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In a recent work, assuming a Beer–Lambert optical absorption and a Gaussian laser time profile, it was shown that the exact solutions for a 1D photoacoustic (PA) boundary-value-problem predict a null pressure for optically strong absorbent materials. In order to overcome this inconsistency, a heuristic correction was introduced by assuming that heat flux travels a characteristic length during the duration of the laser pulse [M. Ruiz-Veloz et al., J. Appl. Phys. 130, 025104 (2021)] τ p. In this work, we obtained exact analytical solutions in the frequency domain for a 1D boundary-value-problem for the Dual-Phase-Lag (DPL) heat equation coupled with a 1D PA-boundary-value-problem via the acoustic wave equation. Temperature and pressure solutions were studied by assuming that the sample and its surroundings have a similar characteristic thermal lag response time τ T ; therefore, the whole system is assumed to have a similar thermal relaxation. A second assumption for τ T is that it is considered as a free parameter that can be adjusted to reproduce experimental results. Solutions for temperature and pressure were obtained for a one-layer 1D system. It was found that for τ T < τ p , the DPL temperature has a similar thermal profile of the Fourier heat equation; however, when τ T ≥ τ p , this profile is very different from the Fourier case. Additionally, via a numerical Fourier transform, the wave-like behavior of DPL temperature is explored, and it was found that as τ T increases, thermal wave amplitude is increasingly attenuated. Exact solutions for pressure were compared with experimental PA signals, showing a close resemblance between both data sets, particularly in time domain, for an appropriated value of τ T ; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup. [ABSTRACT FROM AUTHOR]

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model. [ABSTRACT FROM AUTHOR]

In living and synthetic active matter systems, the constituents can self-propel and interact with each other and with the environment through various physicochemical mechanisms. Among these mechanisms, chemotactic and auto-chemotactic effects are widely observed. The impact of (auto-)chemotactic effects on achiral active matter has been a recent research focus. However, the influence of these effects on chiral active matter remains elusive. Here, we develop a Brownian dynamics model coupled with a diffusion equation to examine the dynamics of auto-chemotactic chiral active droplets in both quasi-two-dimensional (2D) and three-dimensional (3D) systems. By quantifying the droplet trajectory as a function of the dimensionless Péclet number and chemotactic strength, our simulations well reproduce the curling and helical trajectories of nematic droplets in a surfactant-rich solution reported by Krüger et al. [Phys. Rev. Lett. 117, 048003 (2016)]. The modeled curling trajectory in 2D exhibits an emergent chirality, also consistent with the experiment. We further show that the geometry of the chiral droplet trajectories, characterized by the pitch and diameter, can be used to infer the velocities of the droplet. Interestingly, we find that, unlike the achiral case, the velocities of chiral active droplets show dimensionality dependence: its mean instantaneous velocity is higher in 3D than in 2D, whereas its mean migration velocity is lower in 3D than in 2D. Taken together, our particle-based simulations provide new insights into the dynamics of auto-chemotactic chiral active droplets, reveal the effects of dimensionality, and pave the way toward their applications, such as drug delivery, sensors, and micro-reactors. [ABSTRACT FROM AUTHOR]

In this work, we have performed a detailed study of the defects created in the bulk of 4H-SiC after thermal oxidation and post oxidation annealing using deep level transient spectroscopy and minority carrier transient spectroscopy (MCTS). The study reveals the formation of several shallow and deep level majority carrier traps in the bandgap. The ON1 (E C − 0.85 eV), ON2a (E C − 1.05 eV), and ON2b (E c − 1.17 eV) levels are the most dominant and are observed across all the samples (E C denotes the conduction band edge). Three shallow levels Ti(k) (E C − 0.17 eV), E 0.23 (E C − 0.23 eV), and C 1 / 2 (E C − 0.36 / 0.39 eV) are observed in the samples. For most of the majority carrier defects, the highest concentration is observed after an NO anneal at 1300 ° C. This behavior is sustained in the depth profile measurements where the defect concentration after the NO anneal at 1300 ° C is significantly higher than for the rest of the samples. The origin of most of the majority carrier defects has been attributed to C interstitial injection from the interface during thermal oxidation and annealing. MCTS measurements reveal two prominent minority carrier traps, labeled O 0.17 (E V + 0.17 eV) and B (E V + 0.28 eV), where the concentration of O 0.17 is independent of annealing parameters while the concentration of the B level increases after the NO anneal (E V denotes the valence band edge). Furthermore, the depth profiles of the defects are used to evaluate their diffusion parameters by solving the diffusion equation to fit the experimental profiles. The defect concentrations decay exponentially with depth, which evidences that the defects were created at or near the SiO 2 –SiC interface and migrate toward the bulk during oxidation and post-oxidation annealing. [ABSTRACT FROM AUTHOR]

