Differential Equations In Time Research Articles

Adaptive identification of linear infinite-dimensional systems

We propose an adaptive algorithm for identifying the unknown parameter (possibly vector-valued) in a linear stable single-input single-output infinite-dimensional system. We assume that the transfer function of the infinite-dimensional system can be expressed as a ratio of two infinite series in s (the Laplace variable). We also assume that certain identifiability conditions, which include a persistency of excitation condition, hold. For a fixed integer n, we propose an adaptive update law (a differential equation in time) driven by real-time input-output data for estimating the first n + 1 coefficients in the numerator and the denominator of the transfer function. We show that the estimates for the transfer function coefficients generated by the update law are close to the true values at large times provided n is sufficiently large (the estimates converge to the true values as time and n tend to infinity). The unknown parameter can be reconstructed using the transfer function coefficient estimates obtained with n large and the algebraic expressions relating the transfer function coefficients to the unknown parameter. We also provide a numerical scheme for verifying the identifiability conditions and for choosing n sufficiently large so that the value of the reconstructed parameter is close to the true value. The class of systems to which our approach is applicable includes many partial differential equations with constant/spatially-varying coefficients and distributed/boundary input and output. We illustrate the efficacy of our approach using three examples: a delay system with four unknown scalars, a 1D heat equation with two unknown scalars and a 1D wave equation with an unknown linearly-varying (in space) coefficient.

The chemo-mechanics of initiation of crack advance in Niobium Tungsten Oxide single crystals due to delithiation

Niobium tungsten oxide (NWO) micron-sized single crystals show promise as the active particles in the anodes of lithium ion batteries due to their fast charge/discharge and high storage capabilities. Within the “block” phases such as Nb16W5O55, Nb14W3O44 and the HNb2O5 polymorph, Li ion diffusion occurs in a unidirectional manner along a single axis of the crystal and lithiation/delithiation results in significant length changes in this direction. A chemo-mechanics analysis is developed for delithiation of an NWO “block” phase single crystal that contains a pre-existing flaw in the form of a through-thickness edge crack. Galvanostatic delithiation occurs by the flux of Li ions through the free ends of the crystal and also through the faces of the crack, with concomitant shrinkage of the surface layer over a depth equal to that of the crack. The induced tensile stress in the surface layer can be of sufficient magnitude to advance the crack in the NWO crystal, consistent with recent experimental observations for Nb14W3O44. A 2D full-field finite element version of the chemo-mechanics model is used to explore the sensitivity of cracking to crack length, delithiation rate of the anode, diffusivity of Li ions within the NWO single crystal storage particle and to the kinetics of Li ion transfer across the particle surface-electrolyte interface. Additionally, a simplified, lumped parameter model is developed by assuming that the Li ion occupancy is spatially uniform in a surface layer that extends to a depth of the pre-existing crack, and has a spatially uniform but different value than that of the underlying core of the particle. This simplified model can be expressed as a set of first order differential equations in time and gives the main features of delithiation and the associated transient stress state in the single crystal. Both the full model and the lumped parameter version predict that crack advance from a pre-existing defect is triggered by a high delithiation rate, consistent with the experimental literature.

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

Modal Discontinuous Galerkin Simulations for Grad’s 13 Moment Equations: Application to Riemann Problem in Continuum-Rarefied Flow Regime

For real-world engineering applications, there is a great deal of interest in developing effective models and simulations of continuum-rarefied gas flows. In this study, the numerical simulations of Grad’s 13 (G13) moment equations with application to the Riemann problem in a wide range of continuum-rarefied flow regimes are presented. This work emphasizes numerical robustness and wave phenomena in the G13 system to provide a building block for regularized 13 moment systems, and high-order Grad’s models. For this purpose, a high-order modal discontinuous Galerkin solver is developed for solving one-dimensional G13 moment equations. For spatial discretization, hierarchical modal basis functions premised on orthogonal-scaled Legendre polynomials are used. The proposed approach reduces the G13 systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order SSP Runge-Kutta algorithm. Three Riemann test cases, including the shock tube, two shock waves and two rarefaction waves, are examined in continuum-rarefied flow regimes. In the G13 system, the arising characteristic waves and dissipation phenomena are investigated in depth. The numerical results demonstrate that every Riemann problem does not have a solution for the G13 system because of loss of hyperbolicity of the system.

A numerical scheme based on the Taylor expansion and Lie product formula for the second‐order acoustic wave equation and its application in seismic migration

AbstractWe have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration.

A modification of explicit time integrator scheme for unsteady power-law nanofluid flow over the moving sheets

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations’ precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids.

On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems

Nonlinear coupled reaction–diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray–Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge–Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications.

Scaled boundary perfectly matched layer for wave propagation in a three-dimensional poroelastic medium

In this paper, we introduce a novel direct time-domain artificial boundary condition, called the scaled boundary perfectly matched layer, for wave problems in 3D unbounded fluid-saturated poroelastic medium. The poroelastic medium is conceptualized as a two-phase medium composed of a solid skeleton and an interstitial fluid based on the u-p formulation of Biot's theory. The scaled boundary perfectly matched layer proposed in this work originates from an extension of a recently developed perfectly matched layer for wave problems in a single-phase solid medium. The dependence of the conventional perfectly matched layer on the global coordinate system is overcome in the present method. As a result, this method permits the utilization of an artificial boundary with general geometry (not necessarily convex) and can consider planar physical surfaces and interfaces extending to infinity. The proposed method is formulated by a mixed unsplit-field form, which includes both the mixed displacement-stress field and pore pressure-relative displacement field. The resulting scaled boundary perfectly matched layer formulation can be directly described by a system of third-order ordinary differential equations in time, which can be easily integrated into the standard finite-element framework of poroelastic theory. Finally, numerical benchmark examples are presented to demonstrate the accuracy and robustness of the proposed scaled boundary perfectly matched layer, as well as the versatility in practical engineering problems.

Construction of a numerical model and algorithm for solving two-dimensional problems of filtration of multicomponent liquids, taking into account the moving “oil-water” interface

The paper considers a two-dimensional mathematical model of the filtration of a viscous, incompressible fluid in a deformable porous medium. The article describes a mathematical model of the problem at the “oil-water” interface with a system of parabolic differential equations. The problem posed is solved numerically using the differential-difference method based on the longitudinal-transverse scheme and the differential sweep method to determine the unknown boundary. To obtain a differential-difference problem, an algorithmic representation of a hidden scheme of alternating directions (longitudinal-transverse scheme) is used. The resulting system of differential-difference equations with initial conditions is known for each time layer along straight lines along the x axis, and then along each y axis it is solved by a differential sweep along a straight line, where the values just found correspond to time, i.e., the layer is taken as initial conditions. By approximating the differential equations in time, the position of the interface is determined for each time layer.

Computational elastodynamics of functionally graded thick-walled cylinders and annular coatings subjected to pressure shocks

A computational technique based on domain-boundary element method (D-BEM) is developed for elastodynamic analysis of functionally graded thick-walled cylinders and annular coatings subjected to pressure shock type of loadings. The formulation is built on the wave equation, which is derived in accordance with plane elastodynamics. Weighted residual statement for the wave equation is expressed by using the static fundamental solution as the weight function. Applying integration by parts and incorporating the boundary conditions, the problem is reduced to an integral equation. Problem domain is discretized by quadratic cells to transform the integral equation into a system of ordinary differential equations in time. Equation system is solved numerically applying Houbolt's method. Developed procedure is verified through comparisons to the analytical results available in the literature. Parametric analyses are carried out considering short-time ramp and exponential variation types of pressure shocks. Presented numerical results illustrate the influence of material property gradation on time histories and spatial distributions of displacement and stress components in FGM thick-walled cylinders and annular coatings.

Electromagnetic, Gravitational, and Plasma-Related Perturbations of Locally Rotationally Symmetric Class II Spacetimes

We investigate electromagnetic, gravitational, and plasma-related perturbations to the first order on homogeneous and hypersurface orthogonal locally rotationally symmetric (LRS) class II spacetimes. Due to the anisotropic nature of the studied backgrounds, we are able to include a non-zero magnetic field to the zeroth order. As a result of this inclusion, we find interesting interactions between the electromagnetic and gravitational variables already of the first order in the perturbations. The equations governing these perturbations are found by using the Ricci identities, the Bianchi identities, Einstein’s field equations, Maxwell’s equations, particle conservation, and a form of energy-momentum conservation for the plasma components. Using a 1+1+2 covariant split of spacetime, the studied quantities and equations are decomposed with respect to the preferred directions on the background spacetimes. After linearizing the decomposed equations around an LRS background, performing a harmonic decomposition, and imposing the cold magnetohydrodynamic (MHD) limit with a finite electrical resistivity, the system is then reduced to a set of ordinary differential equations in time and some constraints. On solving for some of the harmonic coefficients in terms of the others, the system is found to decouple into two closed and independent subsectors. Through numerical calculations, we then observe some mechanisms for generating magnetic field perturbations, showing some traits similar to previous works using Friedmann–Lemaître–Robertson–Walker (FLRW) backgrounds. Furthermore, beat-like patterns are observed in the short wave length limit due to interference between gravitational waves and plasmonic modes.

Parallel-in-time preconditioner for the Sinc-Nyström systems

The sinc-Nyström method is a high-order numerical method based on sinc basis functions for discretizing evolutionary differential equations in time. But in this method we have to solve all the time steps in one shot (i.e., all-at-once), which results in a large-scale nonsymmetric dense system that is expensive to handle. In this paper, we propose and analyze a parallel-in-time preconditioner for such a dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and it has two advantages. First, we show that the eigenvalues of the preconditioned system are highly clustered with some uniform bounds which are independent of the mesh parameters. Second, the preconditioner can be computed in parallel for all the sinc time points via a block diagonalization procedure. Such a parallel potentials owing to the fact that the eigenvector matrix of the diagonalization is well conditioned. In particular, we show that the condition number of the eigenvector matrix only mildly grows as the number of sinc time points increases, and thus the roundoff error arising from the diagonalization procedure is controllable. The effectiveness of our proposed parallel-in-time preconditioners is verified by the observed mesh-independent convergence rates of the preconditioned GMRES in several numerical examples.

Analytical assessment of nonlinear forced vibration of functionally graded porous higher order hinged beams

This study focuses on the nonlinear free and forced vibration behavior of functionally graded porous beams considering high-order bidirectional porosity distributions. It is assumed that the closed-cell voids are distributed non-uniformly along the thickness and length of the beam. A nonlinear formulation is derived based on the Reddy beam theory, von Karman geometrical nonlinearity using the Hamilton principle. Subsequently, Galerkin technique is implemented to reduce the partial differential equations to a set of nonlinear ordinary differential equations in time. The primary resonance of the system is inspected when the beam is under a combination of a harmonic transverse load and a static compressive axial load. Hence, the harmonic balance method and method of multiple scales are used to develop closed form approximate solutions. It is shown, the method of multiple scales underestimates the amplitude of vibration when either amplitude of vibration or axial load, or both are high. Effects of beam’s aspect ratio, porosity distributions, beam’s shear deformation and porosity volume fraction are studied. Results revealed, the proposed porosity distributions are more effective than the conventional porosity distributions. Furthermore, comparing beams with the same mass reveals that beams with more voids at the center have lowest amplitude of vibration.

Continuum multiscale modeling of absorption processes in micro- and nanocatalysts

In this paper, we propose a novel, semi-analytic approach for the two-scale, computational modeling of concentration transport in packed bed reactors. Within the reactor, catalytic pellets are stacked, which alter the concentration evolution. Firstly, the considered experimental setup is discussed and a naive one-scale approach is presented. This one-scale model motivates, due to unphysical fitted values, to enrich the computational procedure by another scale. The computations on the second scale, here referred to as microscale, are based on a proper investigation of the diffusion process in the catalytic pellets from which, after continuum-consistent considerations, a sink term for the macroscopic advection–diffusion–reaction process can be identified. For the special case of a spherical catalyst pellet, the parabolic partial differential equation at the microscale can be reduced to a single ordinary differential equation in time through a semi-analytic approach. After the presentation of our model, we show results for its calibration against the macroscopic response of a simple standard mass transport experiment. Based thereon, the effective diffusion parameters of the catalyst pellets can be identified.

Joint localization algorithm of TDOA and FDOA considering the position error of underwater sensors

The joint localization method of TDOA and FDOA is widely used because it can estimate the underwater moving source well. Usually, a two-step weighted least squares (TSWLS) algorithm is used to estimate the source. However, the noise square term is always neglected in the TSWLS algorithm. In order to improve this situation, an improved TSWLS algorithm is proposed under the circumstance of sensor position inaccuracy. First, a new localization equation is constructed by the time difference equation, and the source position and velocity are initially estimated. The intermediate variables are then Taylor expanded and inserted into the localization equation of the algorithm's first step to modify the initial estimate value. Finally, the source's precise position and velocity are determined, and the proposed method was demonstrated to achieve the Cramer–Rao lower bound (CRLB) accuracy through simulations.

Theoretical Model of Extracellular Potassium Diffusion as a Signaling Mechanism in Neurovascular Coupling

Neurovascular coupling (NVC) is the process by which neuronal activity causes changes in cerebral blood flow within seconds of the onset of activity. The goal of this study is to explore the potential role of extracellular K+ diffusion as a mechanism for NVC. A mathematical model is used to simulate K+ kinetics in the extracellular space (ECS), including effects of K+release during neuronal activity, K+ uptake by astrocytic and neuronal Na+/K+ ATPase, and K+ diffusion in the ECS. The model consists of a partial differential equation in time and one spatial dimension. Parameter values are derived from experimental literature or from previous models. The effective K+ diffusivity is 937.5 μm2/s. Astrocytic and neuronal K+ uptake is assumed to have Michaelis‐Menten kinetics with maximum rate 84.7 mM/s and Michaelis constant 1.5 mM. The average distance from a neuron to the closest microvessel is 15 μm. A domain of length 15 μm is therefore assumed, with one end representing the neuron and the other the blood vessel. At rest, ECS K+ concentration is 3 mM. One spike is estimated to change the extracellular K+ concentration by 0.11 mM from rest. The neuron is assumed to fire continuously for 10 s at 100 Hz. Equilibrium values of K+ in the ECS are reached within 10 s, with values of 11 mM at the neuron and 9.6 mM at the vessel. The increase in extracellular K+ concentration at the vessel lags approximately 0.3 s behind the increase at the neuron. These results suggest that K+ diffuses fast enough to the vasculature to be a signal for NVC. Although uptake of K+ by Na+/K+ ATPase limits the concentration that reaches the vasculature, the increase in K+ concentration around the vasculature is sufficient to elicit a vascular response. Limitations of the model are that a simplified, one‐dimensional geometry is assumed and possible effects of K+ release at astrocytic endfeet are not included.

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

A time-explicit Runge–Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection–reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ⩾ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results indicate that the proposed scheme is more robust and accurate in comparison to existing approaches based on the continuous Galerkin finite element method or the fast Fourier transform method.

Conservation Laws -- a Source for Distortionless Propagation and Time Delays

Since the very first paper of J. Bernoulli in 1728, a connection exists between initial boundary value problems for hyperbolic Partial Differential Equations (PDE) in the plane (with a single space coordinate accounting for wave propagation) and some associated Functional Equations (FE). From the point of view of dynamics and control (to be specific, of dynamics for control) both type of equations generate dynamical and controlled dynamical systems. The functional equations may be difference equations (in continuous time), delay-differential (mostly of neutral type) or even integral/integro-differential. It is possible to discuss dynamics and control either for PDE or FE since both may be viewed as self contained mathematical objects. A more recent topic is control of systems displaying conservation laws. Conservation laws are described by nonlinear hyperbolic PDE belonging to the class ``lossless'' (conservative); their dynamics and control theory is well served by the associated energy integral. It is however not without interest to discuss association of some FE. Lossless implies usually distortionless propagation hence one would expect here also lumped time delays. The paper contains some illustrating applications from various fields: nuclear reactors with circulating fuel, canal flows control, overhead crane, drilling devices, without forgetting the standard classical example of the nonhomogeneous transmission lines for distortionless and lossless propagation. Specific features of the control models are discussed in connection with the control approach wherever it applies.

Efficient computation of coupled bending and torsion vibrations of slender structures with an axially moving eccentric load

AbstractFrequently slender structures are excited by moving loads and show the corresponding transient vibrations. A model is derived using a simply supported Bernoulli‐Euler beam and the coupled bending and torsion vibration equations are derived. An analytical solution of the decoupled partial differential equations is derived based on the eigenmode expansion considering the boundary and initial conditions. For the pure bending and the pure Saint‐Venant torsion the analysis is performed and the beam vibrations are calculated. The coupled bending and torsion deformations are computed using a modal transformation, where it is essential that the eigenmodes are the same for both kinds of vibrations. The resulting modal decoupling yields two separated ordinary differential equations in time with the assumed constant velocity of the axially moving excitation load. Closed form solutions are used which show a very good convergence. A transformation into the original coordinates results in the solutions of the coupled equations. The results are plotted in a dimensionless form. The evaluation and comparison with a suitable numerical calculation using a finite element model shows a good agreement of the results. It turns out that a fine discretization is necessary and the computation effort for computing the transient dynamic solution is very much higher.

Transient dynamic analysis of functionally graded micro-beams considering small-scale effects

A domain-boundary element method, based on modified couple stress theory, is developed for transient dynamic analysis of functionally graded micro-beams. Incorporating static fundamental solutions as weight functions in weighted residual expressions, governing partial differential equations of motion are converted to a set of coupled integral equations. A system of ordinary differential equations in time is obtained by domain discretization and solved using the Houbolt time marching scheme. Developed procedures are verified through comparisons to the results available in the literature for micro- and macro-scale beams. Numerical results illustrate elasto-dynamic responses of graded micro-beams subjected to various loading types. It is shown that metal-rich micro-beams and those with a smaller length scale parameter ratio undergo higher displacements and are subjected to larger normal stresses.