Generalized Diffusion Equation Research Articles

Morphology-Transport Coupling and Dissipative Structures in PEO-PS+LiTFSI Electrolytes In-Operando Conditions.

A Single-Chain-in-Mean-Field (SCMF) algorithm was introduced to study block copolymer electrolytes in nonequilibrium conditions. This method self-consistently combines a particle-based description of the polymer with a generalized diffusion equation for the ionic fluxes, thus exploiting the time scale separation between fast ion motion and the slow polymer relaxation and self-assembly. We apply this computational method to study ion fluxes in electrochemical cells containing poly(ethylene oxide)-polystyrene (PEO-PS) block copolymers with added lithium salt. Blocking of the anion fluxes by the electrodes in-operando conditions polarizes the cells and results in an inhomogeneous salt-concentration profile. This gradient of salt concentration triggers lamellae-to-disorder and disorder-to-lamellae transitions near the electrodes, in good agreement with previous experimental observations. The effects of the selectivity of the electrode surface, the salt concentration and the voltage applied to the cell are systematically studied. For PEO-selective surfaces, the lamellae parallel to the electrode that forms at low applied potentials transition to a bicontinuous morphology at high applied potentials in order to allow ion transport through the insulating PS layers. The formation of this dissipative structure, which is unexpected considering the equilibrium behavior of the material, is in line with the principle of maximum entropy production. In summary, the transport and morphology in PEO-PS electrolytes are strongly coupled: ionic currents influence self-assembly, which in turn modulates the ionic fluxes in the cell.

Continuity of solutions for tempered fractional general diffusion equations driven by TFBM

Continuity of solutions for tempered fractional general diffusion equations driven by TFBM

Solvability of transmission problems with generalized diffusion equation in L^p-spaces

We study a transmission problem in population dynamics between two juxtaposed habitats. In each habitat, we consider a generalized diffusion equation composed by the Laplace operator and a btocould be negative or null. Using semigroup theory and functional calculus, we give some relation between coefficients to obtain the existence and uniqueness of a classical solution in \(L^p\)-spaces. For more information see https://ejde.math.txstate.edu/Volumes/2024/78/abstr.html

Feynman-Kac formula for tempered fractional general diffusion equations driven by TFBM

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u ( t , x ) = L u ( t , x ) + u ( t , x ) B ˙ H , λ ( t ) , ( t , x ) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ ( 0 , 1 ) and tempered parameter η > 0, 𝔏 is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by P t f(x): = 𝔼 x [f(X t )], and { B H , λ ( t ) } t ≥ 0 is a TFBM with Hurst index H ∈ ( 1 2 , 1 ) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator D β, η (t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM B H, λ (t) and the inverse tempered β-stable subordinator D β, η (t), we show that the Feynman-Kac representation u ( t , x ) = E D [ P D β , η ( t ) f ( x ) e ∫ 0 t ∫ 0 t δ ( s − r ) d D β , η ( s ) d B H , λ ( r ) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator D β, η (t) and the novel estimate of stochastic integrals with respect to B H, λ (t) and D β, η (t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution.

Anomalous Diffusion and Non-Markovian Reaction of Particles near an Adsorbing Colloidal Particle

We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid’s surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles’ behavior on the colloidal particle’s surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better.

Direct and some inverse problems for a generalized diffusion equation with variable coefficients

Direct and two inverse problems for a Legendre equation involving integral convolution in time are studied. The inverse problems are ill-posed in the sense of Hadamard. The analytical series solutions of the problems are constructed by using method of variable separation. The determination of only u(x, y) is studied in the direct problem, the recovery of a pair of functions, i.e., {u(x,y),f(x)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u(x,y), f(x)\\}$$\\end{document} with appropriate addition data at some T is investigated in the 1st inverse problem while the identification of a pair of functions, i.e., {u(x,y),q(y)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u(x,y), q(y)\\}$$\\end{document} with an integral type data is considered in the 2nd inverse problem. By imposing certain regularity conditions, the unique existence of series solutions is developed. We provided some numerical examples to illustrate our results for the inverse problems.