A dynamic Monte Carlo (MC) method recently proposed by us [Nagai et al., J. Chem. Phys. 156, 154506 (2022)] to describe single-particle diffusion of a molecule in a heterogeneous space with position-dependent diffusion coefficient and free energy is generalized here to n-particle dynamics, where n molecules diffuse in heterogeneous media interacting via their intermolecular potential. Starting from the master equation, we give an algebraic proof that the dynamic MC transition probabilities proposed here produce particle trajectories that satisfy the n-particle diffusion equation with position-dependent diffusion coefficient D 0 i ( r i ) , free energy F 1 i ( r i ) , and intermolecular interactions Vij(ri, rj). The MC calculations based on this method are compared to molecular dynamics (MD) calculations for two-dimensional heterogeneous Lennard-Jones test systems, showing excellent agreement of the long-distance global diffusion coefficient between the two cases. Thus, the particle trajectories produced by the present MC transition probabilities satisfy the n-particle diffusion equation, and the diffusion equation well describes the long-distance trajectories produced by the MD calculations. The method is also an extension of the conventional equilibrium Metropolis MC calculation for homogeneous systems with a constant diffusion coefficient to the dynamics in heterogeneous systems with a position-dependent diffusion coefficient and potential. In the present method, interactions and dynamics of the real systems are coarse-grained such that the calculation cost is drastically reduced. This provides an approach for the investigation of particle dynamics in very complex and large systems, where the diffusing length is of sub-micrometer order and the diffusion time is of the order of milliseconds or more. [ABSTRACT FROM AUTHOR]

Focused ultrasound has experimentally been found to enhance the diffusion of nanoparticles; our aim with this work is to study this effect closer using both experiments and non-equilibrium molecular dynamics. Measurements from single particle tracking of 40 nm polystyrene nanoparticles in an agarose hydrogel with and without focused ultrasound are presented and compared with a previous experimental study using 100 nm polystyrene nanoparticles. In both cases, we observed an increase in the mean square displacement during focused ultrasound treatment. We developed a coarse-grained non-equilibrium molecular dynamics model with an implicit solvent to investigate the increase in the mean square displacement and its frequency and amplitude dependencies. This model consists of polymer fibers and two sizes of nanoparticles, and the effect of the focused ultrasound was modeled as an external oscillating force field. A comparison between the simulation and experimental results shows similar mean square displacement trends, suggesting that the particle velocity is a significant contributor to the observed ultrasound-enhanced mean square displacement. The resulting diffusion coefficients from the model are compared to the diffusion equation for a two-time continuous time random walk. The model is found to have the same frequency dependency. At lower particle velocity amplitude values, the model has a quadratic relation with the particle velocity amplitude as described by the two-time continuous time random walk derived diffusion equation, but at higher amplitudes, the model deviates, and its diffusion coefficient reaches the non-hindered diffusion coefficient. This observation suggests that at higher ultrasound intensities in hydrogels, the non-hindered diffusion coefficient can be used. [ABSTRACT FROM AUTHOR]

