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This paper presents an exploration of a specific type of a generalized multicomponent solution model, which appears to be first given by Saulov in the current explicit form. The assumptions of the underlying theory and a brief derivation of the main equation have been provided preliminarily for completeness and notational consistency. The resulting formulae for the Gibbs free energy of mixing and the chemical potentials are multivariate polynomials with physically meaningful coefficients and the mole fractions of the components as variables. With one additional assumption about the relative magnitudes of the solvent–solute and solute–solute interaction exchange energies, combining rules were obtained that express the mixed coefficients of the polynomial in terms of its pure coefficients. This was done by exploiting the mathematical structure of the asymmetric form of the solvent chemical potential equation. The combining rules allow one to calculate the thermodynamic properties of the solvent with multiple solutes from binary mixture data only (i.e., each solute with the solvent), and hence, are of practical importance. Furthermore, a connection was established between the osmotic virial coefficients derived in this work and the original osmotic virial coefficients of Hill found by employing a different procedure, illustrating the equivalency of what appears to be two different theories. A validation of the combining rules derived here has been provided in a separate paper where they were successfully used to predict the freezing points of ternary salt solutions of water. [ABSTRACT FROM AUTHOR]

We consider the stochastic Cahn-Hilliard-Navier-Stokes with surfactant on a bounded domain, driven by a multiplicative noise. The resulting system of partial differential equations consists of a Navier-Stokes systems for the (average) velocity, a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth–order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system with a multiplicative noise of Gaussian type for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients in both two and three dimensional bounded domains. In the two dimensional case and when the viscosity of the mixture is assumed to be constant, we prove the pathwise uniqueness of the weak solution, and using the Yamada-Watanabe classical result to derive the existence of a strong solution (in probabilistic sense). [ABSTRACT FROM AUTHOR]

In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

We construct a positivity-preserving Lie–Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate 1/4 in time and rate 1/2 in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity. [ABSTRACT FROM AUTHOR]

In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington–Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi's replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed p-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states. [ABSTRACT FROM AUTHOR]

Image inpainting is the process of restoring damaged areas in an image using information available from neighboring regions. In this paper, we present a novel, efficient, and simple local image inpainting algorithm based on the Allen–Cahn (AC) equation with a fidelity term. We utilize the phase separation property of the AC equation and introduce a new phase-dependent fidelity parameter to preserve the original values in the neighboring regions of an inpainting region. The governing partial differential equation is solved using the finite difference method, with the values of the neighboring cells serving as the Dirichlet boundary condition. The proposed algorithm is both local and explicit, making it is fast and easy to implement. We demonstrate the performance of the proposed model through several numerical experiments. Furthermore, comparing this method to other image inpainting methods demonstrates its superiority in image inpainting. [ABSTRACT FROM AUTHOR]

The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation. [ABSTRACT FROM AUTHOR]

In this work, we consider a nonconforming mixed finite element method (MFEM) for a parabolic system consisting of the Cahn-Hilliard equations and the Navier-Stokes equations, which describes a diffuse interface model for the flow of two incompressible and immiscible fluids. We focus on the superconvergence analysis for the corresponding first order convex splitting scheme, where the bilinear element is used to approximate the Cahn-Hilliard equation in space, and the constrained nonconforming rotated (CNR) Q 1 element together with the Q 0 constant element to the Navier-Stokes equations. By use of the high accuracy properties of the elements and some specific techniques, the superclose results of order O (h 2 + τ) for the concentration ϕ and the chemical potential ω in the H 1 -norm, the velocity u in the broken H 1 -norm and the pressure p in L 2 -norm are strictly proved. Here, h is the subdivision parameter and τ , the time step. It is worth to say that we derive a new high accuracy character through the integral identity technique for the chosen elements, which plays a quite significant role in getting the final error results. Furthermore, the global superconvergence results of the corresponding variables are concluded by the interpolated postprocessing technique. Numerical tests are provided to justify the performance of the theoretical analysis. [ABSTRACT FROM AUTHOR]