Stress relaxation following sudden cessation of steady shearing from polymer rotarance theory

Deriving rheological material functions from rotarance theory proceeds in two steps. We first solve the general diffusion equation to get the polymer orientation distribution, then we integrate the result in phase space to get the material functions. Rotarance theory thus relies entirely on polymer orientation to explain the elasticity of a polymeric liquid and, thus, to explain how such complex fluids depart from Newtonian behavior. We are attracted to rotarance theory by virtue of its versatility. The rheological behavior can be deduced, from first principles, from the structure of the macromolecule. However, of the 13 canonical rheological material functions, 5 are still unknown. We devote this paper to stress relaxation following cessation of steady shear flow. We arrive at analytical expressions for the relaxation of the orientation distribution function and then integrate this to get expressions for the relaxation of the shear stress, along with both normal stress differences.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) 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 limt→∞⁡ln⁡‖u(t,⋅)‖pg(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.

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

A Generalized Neural Diffusion Framework on Graphs

Recent studies reveal the connection between GNNs and the diffusion process, which motivates many diffusion based GNNs to be proposed. However, since these two mechanisms are closely related, one fundamental question naturally arises: Is there a general diffusion framework that can formally unify these GNNs? The answer to this question can not only deepen our understanding of the learning process of GNNs, but also may open a new door to design a broad new class of GNNs. In this paper, we propose a general diffusion equation framework with the fidelity term, which formally establishes the relationship between the diffusion process with more GNNs. Meanwhile, with this framework, we identify one characteristic of graph diffusion networks, i.e., the current neural diffusion process only corresponds to the first-order diffusion equation. However, by an experimental investigation, we show that the labels of high-order neighbors actually appear monophily property, which induces the similarity based on labels among high-order neighbors without requiring the similarity among first-order neighbors. This discovery motives to design a new high-order neighbor-aware diffusion equation, and derive a new type of graph diffusion network (HiD-Net) based on the framework. With the high-order diffusion equation, HiD-Net is more robust against attacks and works on both homophily and heterophily graphs. We not only theoretically analyze the relation between HiD-Net with high-order random walk, but also provide a theoretical convergence guarantee. Extensive experimental results well demonstrate the effectiveness of HiD-Net over state-of-the-art graph diffusion networks.

Anomalous heat transport and universality in macroscopic diffusion models

Anomalous diffusion is ubiquitous in nature and relevant for a wide range of applications, including energy transport, especially in bio- and nano-technologies. Numerous approaches have been developed to describe it from a microscopic point of view, and recently, it has been framed within universality classes, characterized by the behaviour of the moments and auto-correlation functions of the transported quantities. It is important to investigate whether such universality applies to macroscopic models. Here, the spectrum of the moments of the solutions of the transport equations is investigated for three continuous PDE models featuring anomalous diffusion. In particular, we consider the transport described by: (i) a generalized diffusion equation with time-dependent diffusion coefficient; (ii) the Porous Medium Equation and (iii) the Telegrapher Equation. For each model, the key features of the source-type solution as well as the analytical results for the moment analysis are revisited and extended via both analytical and numerical approaches. Equivalence of the asymptotic behaviour of the corresponding heat transport is confirmed within the realm of weak anomalous diffusion.

Identifying temperature distribution and source term for generalized diffusion equation with arbitrary memory kernel

A diffusion equation involving integral convolution in time variable with arbitrary kernel and nonlocal boundary conditions is considered. The existence and uniqueness results for two inverse problems of determining source terms (space‐ and time‐dependent sources) along with diffusion concentration from appropriate over‐specified conditions are presented. A bi‐orthogonal system of functions is used to have series representation of the solutions of the inverse problems. Several special cases such as standard diffusion, multi‐term diffusion, and tempered diffusion equations are discussed, and some examples are provided.

Feynman-Kac formula for tempered fractional general diffusion equations with nonautonomous external potential

Feynman-Kac formula for tempered fractional general diffusion equations with nonautonomous external potential

On the solvability of direct and inverse problems for a generalized diffusion equation

This paper delves into both direct and two inverse source problems associated with a diffusion equation featuring integral convolution over time, while considering non-classical boundary conditions. The inverse source problems are shown to exhibit ill-posed characteristics in accordance with Hadamard’s definition. A bi-orthogonal function system is employed to express series solutions for the inverse source problems. By imposing specific conditions on the provided data, we establish the existence of unique series solutions. Several special cases of the diffusion equation are presented, depending on the nature of the memory kernel. Furthermore, to illustrate the findings regarding inverse source problems, we provide specific examples.

