Time-dependent Equation Research Articles

CT-FFR by expanding coronary tree with Newton–Krylov–Schwarz method to solve the governing equations of CFD

Abstract Background A new model of computational fluid dynamics (CFD)-based algorithm for coronary CT angiography (CCTA)-derived fractional flow reserve (FFR) (CT-FFR) analysis by expanding the coronary tree to smaller-diameter lumen (0.8mm) using Newton–Krylov–Schwarz (NKS) method to solve the three-dimensional time-dependent incompressible Navier-Stokes equations has been developed; however, the diagnostic performance of this new method has not been sufficiently investigated. Objectives The aim of this study was to determine the diagnostic performance of a novel CT-FFR technique by expanding the coronary tree in the CFD domain. Methods Six centers enrolled 338 symptomatic patients with suspected or known coronary artery disease (CAD) who were prospectively underwent CCTA and FFR. Stenosis assessment in CCTA and CT-FFR analysis were performed in independent core laboratories. Hemodynamically significant stenosis was defined by an CT-FFR and FFR ≤0.80, and anatomically obstructive CAD was defined as a CCTA with stenosis ≥50%. Diagnostic performance of CT-FFR was evaluated against invasive FFR using receiver operator characteristic (ROC) curve analysis. The correlation between CT-FFR and invasive FFR was analyzed using the Spearman correlation coefficient and Bland-Altman analysis. Intra-observer and inter-observer agreement were evaluated utilizing the intraclass correlation coefficient (ICC). Results In this study, 338 patients with 422 targeted vessels were investigated, revealing hemodynamically significant stenosis in 31.1% (105/338) of patients and anatomically obstructive stenosis in 54.1% of patients. On a per-vessel basis, the area under the receiver-operating characteristic curve for CT-FFR was 0.94 versus 0.76 for CCTA (p < 0.001). Per-vessel accuracy, sensitivity, specificity, positive predictive value, and negative predictive value were 89.8%, 89.3%, 90.0%, 79.0%, and 99.2%, respectively, for CT-FFR and were 68.4%, 82.8%, 62.3%, 48.1%, and 89.6%, respectively, for CCTA stenosis. CT-FFR and FFR were well correlated (r = 0.775, p < 0.001) with a Bland-Altman bias of 0.0011, and limits of agreement from -0.1509 to 0.1531 (p = 0.770). The intraclass correlation coefficients with CT-FFR for intro- and inter-observer agreement were 0.919 (95% CI: 0.866 to 0.952) and 0.909 (95% CI: 0.851 to 0.945), respectively. The average computation time for CT-FFR analysis was maintained at 11.7 minutes. Conclusion This novel CT-FFR model with the inclusion of smaller lumen provides high diagnostic accuracy in detecting hemodynamically significant CAD. Furthermore, the integration of the NKS method ensures that the computation time remains within an acceptable range for potential clinical applications in the future.

Time-dependent electromagnetic scattering from dispersive materials

Abstract This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.

Free evolution in the Ginzburg-Landau equation and other complex diffusion equations

Abstract New ordinary differential equations (ODEs) for the evolution of spectral components are derived from the complex Ginzburg-Landau equation (CGLe) for one-dimensional spatial domains without boundaries (free evolution) and with one fixed boundary (semi-free evolution). For such evolution, a complex or imaginary diffusion term creates a tendency for waves to lengthen. This requires a novel ansatz and auxiliary condition that treat wavenumbers as time-varying. The ansatz consists of a discrete spatial Fourier transform modified with a time-dependent wavenumber for the peak spectral component. The wavenumbers of the other components are fixed relative to this wavenumber. The new auxiliary condition is the terminal condition for complex diffusion (after wavenumbers evolve to zero, they remain at zero). The derived free and semi-free ODEs are solved along characteristic lines located symmetrically about a fixed spatial point. Waves lengthen with time away from this point in both directions. Laboratory experiments on the formation of channel sandbars, theoretically described by the CGLe, show two regions whose evolutionary behaviour is qualitatively predicted by the free and semi-free evolution equations. This analysis applies to other time-dependent partial differential equations with complex or imaginary diffusion terms. New freely evolving solutions are derived for the complex heat equation and Schrödinger equation (linear and nonlinear).