A mathematical analysis is performed for a system consisting of two coupled Cahn-Hilliard equations. These equations incorporate a diffusional mobility that depends on concentration. This modeling approach is often used to describe the process of phase separation in a thin layer of a binary liquid mixture covering a substrate, particularly when one of the components wets the substrate. The analysis establishes the existence of a weak formulation for this problem, which is supported by the use of a Lyapunov functional. Additionally, the analysis provides insights into the regularity properties of the weak formulation. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution. [ABSTRACT FROM AUTHOR]

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings. [ABSTRACT FROM AUTHOR]

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qϵ). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qϵ). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qϵ). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this study, we propose a restoration algorithm for distorted objects using a curvature-driven flow. First, we capture the convex-hull shaped contour of the distorted object using the mean curvature flow. With the supplemented mass on the captured feature, we evolve the constraint mean curvature flow to a steady state, preserving the non-distorted region. With respect to the mass, we select a restorative shape by considering the square of the curvature derivative. The Allen–Cahn and Cahn–Hilliard equations are applied to the generated restored image from the implicit curvature motions represented by the order parameter. We impose the Dirichlet boundary condition for the order parameter and the Neumann boundary for the chemical potential to fix the feature and to inherit the mass conservation, respectively. We provided examples of the restoration of half-circle and parentheses-shaped objects to reconstruct a circle shape. [ABSTRACT FROM AUTHOR]

Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class W p 1 , 2 (Q) × W ν 1 , 2 (Q) . One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, alg-frac_sec-ord-varphi_PHT, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper. [ABSTRACT FROM AUTHOR]

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of N ≥ 2 components occupying a bounded region Ω ⊂ R d , d ≤ 3 . Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if d ≤ 2 , we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d = 3 , we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d = 3 , this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application. [ABSTRACT FROM AUTHOR]

A flexible printing electronical membrane is an electron equipment made by precisely spraying conductive metal ink such as silver on a soft membrane substrate. With its advantages of light weight and flexibility, it can adapt to changing working environments and is widely used in aerospace, wearable electronics and other fields. Nevertheless, during the manufacturing preparation of roll-to-roll printing membranes, the high-speed movement of printing electronical membranes under tension is affected by the impact of hot air from the drying oven and the electrostatic interference generated by friction in transmission, which restricts the overprint accuracy and preparation velocity of flexible electronical membranes. To address this issue, the nonlinear forced vibrational characteristics of a traveling flexible printing electronical membrane on temperature coupling subjected to nonlinear electrostatic force were investigated. The roll-to-roll printed intelligent RFID electron membrane is the research target. On the basis of the energy approach and the heat conduction equation considering the effect of deformation, the nonlinear vibrational equations of an axially traveling flexible printing electronical membrane coupled with temperature under the function of nonlinear electrostatical excitation force were derived. The Bubnov–Galerkin algorithm was applied to discretize the vibration partial differential equations; by making full use of the quartic Runge–Kutta numerical algorithm to calculate the approximate solution of equations, the phase portraits, Poincaré maps, time history diagrams, power spectra, and bifurcation plots of the nonlinear vibrations of the traveling printing electronical membrane were used to explore the effects of movement velocities, electrostatical field, and thermal coupling coefficients. The findings obtained the stable working domain and the divergence instability domain of the traveling flexible printing electronic membrane, which provided a theory fundamental for enhancing the stable craft of a printing electronical membrane. [ABSTRACT FROM AUTHOR]

In this paper, we propose and present a non-overlapping substructuring-type iterative algorithm for the Cahn–Hilliard (CH) equation, which is a prototype for phase-field models. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of CH equation has. Here we present a formulation for the linear and non-linear Dirichlet–Neumann (DN) methods applied to the CH equation and study the convergence behaviour in one and two spatial dimensions in multiple subdomains. We show numerical experiments to illustrate our theoretical findings and effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffuse interface model which describes the interaction between a magnetic field $ {\boldsymbol {B}} $ and two immiscible, conducting, incompressible fluids of (volume) averaged velocity $ {\boldsymbol {u}} $. The model consists of the Cahn–Hilliard equation for the order parameter $ \varphi $ with a singular potential coupled with the equations of resistive magnetohydrodynamics for $ {\boldsymbol {u}} $ and $ {\boldsymbol {B}} $. The resulting evolution system is endowed with initial conditions and suitable boundary conditions. Here we show the existence of a global weak solution which is unique in dimension two. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, the (global) strong solution is strictly separated, namely, $ \varphi $ stays uniformly away from pure phases. This enables us to deduce a continuous dependence estimate. Finally, in dimension two, we establish the instantaneous regularization properties of global weak solutions for $ t>0 $. In particular, we show that $ \varphi $ instantaneously satisfies the strict separation property. This result allows us to establish the convergence to a single equilibrium as well as the existence of a global attractor and the validity of the backward uniqueness property. [ABSTRACT FROM AUTHOR]