General rigid bead-rod theory for steady-shear flow

General rigid bead-rod theory yields uniquely the relation between macromolecular architecture and complex viscosity. For this, it relies on the analytical solution of the general diffusion equation for small-amplitude oscillatory shear flow of Bird et al. [Dynamics of Polymeric Liquids, 2nd ed. (Wiley, New York, 1987), Vols. 1–2]. Unfortunately, this general diffusion equation has yet to be solved for any other flow field. In this paper, we do so for steady-shear material functions, namely, viscosity and first normal stress coefficient. We, thus, explain the non-Newtonian behaviors of macromolecular suspensions of any axisymmetric design in steady-shear flow.

Large-amplitude oscillatory shear flow from general rigid bead-rod theory

Oscillatory shear flow, performed at small-amplitude, interrogates polymeric liquids in their equilibrium states. The fluid responds in sinusoidal shear stress waves whose amplitude and phase lead depend on the dimensionless frequency (called the Deborah number). By contrast, this same flow field, performed at large-amplitude, probes departures from the equilibrium state, and the fluid responds with shear stress in the form of a Fourier series, whose component amplitudes and phase leads depend on both the dimensionless frequency (called the Deborah number) and the dimensionless shear rate amplitude (called the Weissenberg number). The physics of these departures from equilibrium in an oscillatory shear flow may be explained by (i) chain disentanglement or (ii) motion along the polymer chain axes (called reptation) or (iii) macromolecular orientation. Of these radically different and yet otherwise equally effective approaches, only (iii) allows the macromolecular structure to be varied arbitrary so that the effect of molecular architecture on the rheology can be explored. Though much has been written about a large-amplitude oscillatory shear flow, we understand little about the role of molecular structure on the measured behaviors, and this has limited its usefulness. In this work, we explain the higher harmonics of both the shear stress (first and third), the first normal stress differences (zeroth, second, and fourth), and the second normal stress differences (zeroth and second) arriving at analytical expressions for all three. These expressions, written in dimensionless form, express the dimensionless rheological responses in large-amplitude oscillatory shear flow in terms of the ratio of the two principal macromolecular moments of inertia. To get these expressions, we derive the first five terms of the orientation distribution function, by solving the general diffusion equation in Euler coordinates. We then integrate in phase space with this orientation result to arrive at our expression for the first seven terms of the polymer contribution to the extra stress tensor. From this tensor, we next write down the Fourier coefficients for the shear stress responses, and the normal stress difference responses, in large-amplitude oscillatory shear flow for a suspension of macromolecules sculpted from a rigid bead-rod structure of any arbitrary axisymmetric shape.

Diffusion Equation Generalized for Modeling of Chladni Patterns

Random walk of particles during Chladni pattern formation is macroscopically treated as a diffusion process. The corresponding generalized diffusion equation is formulated based upon the generator of vibration driven random walk by following Einstein’s treatment of Brownian motion.

Dynamics of stochastic-constrained particles

Prior studies have focused on the overall behavior of randomly moving particle swarms. However, the characteristics of the stochastic-constrained particles that form ubiquitously within these swarms remain oblivious. This study demonstrates a generalized diffusion equation for stochastic-constrained particles that considers the velocity and location aggregation effects observed from their parent particle swarm (i.e., a completely random particle swarm). This equation can be approximated as the form of Schrödinger equation in the microcosmic case (low relative density) and describe the dynamics of the total mass distribution in the macrocosmic case (high relative density). The predicted density distribution of the particle swarm in the stable aggregation state is consistent with the total mass distribution of massive, relaxed galaxy clusters (at least in the range of r<r_{textrm{s}}), preventing cuspy problems in the empirical Navarro–Frenk–White profile. This study opens a window to observe the dynamics of stochastic-constrained particles from a third perspective, from which the aggregation effect of particles without gravitation can be saw.

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure