Linear Diffusion Research Articles

Spreading behavior of Sn droplets impacting Cu and stainless steel substrates

With the development of droplet 3D printing, thermal spraying, spray molding, and other technologies, the collision diffusion process of metal droplets and metal substrates has become a key aspect in controlling these processes. This study used Sn droplets as the research object. Using a pneumatic pulse jet device, molten Sn droplets were impinged on the surface of copper and stainless steel substrates. The impact process was recorded by high-speed photography. The diffusion diameter and solid-liquid contact angle were measured from these images, revealing the difference in the diffusion of Sn droplets on different surfaces. This explains the difference in the droplet diffusion ability from the perspective of driving energy. The entire impact process and final morphology were considerably similar when the substrate temperature was lower than the melting point of Sn. When the substrate temperature was higher than the melting point of Sn, the entire diffusion behavior of the tin droplet from contacting the substrate to the steady state was investigated in stages. From the driving energy analysis, it was observed that the Sn droplet is mainly driven by its internal kinetic energy in the initial stage of diffusion. When the diffusion diameter approached the maximum value, the kinetic energy of Sn droplets gradually dissipated; the surface energy and viscous force work gradually increased, and the diffusion changed from a linear diffusion state to a slow diffusion state. Simultaneously, the surface energy and viscous force of the droplet played a key role. The adhesion of Sn droplets to the surface of the stainless steel substrate was very small, and the interface did not react, which caused the Sn droplets to shrink significantly on the stainless steel surface during the retraction stage.

An efficient method for predicting long term extreme design forces of wave energy converters

This paper proposes to utilize a modified adaptive KDE (Kernel Density Estimation) method based on linear diffusion process in order to more efficiently and accurately calculate the sea state parameter distribution tails and to extrapolate well. This modified adaptive KDE method is applied in predicting the probability distribution tails of a measured ocean wave dataset at National Data Buoy Center Station 44,009, and its efficiency and accuracy have been clearly substantiated through comparisons with the prediction results via the parametric method and the traditional adaptive KDE method. Subsequently, the modified adaptive KDE method is utilized for deriving an accurate 50-year environmental contour line based on the aforementioned measured wave dataset. The derived environmental contour line and another contour line obtained using a parametric contour approach are then applied for predicting the 50-year design PTO (Power-Take-Off) force values for a point absorber Wave Energy Converter (WEC). The predicted 50-year design PTO force value based on the proposed modified adaptive KDE contour (whose accuracy has been clearly substantiated in this paper) is 20674KN, and the WEC PTO system designed based on this accurate prediction obviously will be safer and more reliable. On the contrary, the forecasted 50-year design PTO force value based on the inaccurate contour via the parametric method is only 9256.3 KN, which is likely to cause failure of the PTO system and the entire Wave Energy Converter.

Asymptotic Stability of Diffusion Waves of a Quasi-Linear Hyperbolic-Parabolic Model for Vasculogenesis

In this paper, we derive the large-time profile of solutions to the Cauchy problem of a hyperbolic-parabolic system modeling the vasculogenesis in $\R^3$. When the initial data are prescribed in the vicinity of a constant ground state, by constructing a time-frequency Lyapunov functional and employing the Fourier energy method and spectral analysis, we show that solution of the Cauchy problem tend time-asymptotically to linear diffusion waves around the constant ground state with algebraic decaying rates under certain conditions on the density-dependent pressure function.

Resurfacing processes constrained by crater distribution on Ryugu

Understanding the geological modification processes on asteroids is fundamental for elucidating their surface evolution. Images of small asteroids from spacecrafts show a depletion in smaller craters. Seismic shaking was considered to be responsible for erasing such small craters and the main driver modifying the geology of asteroids. However, a recent artificial impact experiment on asteroid Ryugu conducted by the Japanese Hayabusa2 spacecraft revealed minimal seismic efficiency on the asteroid. To investigate whether a standard seismic shaking model can reproduce the observed crater record, we analyzed the crater distributions on four asteroids, i.e., Eros, Itokawa, Ryugu, and Bennu, using crater production functions under cohesionless conditions. Crater retention ages were estimated as a function of crater diameter for the four asteroids using the crater size-frequency distribution and crater production function estimated for each asteroid. We obtained the relation between a crater retention age t and crater diameter D in the form of power-law function (t ∝ Da).We found that the power-law indices a are inconsistent with diffusion processes (e.g., seismic shaking, where a = 2). This result suggests that seismic shaking models based on a linear diffusion equation cannot explain the crater distribution on the small asteroids. Alternative crater obliteration processes include surface flows suggested by geomorphological and spectral features of Ryugu. Using the crater statistics, we also show that the vertical mixing of the Ryugu material at the depths shallower than 1 m occurs in 103–105 yr by cratering. This rapid resurfacing and replacement of the surface layer in the short timescale would account for the decrease in space weathering rate suggested by previous studies. Furthermore, the timescale required for vertically transporting Ryugu materials to the depths of 2–4 m (104–106 yr), where cosmic rays would not reach, can be compared with the cosmic-ray exposure ages of returned samples. This comparison can be used to constrain the distribution of impactors that collide with Ryugu.

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0,where θ > 0 and ν∈R are two parameters. When θ < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.

Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ < 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

Branched Supramolecular Copolymers: Inducing Branching in Bisurea-Based Monomers Using Multi-Sulfonate Molecules

Supramolecular polymers are based on noncovalent interactions, which impart unique properties such as dynamic behavior, concentration-dependent degree of polymerization, and environmental responsiveness. While linear supramolecular polymers are ubiquitous and have been extensively studied, branched polymers that are based exclusively on supramolecular interactions are much less abundant, and a fundamental understanding of their molecular-level structure is still lacking. We report on the preparation of branched, all-supramolecular polymers based on a combination of a bisureidotoluene building block [N,N′-2,4-bis((2-ethylhexyl)ureido)toluene (EHUT)], which is associated with four-point hydrogen bonding, and three anionic co-monomers featuring one, two, or three sulfonate groups. The co-monomers were designed to serve as a chain stopper, a bifunctional linear co-monomer, and a branch point. Whereas combination of EHUT with the singly functionalized co-monomer led to linear supramolecular chains, diffusion and viscosity data indicate that branched supramolecular polymers were obtained when EHUT was combined not only with the triply functionalized molecules but also with the doubly functionalized molecules. Theoretical analysis based on an adaptation of Flory’s theory of branched polymers suggests that in both cases, the interaction of certain EHUT units with the multiply functionalized co-monomers converted these EHUT units into branch points, which led to substantially reduced viscosities in these systems. The insights gained from this study enable tuning the properties of supramolecular polymers not only by concentration and temperature but also by introducing appropriately designed molecular additives. This may lead to the development of sophisticated smart materials.

Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation

In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the performance of the proposed approach in the case that the hidden signal is governed by the two-dimensional stochastic advection-diffusion equation.

An integrated study of wave attenuation by vegetation

Coastal protection is an important issue. For many years, protection of coastal areas has been approached from an engineering perspective, creating hard defences to withstand wave forces. However, ‘soft engineering’ alternatives that blend in with the natural environment. Vegetation such as seagrass can dissipate wave energy, acting as a natural barrier. In contrast to hard defences, there are currently limited design guidelines for designing soft defences based on vegetation. Here, we present a mathematical model of wave attenuation by vegetation. Of interest is whether a relatively simple wave model is capable of capturing the attenuation of wave energy by vegetation. Specifically, whether the attenuation of short and long waves can be mimicked through the introduction of a linear drag or diffusion term in the momentum equation. The model is based on the shallow water equations. To accommodate the dispersive effect that appears in short waves, we modified the model further to include the hydrodynamic pressure. The equations are solved numerically using a free-damping-error method. Analytical solutions derived for the case where the seabed is flat are used to verify the computational solutions. In addition, results of laboratory experiments of wave attenuation by artificial seagrass provide additional testing of the computational model. Extremely good agreement between computations and analytical solutions was found. In comparisons with the laboratory experiments, the best results with the computational model were found using diffusion for long waves and using a simple drag term for short waves. Wave attenuation was also found to depend strongly on the dimension of the vegetated area.

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

<p style='text-indent:20px;'>In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and <inline-formula><tex-math id="M2">\begin{document}$ 1-d $\end{document}</tex-math></inline-formula> unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.</p>

DIGITAL IMAGE RESTORATION USING LINEAR PDE-BASED FILTERING MODELS

The linear partial differential equation (PDE) - based models for Gaussian noise removal are discussed in this paper. The digital image denoising and restoration solutions based on linear diffusion equations are surveyed first. Then, our own contributions in this image processing domain, representing some effective linear PDE-based filtering models based on hyperbolic and stochastic differential equations, are presented. The results of our denoising experiments are also provided in this article.