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with the convective Cahn-Hilliard equation. This system describes the evolution of an incompressible isothermal Newtonian mixture of binary fluids. In this article we investigate a variant of this model, which consists of the nonhomogeneous Kelvin-Voigt equations coupled with the Cahn-Hilliard equations. We prove the existence of global weak and strong solutions in a two and three dimensions. Furthermore, we prove some regularity results for the strong solutions and show their uniqueness in both two or three-dimensional bounded domains. Lastly, we also solve a problem of uniqueness of regular solutions that was left open by Giorgini and Temam (2020) [5]. [ABSTRACT FROM AUTHOR]

We propose a method for controlling the formation of microarray structures on the surface by using an electric field. In this study, a phase field model is employed to describe the formation of microstructures under the influence of the electric field. Initially, we utilize Cahn–Hilliard equation, Navier–Stokes equation, and Gaussian theory to construct the phase field model. Subsequently, we obtain the theoretical solution for the partial differential equations and apply numerical methods such as semi-implicit Fourier spectral and semi-implicit difference method. Finally, we establish the numerical model to observe the morphological changes on the polymer surface by changing the electric field strengths, the distribution and the radius of the electrode plate. Simulated observations show that the electric field induce the formation of microarray structures on the polymer surface successfully, and the microarray structures match the shape of the electric field. Additionally, higher electric field strength promotes the formation of morphological changes obviously. The distribution of microarray structures is controlled by changing the electric field distribution. These studies provide a theoretical basis for the fabrication of microarray structures. [ABSTRACT FROM AUTHOR]

Due to the significant value that hotel reviews hold for both consumers and businesses, the development of an accurate sentiment classification method is crucial. By effectively distinguishing the authenticity of reviews, consumers can make informed decisions, and businesses can gain insights into customer feedback to improve their services and enhance overall competitiveness. In this paper, we propose a partial differential equation model based on phase-field for sentiment analysis in the field of hotel comment texts. The comment texts are converted into word vectors using the Word2Vec tool, and then we utilize the multifractal detrended fluctuation analysis (MF-DFA) model to extract the generalized Hurst exponent of the word vector time series to achieve dimensionality reduction of the word vector data. The dimensionality reduced data are represented in a two-dimensional computational domain, and the modified Allen–Cahn (AC) function is used to evolve the phase values of the data to obtain a stable nonlinear boundary, thereby achieving automatic classification of hotel comment texts. The experimental results show that the proposed method can effectively classify positive and negative samples and achieve excellent results in classification indicators. We compared our proposed classifier with traditional machine learning models and the results indicate that our method possesses a better performance. [ABSTRACT FROM AUTHOR]

In this report we present two unconditionally stable, second order, decoupled ensemble schemes for computing evolutionary Boussinesq equations: the stabilized scalar auxiliary variable Crank-Nicolson leap-frog ensemble scheme (Stab-SAV-CNLF-En) and the stabilized scalar auxiliary variable BDF2 ensemble scheme (Stab-SAV-BDF2-En). The two ensemble schemes adopt the recently developed stabilized scalar auxiliary variable (SAV) idea and ensemble timestepping to achieve both high efficiency and unconditional stability. Specifically, the stabilized SAV approach makes it possible to devise unconditionally long time stable schemes for which the nonlinear terms and coupling terms in the Boussinesq equations are made fully explicit, leading to linear systems with constant coefficient matrices to be solved after spatial discretization. The ensemble timestepping further improves the efficiency by making the coefficient matrices of all realizations the same, so that efficient block solvers can be applied to solve the corresponding one linear system with multiple right hand sides and thus greatly reduce the computational cost. We prove the proposed schemes are unconditionally stable and present implementation details. Ample numerical tests are performed to show the efficiency and effectiveness of the combined approach. [ABSTRACT FROM AUTHOR]

The stress-strain state that occurs in multilayered structures (for example, beams) significantly depends on the nature of the interaction of their layers. In practice, when calculating such structural elements, an ideal contact of the layers is most often considered, this implies the continuity of displacement vectors and stresses at the contact boundary. Strict elasticity theory solutions in this case inevitably lead to the appearance of infinite tangential stresses on the free surfaces at the angular points of structures at the layer interfaces. In this paper, the problem of the stress-strain state of a multilayered beam is considered in which the interaction of layers is assumed nonideal. For this aim, a layer interaction model with a thin contact layer between the adhesive and adherends is chosen. The equations obtained on the basis of this model allowed us to find non-singular solutions that describe the essentially inhomogeneous stress-strain state in each layer of the beam and at their interfaces. [ABSTRACT FROM AUTHOR]

GaN-based heterostructures have been used in high-power radio frequency applications for a number of years due to the wide bandgap of GaN and high values of spontaneous and piezoelectric polarization, resulting in high breakdown voltages and high 2D carrier concentrations. However, the accurate modeling of low-field electron mobility within such structures remains a topic of interest. This paper presents a comprehensive numerical model for calculating the carrier mobility within a 2D electron gas in Al x Ga 1 − x N/GaN high electron mobility transistors. The model is based on solving the Schrödinger and Poisson equations self-consistently, taking into account the polarization charges at material interfaces and performing semi-classical numerical calculations of low-field electron mobility within the momentum relaxation time approximation, taking into account all relevant scattering mechanisms. Both intra- and inter-subband transitions are considered, and the differences in intra- and inter-subband scattering rates are analyzed for some of the key scattering mechanisms. The importance of including inter-subband transitions in the calculations is demonstrated by comparing the calculated results with experimentally measured mobilities. [ABSTRACT FROM AUTHOR]

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters. [ABSTRACT FROM AUTHOR]

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson--Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter Λ, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As Λ = 0, the PB-steric equation becomes the conventional PB equation, but as Λ > 0, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as Λ goes to infinity. Our theoretical results show that the PB-steric equations (for 0 ≤ Λ ≤ ∞ ) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the nonconvex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak–strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model, we present the estimates in three space dimensions and derive conditional relative energy estimates. [ABSTRACT FROM AUTHOR]

This paper provides a concise description of the free energy principle, starting from a formulation of random dynamical systems in terms of a Langevin equation and ending with a Bayesian mechanics that can be read as a physics of sentience. It rehearses the key steps using standard results from statistical physics. These steps entail (i) establishing a particular partition of states based upon conditional independencies that inherit from sparsely coupled dynamics, (ii) unpacking the implications of this partition in terms of Bayesian inference and (iii) describing the paths of particular states with a variational principle of least action. Teleologically, the free energy principle offers a normative account of self-organisation in terms of optimal Bayesian design and decision-making, in the sense of maximising marginal likelihood or Bayesian model evidence. In summary, starting from a description of the world in terms of random dynamical systems, we end up with a description of self-organisation as sentient behaviour that can be interpreted as self-evidencing; namely, self-assembly, autopoiesis or active inference. [Display omitted] • This paper provides a concise derivation of the free energy principle. • The FEP is a normative account of self-organisation and sentient behaviour: • It describes self-organisation as maximizing Bayesian model evidence, • And sentient behaviour as optimal Bayesian design and decision-making. • Altogether, we start from a description of the world as a random dynamical system, • And end with a Bayesian mechanics that can be read as a physics of sentience. [ABSTRACT FROM AUTHOR]

In this paper, the splitting-up scheme for rough partial differential equations (RPDE) is introduced. The convergence rate of the splitting-up scheme for weak solutions to semilinear partial differential equations with linear rough noises is given. An application to stochastic partial differential equations with fractional Browian motions is considered. [ABSTRACT FROM AUTHOR]

Machine learning has proven exceptionally competent in numerous applications of studying dynamical systems. In this article, we demonstrate the effectiveness of reservoir computing, a famous machine learning architecture, in learning a high-dimensional spatiotemporal pattern. We employ an echo-state network to predict the phase ordering dynamics of 2D binary systems—Ising magnet and binary alloys. Importantly, we emphasize that a single reservoir can be competent enough to process the information from a large number of state variables involved in the specific task at minimal computational training cost. Two significant equations of phase ordering kinetics, the time-dependent Ginzburg–Landau and Cahn–Hilliard–Cook equations, are used to depict the result of numerical simulations. Consideration of systems with both conserved and non-conserved order parameters portrays the scalability of our employed scheme. [ABSTRACT FROM AUTHOR]

An efficient and easy-to-implement second-order Strang splitting approach is mainly applied to study the scalar Allen-Cahn (AC) equation in this paper. Base on the idea of dimensional splitting, a new time dependent function (called exponential integrating factor) is introduced for the scalar AC equation. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the explicit 2-stage strong stability preserving Runge-Kutta(SSP-RK2) method is employed for the nonlinear part. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence of the proposed scheme. Various numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

The aim of this work is to prove the global-in-time existence of weak solutions for a viscoelastic phase separation model in three space dimensions. To this end, we apply the relative energy concept provided by [7]. We consider the case of regular polynomial-type potentials and positive mobilities, as well as the degenerate case with logarithmic potential and vanishing mobility. [ABSTRACT FROM AUTHOR]

We prove the existence of the monotone traveling wave for the isothermal fluid equations with viscous and capillary terms by the planar dynamical system method. We obtain that the monotone traveling wave is asymptotically stable under the suitable perturbation. In the process of establishing the uniform a priori estimate, we dispose the capillary term reasonably according to the feature of the equations, and find the appropriate weighted function to overcome the difficulty caused by the non-convex pressure function. [ABSTRACT FROM AUTHOR]

In fractional calculus, the fractional differential equation is physically and theoretically important. In this article an efficient numerical process has been developed. Numerical solutions of the time fractional fourth order reaction diffusion equation in the sense of Caputo derivative is obtained by using the implicit method, which is a finite difference method and is developed by increasing the number of iterations. The advantage of the implicit difference scheme is unconditionally stable. The stability analysis and convergency have been proven. A numerical example has been presented, and the validity of the method is supported by tables and graphics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the unconditionally optimal H1-error estimate of a fast second-order scheme for solving nonlinear subdiffusion equations on the nonuniform mesh. We use the Galerkin finite element method (FEM) to discretize the spacial direction, the Newton linearization method to approximate the nonlinear term and the sum-of-exponentials (SOE) approximation to speed up the evaluation of Caputo derivative. Our analysis of the unconditionally optimal H1-error estimate involves the temporal-spatial error splitting approach, an improved discrete fractional Grönwall inequality and error convolution structure. In order to find a suitable test function to estimate H1-error, we here consider two cases: linear and high-order FEM space, using the time-discrete operator and Laplace operator in the test function respectively. Numerical tests are provided demonstrate the effectiveness and the unconditionally optimal H1-error convergence of our scheme. [ABSTRACT FROM AUTHOR]

In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in [1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided. [ABSTRACT FROM AUTHOR]

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat ux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable double obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Besides existence results, we derive first-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary optimality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, unconditionally optimal error estimates in L 2 -norm of two fully discrete schemes, one is backward Euler scheme and the other is a modified Crank-Nicolson scheme, are derived for cubic Schrödinger equation. Firstly, the mass and energy stability of the two schemes are proved rigorously. Secondly, the existence and uniqueness of the numerical solutions of the two schemes are presented. Based on the above priori estimations of the numerical solutions, unconditionally optimal error estimates in L 2 -norm are obtained without any timestep restrictions. Finally, some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

We propose a new class of uniformly accurate splitting methods for the Benjamin–Bona-Mahony equation which converge uniformly in the dispersive parameter ε . The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter ε . The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case P (∂ x) = ∂ x our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in H r for solutions in H r without any loss of derivative. This estimate holds uniformly in ε . In regularizing regimes ε = O (1) we even gain a derivative with our time discretisation at the cost of loosing in terms of 1 ε . Numerical experiments underline our theoretical findings. [ABSTRACT FROM AUTHOR]

