are considered along the flow at one end in two separate cases, while concentration gradient, at non-source end boundary, is supposed to be zero. An analytical solution of the current boundary value problem is obtained using the Laplace Integral Transform Technique (LITT). The results obtained from the proposed problem are demonstrated graphically for a particular time functions in dispersion and groundwater velocity.

A class of supercritical/critical singular stochastic PDEs: Existence, non-uniqueness, non-Gaussianity, non-unique ergodicity

We study the surface quasi-geostrophic equation with an irregular spatial perturbation∂tθ+u⋅∇θ=−ν(−Δ)γ/2θ+ζ,u=∇⊥(−Δ)−1θ, on [0,∞)×T2, with ν⩾0, γ∈[0,3/2) and ζ∈B∞,∞−2+κ(T2) for some κ>0. This covers the case of ζ=(−Δ)α/2ξ for α<1 and ξ a spatial white noise on T2. Depending on the relation between γ and α, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures [38]. Based on purely analytical tools from convex integration and without the need of any probabilistic arguments including renormalization, we prove existence of infinitely many analytically weak solutions in Llocp(0,∞;B∞,1−1/2)∩Cb([0,∞);B∞,1−1/2−δ)∩Cb1([0,∞);B∞,1−3/2−δ) for all p∈[1,∞) and δ>0. We are able to prescribe an initial as well as a terminal condition at a finite time T>0, and to construct steady state, i.e. time independent, solutions. In all cases, the solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times. Moreover, a coming down from infinity with respect to the perturbation and the initial condition holds. Finally, we show that our solutions generate statistically stationary solutions as limits of ergodic averages, and we obtain existence of infinitely many non-Gaussian time dependent ergodic stationary solutions. We also extend our results to a more general class of singular SPDEs.

Strongly supercritical non-Boussinesq sustained gravity currents: Time-dependent and steady-state approximate solutions

An inertial supercritical gravity current sustained by a constant source at $x=0$ propagates toward a free drainage edge at $x={x}_{D}$. After the nose spills over the edge, a steady state is possible. We show that the time-dependent process leading to the steady state is important in applications (e.g., fire on the ceiling) and is strongly affected by the mixing/entrainment model between the current and the ambient.

Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor

We analyze the time-dependent solution of master equations by exploiting fermionic duality, a dissipative symmetry applicable to a large class of open systems describing quantum transport. Whereas previous studies mostly exploited duality relations after partially solving the evolution equations, we here systematically exploit the invariance under the fermionic duality mapping from the very beginning when setting up these equations. Moreover, we extend the resulting simplifications -so far applied to the local state evolution- to non-local observables such as transport currents. We showcase the exploitation of fermionic duality for a quantum dot with strong interaction -covering both the repulsive and attractive case- proximized by contact with a large-gap superconductor which is weakly probed by charge and heat currents into a wide-band normal-metal electrode. We derive the complete time-dependent analytical solution of this problem involving non-equilibrium Cooper pair transport, Andreev bound states and strong interaction. Additionally exploiting detailed balance we show that even for this relatively complex problem the evolution towards the stationary state can be understood analytically in terms of the stationary state of the system itself via its relation to the stationary state of a dual system with inverted Coulomb interaction, superconducting pairing and applied voltages.

An affordable Three-Dimensional (3D) Printed Recording Chamber for Two-Electrode Voltage Clamp Electrophysiology.

Two electrode voltage-clamp (TEVC) electrophysiology in Xenopus oocytes is a common approach to studying the physiology and pharmacology of membrane transport proteins. Undergraduates may learn to use TEVC methodology in neuroscience or physiology courses and/or in faculty-mentored research experiences. Challenges with the methodology include the cost of commercially available recording chambers, especially when a lab needs multiple copies, and the additional time and expertise needed to use agar bridges and to stabilize solution flow and minimize noise from solution aspiration. Offering a low-cost and accessible recording chamber that overcomes these challenges would lower the barriers to success for undergraduates while also supporting publication-quality recordings. To address these issues, we developed a recording chamber using stereolithography, a 3D printing process. The physiology (PhISio) recording chamber features two options for solution aspiration that allow for individual preferences, optimizes placement of pre-made agar bridges to achieve laminar flow and reduce the time delays in initiating daily experiments, and minimizes the challenges of changing solution height and aspiration noise during perfusion. We compared the functionality of the PhISio chamber with a commercially available Warner Instruments RC-1Z chamber in electrophysiological recordings of inwardly rectifying potassium channels expressed in Xenopus oocytes. The PhISio chamber produced equivalent results to the RC-1Z chamber with respect to time-dependent solution changes and has several operational advantages for both new and experienced electrophysiologists, providing an affordable and convenient alternative to commercially available TEVC recording chambers.