We present an approximate analytical approach to the adsorption problem of ABA triblock copolymers confined between two parallel plates in a θ solvent and give the expression of the propagator q(x, t) as a piece-wise function by solving the modified diffusion equation. In this way, the role of separation between the two plates, adsorption energy and block lengths on segment concentration profile, chain conformations, and interaction potential is then investigated, which agrees well with the numerical results. It is demonstrated that there are parallels between lengthening adsorbing A blocks and increasing surface affinity: strong adsorption and long adsorbing blocks favor the formation of loops and bridges, whereas more tails and free chains exist in the case of weak adsorption and short A blocks at large separations. For moderate and strong adsorptions, the bridging fraction begins to plummet at a separation larger than the end-to-end distance of non-adsorbing B block RB and becomes negligible at above 2RB owing to the entropy effect. The depth of the potential well in the interaction potential profile depends on the adsorption energy and A block length, while the location of the potential minimum corresponds to the onset of the sharp decrease in bridges. [ABSTRACT FROM AUTHOR]

Germanium has received increased research interest for use in next-generation CMOS technology as its high carrier mobilities allow for enhanced device performance without further device scaling. Fabrication of high-performance NMOS Ge devices is hindered by high diffusivity and low activation of n-type implanted dopants. While the high solid solubility of P in Ge makes it an ideal dopant, its diffusion mechanism is poorly understood and results in heavy tradeoffs between implanted dopant diffusion and electrical activation. In this study, we demonstrate the suppression of in-diffusion of implanted P via a co-implantation with Ar. Diffusivity of implanted P species and their activation is investigated over a wide range of annealing temperatures and times. P diffusion was explored by secondary-ion-mass-spectrometry and the diffusivity of P was extracted by solving the 2D diffusion equation using the Crank–Nicolson method, and the dopant electrical activation was extracted from the Hall effect measurements. The co-implantation of P with Ar entirely suppresses P in-diffusion up to annealing temperatures as high as 700 °C but at the cost of its reduced electrical activation. Extracted diffusivity reveals a highly correlated exponential relationship with annealing. P activation energy was extracted from Arrhenius behavior. A 450 °C/10 min annealing of P implant shows negligible in-diffusion of P with the activation as high as 70%. RTA processing of the Ar/P co-implanted sample at 750 °C for 1 min results in a negligible P in-diffusion and an electrical activation of 20%. [ABSTRACT FROM AUTHOR]

This article studies the output feedback finite‐time boundary control for unstable heat systems with the spatially varying coefficient. First, a finite‐time observer with switched gains under a state‐dependent switching law is designed in order to estimate the states of the system in a finite‐time only exerting one displacement boundary measurement. Next, an observer‐based linear finite‐time control is planned. Namely, a linear switched control under a state‐dependent switching law is proposed to vanish every solution of an unstable heat partial differential equation with spatially varying coefficients in a finite time. We also present explicit forms for the proposed observer gains and output feedback finite‐time controller. Finally, some numerical simulations are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: \[ \left\{ \begin{aligned} & \partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+};\cr & u(0,x)=f(x),\ x\in\mathbb R^{n}. \end{aligned}\right. \] { ∂ t β u (t , x) = − ν (− Δ) α / 2 u (t , x) , (t , x) ∈ R + 1 + n ; u (0 , x) = f (x) , x ∈ R n. Due to the time-fractional derivative, the semigroup property is invalid for the solutions $ u(\cdot,\cdot) $ u (⋅ , ⋅) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $ u(\cdot,\cdot) $ u (⋅ , ⋅) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations. [ABSTRACT FROM AUTHOR]

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods. [ABSTRACT FROM AUTHOR]

In this article, we examine the general space‐time fractional diffusion equation for left‐invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation, the space‐time fractional diffusion equation when diffusion is under the influence of sub‐Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well‐posedness of the Cauchy problem for the general space‐time fractional diffusion equation of the Rockland operator on a graded Lie group in the associated Sobolev spaces and also develop some regularity estimates for it. [ABSTRACT FROM AUTHOR]

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo‐differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo‐differential operators are derived. It is shown that pseudo‐differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation. [ABSTRACT FROM AUTHOR]

The optimality of decay properties of the one-dimensional damped wave equations with potentials belonging to a certain class is discussed. The typical ingredient is a variant of Nash inequality which involves an invariant measure for the corresponding Schrödinger semigroup. This enables us to find a sharp decay estimate from above. Moreover, the use of a test function method with the Nash-type inequality provides the decay estimate from below. The diffusion phenomena for the damped wave equations with potentials are also considered. [ABSTRACT FROM AUTHOR]