Deformation evolution and diffusion characteristics of polymethyl methacrylate under impact loading

Dynamic compression experiments on cubic polymethyl methacrylate (PMMA) specimens and two kinds of trapezoid PMMA specimens are carried by changing the transmission bars into steel bar and aluminum bar on the experimental device of split Hopkinson pressure bar (SHPB). The compression processes of PMMA specimens are recorded by high-speed photography, and the breakage processes of PMMA specimens are analyzed based on the force displacement curves and high-speed images. The evolutions of deformation and diffusion resistances of PMMA specimens under impact loading are discussed. The results show that the failure of the sample is caused mainly by the partial failure front at the contact end, and then the failure front propagates to the inside of the sample, leading the sample to break. The failure front of cubic sample is generated preferentially at the transmission end under low speed impact and at the incident end under the higher speed impact. After changing the shape of the specimen and the material of the transmission bar, the relaxation phenomenon is prominent, and the failure front occurs only at the incident end. The compressive deformation of the trapezoid sample before breakage is non-uniform, and the stress and strain in the sample gradually decrease with the increase of the cross section, and show a linear diffusion distribution. The strain distribution and shear activation diffusion equation are used to obtain the generalized diffusion resistance distribution of the failure front. The generalized diffusion resistance increases first in front of the failure front and decreases after the failure front, and the amplitude of the generalized diffusion resistance is related to the release of local strain energy.

Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon

We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them continuation bays and stopping spikes). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.

Using Leray-Schauder topological degree to solve a linear diffusion parabolic equation with periodic initial conditions

Throughout this manuscript, we show time periodic solutions to a linear diffusion parabolic equation with Diriclet condition. Based on the topological degree theorem, we prove a time periodic solutions of the system such that we found the fixed point when the domain of the solution is sufficiently small.

A fast method for solving time-dependent nonlinear convection diffusion problems

<abstract><p>In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.</p></abstract>

Examining the Performance of MATLAB’s Matrix Capabilities, Testing on Euler’s Method Applied on the Diffusion Equation

When one develops, tests and uses numerical methods to solve a differential equation, the performance of the method depends on the concrete way how the method is implemented and coded. In this tentative work, we solve the linear diffusion equation by the simplest explicit Euler method implemented with for loops as well as the built-in matrix operations of MATLAB. We obtain that the for loop performs better in one space dimension, but the matrix operations are faster in two space dimensions.

Study on deformation evolution and diffusion characteristics of PMMA under impact loading

Dynamic compression experiments were carried out on cubic PMMA specimens and two kinds of trapezoid PMMA specimens by changing the transmission bars to steel and aluminum bar with the Loading device of split Hopkinson pressure bar (SHPB). The compression process of PMMA specimens was recorded by high-speed photography, and the breakage process of PMMA specimens was analyzed based on the force displacement curves and high-speed images. The evolution of deformation and diffusion resistance of PMMA specimens under impact loading was discussed. The results show that the failure of the sample is mainly caused by the partial failure front at the contact end, and then the failure front propagates to the inside of the sample and leads to the breakage of the sample. The failure front of cubic samples is preferentially generated at the transmission end under low speed impact and generated at the incident end under the higher speed impact. After changing the shape of the specimen and material of the transmission bar, the relaxation phenomenon is obvious, and the failure front only occurs at the incident end. The compressive deformation of the trapezoid sample before breakage is non-uniform, and the stress and strain in the sample gradually decrease with the increase of the cross section, and show a linear diffusion distribution. The strain distribution and shear activation diffusion equation were used to obtain the generalized diffusion resistance distribution of the failure front. The generalized diffusion resistance increases first in front of the failure front and decreases after the failure front and the amplitude of the generalized diffusion resistance is related to the release of local strain energy.

Modeling the volatity of cryptocurrency markets

The application of the model of geometric Brownian motion (GBM) for the problem of modeling and forecasting prices for cryptocurrencies is analyzed. For prediction the solution of the stochastic differential equation of the GBM model is used, which has a linear drift and diffusion coefficients. Different scenarios of price movement are considered.
 Keywords: geometric Brownian motion (GBM), modeling, forecasting, cryptocurrency.