The modified phase-field crystal (MPFC) equation was extended to the MPFC equation with a strong nonlinear vacancy potential (VMPFC) to include vacancies. In this paper, we develop a linear, second-order, and unconditionally energy stable scheme for solving the VMPFC equation by proposing a new linear convex splitting, using the Crank–Nicolson formula to the contractive part and the second-order Adams–Bashforth formula to the expansive part, and adding a second-order Douglas–Dupont regularization. The unconditional energy stability of the developed scheme is proved. By solving the VMPFC equation, vacancies which exist in real materials can be observed in numerical simulations. [ABSTRACT FROM AUTHOR]

Interfaces separating pure materials and mixtures tend to be severely smeared with interface-capturing methods for compressible multi-material flows, necessitating the requirement of various interface-sharpening techniques. However, these techniques have various problems related to consistency, conservation, and thermodynamic compatibility. In this work, we derive a general theoretical formulation of interface-sharpening techniques for the generic five-equation model. This theoretical formulation is not only conservative in mass, momentum, and total energy but also asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed, and is independent of various specific numerical algorithms. We further propose a general numerical method to solve this theoretical formulation. The proposed method is consistent and conservative, and it prevents spurious errors at the interfaces. Examples of one- and two-dimensional multimaterial compressible flow problems, including shocks and interfaces, are considered to verify the analysis and demonstrate the efficiency of the method. • We first derived a general theoretical formulation of interface-sharpening techniques for the generic five-equation model. • A multi-material artificial compression method (MMACM) is proposed. • The proposed method can handle more complicated equations of state, and materials that is larger than two. [ABSTRACT FROM AUTHOR]

In this article, we devote to the finite dimensional reduction for the hyperviscous Navier–Stokes equation with L2$$ {L}^2 $$ force and prove the existence of an N$$ N $$‐dimensional inertial manifold for this problem in 핋3. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard equation is a fundamental model that describes the phase separation process in multi-component mixtures. It has been successfully extended to different contexts in various scientific fields. In this survey article, we briefly review the derivation, structure as well as some analytical issues for the Cahn–Hilliard equation and its variants. Our focus will be placed on the well-posedness as well as long-time behavior of global solutions for the Cahn–Hilliard equation in the classical setting and recent progresses on the dynamic boundary conditions that describe non-trivial boundary effects. [ABSTRACT FROM AUTHOR]

In this work, we propose and analyze a stabilized linear Crank-Nicolson scheme for the Swift-Hohenberg equation. More precisely, we treat the nonlinear term explicitly while two second-order stabilization terms are added to improve the stability of the scheme. We show that our scheme satisfies the discrete energy dissipation. We prove rigorously that our scheme is second-order accurate in time. Moreover, we adopt a spectral-Galerkin approximation for the spacial variables and establish error estimates for the fully discrete scheme. Numerical experiments are presented to show the accuracy and energy stability of our scheme. [ABSTRACT FROM AUTHOR]

It is well known that the classic Allen–Cahn equation satisfies the maximum bound principle (MBP), that is, the absolute value of its solution is uniformly bounded for all time by certain constant under suitable initial and boundary conditions. In this paper, we consider numerical solutions of the modified Allen–Cahn equation with a Lagrange multiplier of nonlocal and local effects, which not only shares the same MBP as the original Allen–Cahn equation but also conserves the mass exactly. We reformulate the model equation with a linear stabilizing technique, then construct first‐ and second‐order exponential time differencing schemes for its time integration. We prove the unconditional MBP preservation and mass conservation of the proposed schemes in the time discrete sense and derive their error estimates under some regularity assumptions. Various numerical experiments in two and three dimensions are also conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce an entropic-regularisation of the scheme. We prove the convergence of both schemes to a weak solution of the evolutionary equation. We illustrate the generality of our work by providing a number of examples, including the kinetic Fokker-Planck equation and the (regularized) Vlasov-Poisson-Fokker-Planck equation. [ABSTRACT FROM AUTHOR]

Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized- α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems. [ABSTRACT FROM AUTHOR]