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper. [ABSTRACT FROM AUTHOR]

Innovative methods for substrate patterning provide intriguing possibilities for the development of devices based on ordered arrays of semiconductor nanowires. Control over the nanostructures' morphology in situ can be obtained via extensive theoretical studies of their formation. In this paper, we carry out an investigation of the ordered nanowires' formation kinetics depending on the growth mask geometry. Diffusion equations for the growth species on both substrate and nanowire sidewalls depending on the spacing arrangement of the nanostructures and deposition rate are considered. The value of the pitch corresponding to the maximum diffusion flux from the substrate is obtained. The latter is assumed to be the optimum in terms of the nanowire elongation rate. Further study of the adatom kinetics demonstrates that the temporal dependence of a nanowire's length is strongly affected by the ratio of the adatom's diffusion length on the substrate and sidewalls, providing insights into the proper choice of a growth wafer. The developed model allows for customization of the growth protocols and estimation of the important diffusion parameters of the growth species. [ABSTRACT FROM AUTHOR]

We consider the fully nonlocal diffusion equations with nonnegative L1$L^1$‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations. [ABSTRACT FROM AUTHOR]

We study the time fractional diffusion equation ∂tu=∂t1−αAu+G(u)$\partial _t u = \partial _t^{1-\alpha } \mathcal {A} u + G(u)$, 0<α<1$0<\alpha <1$, in a bounded domain Ω⊂RN$\Omega \subset \mathbb {R}^N$ with an elliptic operator A$\mathcal {A}$ and a locally Lipschitz nonlinearity G$G$ on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time t=0$t=0$, but at a final time T>0$T>0$, that is, u(T)$u(T)$ is given instead of u(0)$u(0)$. The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of G$G$. Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial L∞$L^\infty$‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and Lr−Ls$L^r-L^s$ estimates for heat semigroup. [ABSTRACT FROM AUTHOR]

The global dynamics of a nonlocal predator–prey model with mutation are investigated in this paper. First, by building a new comparison principle and constructing monotone iterative sequences, we give the existence of the solution for the corresponding initial boundary value problem. Then the uniqueness and uniformly boundedness of the solution are established by using the fundamental solution and quasi-fundamental solution of the heat equation, Grönwall's inequality and auxiliary functions. Finally, under the truncation function method, we discuss the asymptotic behavior of the solution. It is worth mentioning that the asymptotic behavior here is different from the conclusion of the classical model and we use more refined analysis and estimation in the research process. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous heat equation on the half-space $ \mathbb R_{+}^{d} $ R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).]. [ABSTRACT FROM AUTHOR]

Forward‐backward parabolic equations have been studied since the 1980s, but a mathematically rigorous picture is still far from being established. As quite a number of new papers have appeared recently, we review in this work the current state of the art. We focus our analysis on the status quo regarding the three most common types of regularizations, namely semidiscretization, the viscous approximation, and regularization with higher order spatial derivatives. [ABSTRACT FROM AUTHOR]

Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

The second-moment (SM) method is a linear variant of the quasi-diffusion (QD) method for accelerating the iterative convergence of Sn source calculations. It has several significant advantages relative to the QD method, diffusion synthetic acceleration, and nonlinear diffusion acceleration. Here, we define a variant of this method for k-eigenvalue calculations that retains the advantages of the original method, and we computationally demonstrate the efficacy of the method for simple example calculations. In particular, this method has two important properties. First, it is a linear acceleration scheme requiring only the solution of a pure k-eigenvalue diffusion equation with a corrective source term as opposed to a k-eigenvalue drift-diffusion equation. Second, unconditional stability is achieved even when the diffusion equation is not discretized in a manner consistent with the Sn spatial discretization. We are unaware of any other scheme that has these properties. We also show a connection between our method and the k-eigenvalue acceleration technique of Barbu and Adams, which motivated us to develop our SM method. [ABSTRACT FROM AUTHOR]

In contrast with the diffusion equation which smoothens the initial data to C^\infty for t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods. [ABSTRACT FROM AUTHOR]

The performance of many biomedical assays strongly depends on the transport of analyte molecules to the surface where the binding reactions occur. In some experimental systems, liquid flow can drastically improve the detection limit and decrease the analysis time. This is true for many membrane-based analyses, such as dot-blotting and immunofiltration, in which the sample flows through a porous membrane and adsorbs onto the polymer surface in the pores. Here, we use the convection-diffusion-adsorption equation to calculate the amount of the adsorbed analyte as a function of time. The analyte-binding efficiency is compared for the two sample loading procedures—pressure-driven flow and incubation. Computation shows that when the liquid flow rate reaches a certain threshold, the pressure-driven flow setup ensures higher analyte adsorption than incubation. The convection can drastically decrease the time necessary to capture a small detectable amount of the analyte. However, the infinite increase of the flow rate seems useless because it cannot overcome the kinetic limitation of the adsorption. The model proposed here is in good agreement with the experimental data and can be valuable for the development of membrane-based biosensors. [ABSTRACT FROM AUTHOR]

Let L = - Δ ℍ n + V be a Schrödinger operator on Heisenberg groups ℍ n , where Δ ℍ n is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 풬 / 2 . Here 풬 is the homogeneous dimension of ℍ n . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 , α > 0 , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ (⋅ , ⋅) , respectively. As an application, we characterize the space BMO L γ ⁢ (ℍ n) via { e - t ⁢ L α } t > 0 . [ABSTRACT FROM AUTHOR]

The operation of machine tools often demands a highly accurate knowledge of the tool center point's (TCP) position. The displacement of the TCP over time can be inferred from coupled thermal models, which comprise a set of geometrically coupled heat equations. Each of these equations represents the temperature in part of the machine, and they are often formulated on complicated geometries. The accuracy of the TCP prediction depends highly on the accuracy of the model parameters, such as heat exchange parameters, and the initial temperature. Thus it is of utmost interest to determine the influence of these parameters on the TCP displacement prediction. In turn, the accuracy of the parameter estimate is essentially determined by the measurement accuracy and the sensor placement. Determining the accuracy of a given sensor configuration is a key prerequisite of optimal sensor placement. In this paper, we develop a thermal model for the discretization of realistic machine tool. On top of this model we propose two numerical algorithms to evaluate any given thermal sensor configuration with respect to its accuracy. We compute the posterior variances from the posterior covariance matrix with respect to an uncertain initial temperature field. The full matrix is dense and potentially very large, depending on the size of the discretized model. We first present a way to compute this approximation, which relies on the computation of the model sensitivities with respect to the initial values. Additionally, we present a low-rank tensor method which exploits the underlying system structure. We compare the efficiency of both algorithms with respect to runtime and memory requirements and discuss their respective advantages with regard to optimal sensor placement problems. [ABSTRACT FROM AUTHOR]

This paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop H1$$ {H}&#x0005E;1 $$ stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally. [ABSTRACT FROM AUTHOR]

This paper investigates the longtime behavior of solutions to a temporally discrete diffusion equation with a fixed boundary and a free boundary respectively in one space dimension. Such equation can be equivalent in any case to an integro-difference equation, another important time discrete equation that provides powerful tools for the study of dispersal phenomena. In this paper, we first discuss the global dynamics of the equation in a fixed bounded domain. With a Stefan type free boundary, we then give a new well-posedness proof and the regular spreading-vanishing dichotomy for the corresponding problem. Moreover, a modified comparison principle for the time discrete free boundary problem is proved in an effort to provide the sufficient conditions for dichotomy. It is the first attempt to study the temporally discrete diffusive phenomenon with a free boundary. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following 2D incompressible chemotaxis‐Navier–Stokes equations with the fractional diffusion ∂tn+u·∇n−Δn=−∇·(n∇c)+n(1−n)(n−a),∂tc+u·∇c−Δc=−cn,∂tu+u·∇u+∧2αu+∇P=−n∇ϕ,$$ \left\{\begin{array}{l}{\partial}_tn&#x0002B;u\cdotp \nabla n-\Delta n&#x0003D;-\nabla \cdotp \left(n\nabla c\right)&#x0002B;n\left(1-n\right)\left(n-a\right),\\ {}{\partial}_tc&#x0002B;u\cdotp \nabla c-\Delta c&#x0003D;- cn,\\ {}{\partial}_tu&#x0002B;u\cdotp \nabla u&#x0002B;{\wedge}&#x0005E;{2\alpha }u&#x0002B;\nabla P&#x0003D;-n\nabla \phi, \end{array}\right. $$ where ∧:=(−Δ)12$$ \wedge :&#x0003D; {\left(-\Delta \right)}&#x0005E;{\frac{1}{2}} $$ and α∈[12,1]$$ \alpha \in \left[\frac{1}{2},1\right] $$. The gravitational potential ϕ$$ \phi $$ is generally supposed to be sufficiently smooth given functions. By establishing the new priori estimates, we get the global well‐posedness for the above system with large initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion { ∂ t n + u ⋅ ∇ n − Δ n = − ∇ ⋅ (n ∇ c) − n 3 , ∂ t c + u ⋅ ∇ c − Δ c = − c + n , ∂ t u + u ⋅ ∇ u + ∧ 2 α u + ∇ P = − n ∇ ϕ , where ∧ : = (− Δ) 1 2 and α ∈ [ 1 2 , 1 ] . We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions. [ABSTRACT FROM AUTHOR]

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier–Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside L^2(\mathbb {T}^d) such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier–Lebesgue \gamma-radonifying regularity in the construction of the Wick powers of the associated stochastic convolution. [ABSTRACT FROM AUTHOR]

The present paper deals with self‐heating phenomenon under cyclic loading at high loading frequency 20 kHz and low stress amplitude (i.e., far below the yield stress). The main goal is to propose two different methods to determine the heat source field at the origin of this temperature change. To do that the local expression of the 1D heat diffusion equation and experimental thermal data fields obtained by an infrared camera are used. The proposed methodology is applied to experimental results from tests performed on a ML340 maraging steel. [ABSTRACT FROM AUTHOR]

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer's function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay. [ABSTRACT FROM AUTHOR]

This study uses the cubic spline method to solve the one-dimensional (1D) (one spatial and one temporal dimension) heat problem (a parametric linear partial differential equation) numerically using both explicit and implicit strategies. The set of simultaneous equations acquired in both the explicit and implicit method may be solved using the Thomas algorithm from the tridiagonal dominating matrix, and the spline offers a continuous solution. The results are implemented with very fine meshes and with relatively small-time steps. Using mesh refinement, it was possible to find better temperature distribution in the thin bar. Five numerical examples are used to support the efficiency and accuracy of the current scheme. The findings are also compared with analytical results and other results in terms of error and error norms L 2 and L ∞ . The Von-Neuman technique is used to analyse stability. The truncation error of both systems is calculated and determined to have a convergence of order O h + Δ t 2 . [ABSTRACT FROM AUTHOR]

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical Q 1 -finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter ω K. The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific ω K to ensure the coercivity result of the proposed schemes on arbitrary parallelogram, quasi-parallelogram, trapezoidal and some general convex quadrilateral meshes. More interesting is that, the parameter ω K can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal H 1 error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples. [ABSTRACT FROM AUTHOR]

This paper derives well-posedness and asymptotic results that provide qualitative information about the behavior, mechanism and strategies used by living organisms to navigate their biological networks. Chemical driven swimming is a captivating phenomenon that is observed in various living organisms like bacteria and protozoa but the problem in weighted networks is more complex, since the equations of parabolic-parabolic Keller-Segel model coupled with incompressible Navier-Stokes equations must be reformulated in a discrete setting. The starting point is to transpose the coupled system from the Euclidean case to the connected networks with certain network-theoretic simplified approaches, which yields fruitful key results. These results not only enable the construction of global solutions but also serve as a foundation for determining information about the stability and large time behavior of the system. Then, decay rates are well established to predict important features, such as how quickly weak solutions at a given point decrease over time due to dissipative processes. Additionally, the L 1 - convergence of cell densities towards the self-similar Gaussian solution of the heat equation is well proved by time dependent scaling, which shows that the solution maintains its shape and only scales in time and space as time evolves. Finally, this paper includes many numerical tests through a recent robust numerical scheme to illustrate the theoretical results and to develop computational control and prediction of living organisms' trajectories around central nodes in networked flows. [ABSTRACT FROM AUTHOR]