Evolution of fermion resonance in thick brane

Abstract In this work, we investigate the numerical evolution of massive Kaluza–Klein (KK) modes of a Dirac field on a thick brane. We deduce the Dirac equation in five-dimensional spacetime, and obtain the time-dependent evolution equation and Schrödinger-like equation of the extra-dimensional component. We use the Dirac KK resonances as the initial data and study the corresponding dynamics. By monitoring the decay law of the left- and right-chiral KK resonances, we compute the corresponding lifetimes and find that there could exist long-lived KK modes on the brane. Especially, for the lightest KK resonance with a large coupling parameter and a large three momentum, it will have an extremely long lifetime.

RKFD Scheme for Quantum Reflection Model of Bose-Einstein Condensates (BEC) from Silicon Surface

We applied the numerical combination of Runge-Kutta and Finite Difference (RKFD) scheme for a quantum reflection model of Bose-Einstein condensate (BEC) from a silicon surface. It is by the time-dependent Gross-Pitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) in the context of quantum mechanics. The role of cut-off potential δ and negative imaginary potential Vim is essential to estimating non-interacting BEC reflection models. Relying on these features, we performed a numerical simulation of the BEC quantum reflection model and calculated the effect of reflection probability R versus incident speed vx. The model is based on the three rapid potential variations: positive-step potential +Vstep, negative-step potential -Vstep, and Casimir-Polder potential VCP. As a result, the RKFP numerical scheme was successfully set up and applied to the quantum reflection model of BEC from the silicon surface. The numerical simulations results show that the reflection probability R decays exponentially to the incident speed vx.

An Effective Alternative to Current Mathematics

If you don't understand mathematics, ask yourself if I'm right, because others don't understand mathematics either. By effective alternative to current mathematics, we mean working in a more complete mathematical space than the classical 3D+t variety which is inadequate for generating well-defined definitions and hypotheses as well as its limited ability to solve time-dependent partial differential equations. The current classical discrete 3D+t space PDE, in which time is an external controller and not integrated into the 3D geometric space, cannot be integrated digitally. This space is logically incomplete and misleading in the production of definitions and hypotheses as well as in the resolution itself of time- dependent PDEs. It is no wonder that these definitions/assumptions are confusing and result in weak or intractable mathematics, leading to all kinds of misunderstandings, from horrible notations to undisciplined length of theorems containing a considerable amount of black magic and ending with a gray nature of the mathematical result obtained. In this article, we present some of the most inaccurate assumptions and definitions in current classical mathematics that arise from using the 3D+t manifold space to specify initial conditions, boundary conditions, and the source/sink term. Fortunately, these inaccurate assumptions that start with inadequate space for boundary conditions, initial conditions, and source/sink term can be spotted and analyzed via 4D unitary numerical statistical theory called Cairo techniques in the format of transition chains of matrix B to complete what is missing. In other words, we present how to spot some of the worst mathematical conclusions of classical 3D geometry plus t as an external control numerical space, and then show how to correct them via the 4D unit space which is the subject of this article.

Numerical Modeling of Vortex-Based Superconducting Memory Cells: Dynamics and Geometrical Optimization.

The lack of dense random-access memory is one of the main obstacles to the development of digital superconducting computers. It has been suggested that AVRAM cells, based on the storage of a single Abrikosov vortex-the smallest quantized object in superconductors-can enable drastic miniaturization to the nanometer scale. In this work, we present the numerical modeling of such cells using time-dependent Ginzburg-Landau equations. The cell represents a fluxonic quantum dot containing a small superconducting island, an asymmetric notch for the vortex entrance, a guiding track, and a vortex trap. We determine the optimal geometrical parameters for operation at zero magnetic field and the conditions for controllable vortex manipulation by short current pulses. We report ultrafast vortex motion with velocities more than an order of magnitude faster than those expected for macroscopic superconductors. This phenomenon is attributed to strong interactions with the edges of a mesoscopic island, combined with the nonlinear reduction of flux-flow viscosity due to the nonequilibrium effects in the track. Our results show that such cells can be scaled down to sizes comparable to the London penetration depth, ∼100 nm, and can enable ultrafast switching on the picosecond scale with ultralow energy per operation, ∼10-19 J.

On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift

We prove the existence of weak solutions of Itô’s stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.

A stabilized finite volume method based on the rotational pressure correction projection for the time-dependent incompressible MHD equations

This paper presents an efficient numerical scheme for solving the time-dependent incompressible Magnetohydrodynamics (MHD) equations based on the rotational pressure correction projection method within the finite volume framework. Utilizing the lowest equal-order mixed finite element pair (P1−P1−P1) to approximate the velocity, magnetic and pressure fields, our numerical scheme satisfies the discrete inf-sup condition by the pressure projection stabilization. To tackle the coupling inherent in the time-dependent incompressible MHD equations,the rotational pressure correction projection method is introduced to split the original problem into several linear subproblems. The unconditional stability of numerical schemes are provided,optimal error estimates in both L2 and H1-norms of numerical solutions are also presented.Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical schemes.

Dynamic of time-independent and time-dependent asymmetric Gross-Pitaevskii equation around exceptional point

Systems operating at exceptional points (EPs) are highly sensitive to small perturbations, making it challenging to work near an EP. Eigenvalue analysis of the Gross-Pitaevskii equation has shown that asymmetric nonlinearity can compensate for detuning effects. However, an experimentally feasible system based on asymmetric nonlinear coupled resonators has not yet been explored. Additionally, some intriguing features of such a system are hidden in time domain analysis, which is rarely investigated. In this study, we demonstrate this feature using a full-wave simulation of an asymmetric nonlinear coupled resonator based on the finite-element method in Comsol. The time-dependent analysis reveals that detuning can shift the system from PT-symmetric to broken PT-symmetric (or vice versa), and nonlinearity can reverse this dynamic. This study provides an experimental framework for examining exceptional points (EPs) in nonlinear detuned coupled resonators and opens up new avenues for fundamental research into the influence of nonlinearity and detuning on the system’s state during EP encirclement.

Fokker–Planck equation and Feynman–Kac formula for multidimensional stochastic dynamical systems with Lévy noises and time-dependent coefficients

The aim of this paper is to establish a version of the Feynman–Kac formula for the time-dependent multidimensional nonlocal Fokker–Planck equation corresponding to a class of time-dependent stochastic differential equations driven by multiplicative symmetric (or asymmetric) α-stable Lévy noise. First the forward nonlocal Fokker–Planck equation is derived by the adjoint operator method, overcoming the challenges posed by time-dependent multidimensional nonlinear symmetric α-stable Lévy noise. Subsequently, the Feynman–Kac formula for the forward multidimensional time-dependent nonlocal Fokker–Planck equation is established by applying techniques for the backward nonlocal Fokker–Planck equations, which is associated with the backward stochastic differential equation driven by the multiplicative symmetric α-stable Lévy noise. Notably, in the case of asymmetric α-stable Lévy noise case, the presence of the characteristic function in the nonlocal operator adds complexity to the analysis. Using the Feynman–Kac formula, it is demonstrated that the solution of the forward nonlocal Fokker–Planck equation can be readily simulated through Monte Carlo approximation, especially in scenarios involving long-time simulation settings with large steps. These concepts are illustrated with intriguing examples, and the dynamic evolution of the probability density function corresponding to the stochastic SIS model and the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, ComK and ComS) are investigated over an extended period.

A new spline method on graded mesh for fourth-order time-dependent PDEs: application to Kuramoto-Sivashinsky and extended Fisher-Kolmogorov equations

In this research, we introduce a two-tier non-polynomial spline approach with graded mesh discretization for addressing fourth-order time-dependent partial differential equations, which find applications in various physical scenarios like the nonlinear Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation. Our method involves considering three spatial points at each time step for scheme development, achieving spatial accuracy of three and temporal accuracy of two. Notably, our approach offers the advantage of directly tackling singular biharmonic problems without the need for discretizing boundary conditions or incorporating fictitious points. It demonstrates unconditional stability when tested on fourth-order problems and outperforms previous methods, as evidenced by comparative analyses. Additionally, we present 2D and 3D plots illustrating numerical solutions for different benchmark scenarios, including the depiction of the chaotic nature of the Kuramoto-Sivashinsky equation under Gaussian initial conditions at various time levels.

On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi equations

We show in this paper that maximal Lq-regularity for time-dependent viscous Hamilton-Jacobi equations with unbounded right-hand side and superquadratic γ-growth in the gradient holds in the full range q>(N+2)γ−1γ. Our approach is based on new γ−2γ−1-Hölder estimates, which are consequence of the decay at small scales of suitable nonlinear space and time Hölder quotients. This is obtained by proving suitable oscillation estimates, that also give in turn some Liouville type results for entire solutions.

Analysis of dynamic disturbance and multistage shear creep damage evolution law of the weak intercalated layers in slope under the influence of coupled damage effect

Based on the damage characteristics of multistage shear creep in weak intercalated layers (carbonaceous mud shale) of slopes under the influence of dynamic disturbance, the effective bearing area method was used. A new coupled damage equation (dynamic disturbance damage, shear creep damage, and initial damage) was established through further derivation, and its applicability was demonstrated. The calculation method for the relevant coupled damage degrees was also provided. Furthermore, by targeting the three coupled damage factors and extending the Kachanov damage law, a time-dependent damage evolution equation for weak intercalated layers under the influence of the three coupled damage effects was established. The influence of different dynamic disturbance intensities on the evolution of multistage shear creep damage in weak intercalated layers of slopes under the influence of coupled damage effects was analysed. The results show that the damage to the rock mass caused by dynamic disturbance mainly occurs in the low-frequency stage (40–80 Hz). The instantaneous damage caused by dynamic disturbance to the shear plane of weak intercalated layers is not only affected by the intensity of the dynamic disturbance but also limited by the magnitude of the shear creep load. The influence of the dynamic disturbance intensity on the entire process of multistage shear creep damage of weak intercalated layers was analysed. With increasing of dynamic disturbance intensity, the cumulative coupled damage at the end of shear creep at all levels gradually exhibits linear evolution. The time-dependent coupling damage evolution process of weak intercalated layers was quantitatively characterized.

Dynamic density functional theory of polymers with salt in electric fields.

We present a dynamic density functional theory for modeling the effects of applied electric fields on the local structure of polymers with added salt (polymer electrolytes). Time-dependent equations for the local electrostatic potential and volume fractions of polymer, cation, and anion of added salt are developed using the principles of linear irreversible thermodynamics. For such a development, a field theoretic description of the free energy of polymer melts doped with salts is used, which captures the effects of local variations in the dielectric function. Connections of the dynamic density functional theory with experiments are established by relating the three phenomenological Onsager's transport coefficients of the theory to the mutual diffusion of electrolyte, ionic conductivity, and transference number of one of the ions. The theory is connected with a statistical mechanical model developed by Bearman and Kirkwood [J. Chem. Phys. 28, 136 (1958)] after relating the three transport coefficients to friction coefficients. The steady-state limit of the dynamic density functional theory is used to understand the effects of dielectric inhomogeneity on the phase separation in polymer electrolytes. The theory developed here provides not only a way to connect with experiments but also to develop multi-scale models for studying connections between local structure and ion transport in polymer electrolytes.

Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation

In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in l2-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.

Development of multi-physics calculation method in lead cooled fast reactor code system MOSASAUR

A full-core three-dimensional multi-physics calculation method was researched based on the Lead cooled fast reactor (LFR) code system MOSASAUR, which is focused on both core steady-state and transient analyses of LFR. In the previous version of MOSASAUR, cross-sections generation module and core steady-state simulation module have been developed. In this research, the stiffness confinement method is employed to solve the time-dependent multi-group neutron transport equation to expand the capabilities of core transient analyses. And the thermal hydraulic calculation method is developed as thermal feedback module for core steady-state and transient analyses. Based on the fixed-point iteration scheme, the neutron module and thermal hydraulic module are coupled at each time step. The LMW benchmark and the problem based on ORAL 19-rod bundle are employed to verify the accuracy of transient calculation and thermal hydraulic module respectively. Finally, the multi-physics calculation method is utilized to simulate the transient process of the MicroURANUS core. The simulation results demonstrate the capability of the constructed multi-physics calculation framework to accurately simulate the transient behaviors of LMRs. Numerical results showed the good accuracy of the newly-developed multi-physics calculation module with MOSASAUR.

Switching Response in Organic Electrochemical Transistors by Ionic Diffusion and Electronic Transport.

The switching response in organic electrochemical transistors (OECT) is a basic effect in which a transient current occurs in response to a voltage perturbation. This phenomenon has an important impact on different aspects of the application of OECT, such as the equilibration times, the hysteresis dependence on scan rates, and the synaptic properties for neuromorphic applications. Here we establish a model that unites vertical ion diffusion and horizontal electronic transport for the analysis of the time-dependent current response of OECTs. We use a combination of tools consisting of a physical analytical model; advanced 2D drift-diffusion simulation; and the experimental measurement of a poly(3-hexylthiophene) (P3HT) OECT. We show the reduction of the general model to simple time-dependent equations for the average ionic/hole concentration inside the organic film, which produces a Bernards-Malliaras conservation equation coupled with a diffusion equation. We provide a basic classification of the transient response to a voltage pulse, and the correspondent hysteresis effects of the transfer curves. The shape of transients is basically related to the main control phenomenon, either the vertical diffusion of ions during doping and dedoping, or the equilibration of electronic current along the channel length.

Time Dependent Spectral Ratio Technique for Micro-seismic Exploration in the Niger Delta Region of Nigeria

Abstract: Since the discovery of the hydrocarbon micro-tremors, there have been numerous efforts to understand the causes of these microseisms related to oil and gas. A spectral ratio (SR) time-dependent equation was derived based on geological principles. The equation showed tremendous success in analyzing multi-layered geological terrain and estimating the spectral ratio in the Niger Delta region of Nigeria. This study seeks to develop a reliable model using standard seismic reflection data to analyze microseismic datasets from hydrocarbon reservoirs. Type A-SR acts over a longer duration and can be used to determine hydrocarbon reservoirs. Type B-SR acts in a short time with maximum impact. It can be used to determine gas condensate flow in the hydrocarbon reservoir. It is recommended that this technique be applied in real time to ascertain its accuracy. Keywords: Niger Delta, Microseismic, Amplitude, Special ratio, Geology.

An anisotropic damage visco-hyperelastic model for multiaxial stress-strain response and energy dissipation in filled rubber

In this article, we introduce a novel physically-based anisotropic damage visco-hyperelastic model designed to predict the history-dependent inelastic behavior of multiaxially stretched filled rubber. The model integrates both the anisotropic Mullins effect and intrinsic viscosity through the consideration of internal physics, represented by two distinct networks: an elastic ground network and a superimposed viscous network. The rupture of molecular bonds within the elastic network chain backbone is modeled using statistical mechanics, while the effects of anisotropy-induced chain orientation at the upper scale are addressed through a microsphere-based scale transition method. The intrinsic viscosity is represented by the viscous network, which is governed by time-dependent equations to account for the viscous overstress. The influence of fillers is captured through the concept of strain amplification, applied to the two networks within the rubber matrix. The effectiveness of the model in capturing the biaxial behavior of filled rubber is evaluated by comparing its outputs with experimental data from a filled rubber system. This assessment specifically considers the impact of pre-stretching under various loading conditions and across a wide range of filler concentrations. Notably, it successfully predicts anisotropic stress-strain response and energy dissipation, and the coupled effects of damage and viscosity.