This study provides a detailed numerical assessment of the productivity of single-slope solar stills across key cities in the Caspian region, including Aktau, Atyrau, Astrakhan, Makhachkala, Baku, Tehran, and Turkmenbashi. Employing a mathematical model based on heat balance equations and the Fourth Order Runge–Kutta method, we simulated the distillation process under various climatic conditions. The results reveal that productivity is significantly influenced by geographic location and local meteorological conditions. Tehran demonstrated the highest productivity across all seasons, with values of 1.75 × 103 kg/(m2·year) and an efficiency of 0.53, due to its optimal solar irradiation and ambient temperature. In contrast, Atyrau and Makhachkala exhibited lower productivity, particularly in colder months, highlighting the effect of ambient temperature on solar still efficiency. The analysis identified the optimal water depth at 2 cm and insulation thickness between 4 and 9 cm for enhancing productivity in continental climates like Aktau. Additionally, the lowest cost of distilled water was USD 0.024 per kilogram in Baku. These findings align with the existing literature, validating the numerical model's accuracy. Future research will explore integrating solar stills with other renewable and fossil fuel-based technologies. [ABSTRACT FROM AUTHOR]

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ u (t , x) ∂ t = L u (t , x) + u (t , x) d B H , λ (t) d t , (t , x) ∈ R + × R d , where L is the infinitesimal generator of some time-homogeneous Markov process { X t } t ≥ 0 and { B H , λ (t) } t ∈ R is a tempered fractional Brownian motion with Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formula u (t , x) = E x [ f (X t) ] e B H , λ (t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f , Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that lim t → ∞ ⁡ ln ⁡ ‖ u (t , ⋅) ‖ p g (t) = 0 in the almost surely sense where p ∈ R and the function g : R + → R + grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals. [ABSTRACT FROM AUTHOR]

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Z n whose initial condition is given by a linear code. [ABSTRACT FROM AUTHOR]

Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton–Jacobi–Bellman equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks. [ABSTRACT FROM AUTHOR]

The infra-red video bolometer (IRVB) is a diagnostic equipped with an infra-red camera that measures the total radiated power in thousands of lines of sight within a large field of view. Recently validated in MAST-U [Fderici et al., Rev. Sci. Instrum. 94, 033502 (2023)], it offers a high spatial resolution map of the radiated power in the divertor region, where large gradients are expected. The IRVB's sensing element comprises a thin layer of high Z absorbing material, typically platinum, usually coated with carbon to reduce reflections [Peterson et al., Rev. Sci. Instrum. 79, 10E301 (2008)].Here, the possibility of using a relatively inert material such as titanium, is explored that can be produced in layers up to 1 μm compared to 2.5 μm for Pt and then coat it with Pt of the desired thickness (0.3 μm per side here) and carbon. This leads to a higher temperature signal (about 3 times) and better spatial resolution (about 4 times), resulting in higher accuracy in the measured power [Peterson et al., Rev. Sci. Instrum. 79, 10E301 (2008)]. This assembly is also expected to improve foil uniformity, as the Pt layer is obtained via deposition rather than mechanical processes [Mukai et al., Rev. Sci. Instrum. 87, 2014 (2016)].Given its multi-material composition, measuring the thermal properties of the foil assembly is vital. Various methods using a calibrated laser as a heat source have been developed, analyzing the temperature profile shape [Sano et al., Plasma and Fusion Res. 7, 2405039 (2012)] and [Mukai et al., Rev. Sci. Instrum. 89, 10E114 (2018)] or fitting the calculated laser power for different intensities and frequencies [Fderici et al., Rev. Sci. Instrum. 94, 033502 (2023)]. Here, a simpler approach is presented, which relies on analyzing the separate components of the foil heat equation for a single laser exposure in a given area. This can then be iterated over the entire foil to capture local deviations. [ABSTRACT FROM AUTHOR]

The conformal fractional derivative (CFD) has become a hot research topic since it has a physical interpretation and is easier to grasp and apply to problems compared with other fractional derivatives. Its application to heat transfer, diffusion, diffusion-advection, and wave propagation problems can be found in the literature. Fractional diffusion equations have received great attention recently due to their applicability in physical, chemical, and biological processes and engineering. The diffusion of the pollutants within the ground, which is an important environmental problem, can be modeled with a diffusion equation. Diffusion in some porous materials or soil can be modeled more accurately with fractional derivatives or the conformal fractional derivative. In this study, the diffusion problem of a spilled pollutant leaking into the ground modeled with the conformal fractional time derivative in spherical coordinates has been solved analytically using the Fourier series for a constant mass flow rate and complete symmetry under the assumptions of homogeneous and isotropic soil, constant soil temperature, and constant permeability. The solutions have been simulated spatially and in time. A parametric analysis of the problem has been performed for several values of the CFD order. The simulation results are interpreted. It has also been suggested how to find the parameters of the soil to see whether the CFD can be used to model the soil or not. The approach described here can also be used for modeling pollution problems involving different boundary conditions. [ABSTRACT FROM AUTHOR]

The solution of coupled multi‐field problems by means of commercially available finite element codes such as Abaqus constitutes an important part in the study of industrial processes. Against this background, a comprehensive implementation framework for a gradient‐enhanced continuum damage formulation in a thermo‐mechanically coupled finite deformation setting is proposed in this contribution. To assess the applicability of the implementation framework to industrial processes, a thermo‐mechanically coupled, gradient‐enhanced ductile damage model is exemplarily employed. The resulting three‐field problem, consisting of the balance of linear momentum, the heat equation and the balance of micromorphic momentum, is implemented based on an Abaqus user material by making use of a novel two‐instance formulation. Representative simulation results are discussed, with the specific focus lying on the application to manufacturing processes involving complex contact interactions. The Abaqus framework is made available as an open‐source code on GitHub, see Sobisch et al., Finite Elements in Analysis and Design 232 (2024) 104105. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type t−4$t^{-4}$. Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied. [ABSTRACT FROM AUTHOR]

The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space RN$\mathbb {R}^N$ with N≥3$N\ge 3$. In our work, we suppose the initial condition v0$v_0$ stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution u∈L∞((0,T);Hs(RN)),∇u∈L2((0,T);Hs(RN)),$$\begin{equation*} \hspace*{60pt}u\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla u\in L^2((0,T);H^s(\mathbb {R}^N)), \end{equation*}$$∇v∈L∞((0,T);Hs(RN)),∇2v∈L2((0,T);Hs(RN))$$\begin{equation*} \hspace*{55pt}\nabla v\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla ^2v\in L^2((0,T);H^s(\mathbb {R}^N)) \end{equation*}$$for s>N2−1$s>\frac{N}{2}-1$, asking low regularity assumptions on u0$u_0$ and v0$v_0$. [ABSTRACT FROM AUTHOR]

A wide class of two‐component evolution systems is constructed admitting an infinite‐dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross‐diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above‐mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two‐component evolution systems with an infinite‐dimensional Lie symmetry that possess essentially different structures is presented. [ABSTRACT FROM AUTHOR]

This research aimed to implement a finite element heat transfer model to simulate the thermal cycle of steel ladles using an open‐source tool. FEniCS was selected, as it can solve partial derivative equations such as the heat transfer ones, which describes the thermal state of the steel ladle via finite element method. For this, each step of the steel ladle cycle (preheating, waiting steps, and holding) had its own model, comprising specific combinations of boundary conditions. An energy consumption analysis was carried out based on the calculation of the heat flux for each stage of the ladle cycle according to three selected configurations of the refractory lining chosen for the simulation. The results showed that the solutions obtained by both the open‐source method and by the commercial tool (Abaqus) were equivalent. The comparison of different lining configurations, especially using insulators, presented several advantages from the energy point of view. The attained conclusions can be of great interest to the refractory engineers who seek novel solutions to improve the insulating performance of the steel ladle. The present framework makes it possible to test a multitude of different designs and processing operation conditions, making the proposed tool a cost‐effective solution to aid the required improvements to reduce the carbon footprint of critical industrial processes such as steelmaking. [ABSTRACT FROM AUTHOR]