Finite element approximation of a phase field model arising in nanostructure patterning

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

Boundary value problems are considered for a system of stationary partial differential equations that stems from the models describing the implantation of impurities into semiconductor devices and is similar to the equations of carrier transport. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

This paper is concerned with the Abels-Garcke-Gr\"{u}n (AGG) model for two-phase flows of two viscous incompressible system with different densities. The AGG model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a mixture density depending on a volume fraction and an additional flux term due to interface diffusion. We establish the global existence of weak solutions to the initial boundary value problem with the mixed boundary conditions, including the velocity, pressure and Navier-slip conditions for fluid, and the Dirichlet and Neumann conditions for volume fraction and chemical potential, together, in a cubic domain of $\mathbb{R}^3$.

The finite element approximation in parabolic quasi-variational inequalities related to impulse control problem with mixed boundary conditions

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

The correlation between molecular structure and photovoltaic performance is lagging for constructing high-performance indoor organic photovoltaic (OPV) cells. Herein, this relationship is investigated in depth by employing two medium-bandgap nonfullerene acceptors (NFAs). The newly synthesized NFA of FTCCBr exhibits a similar bandgap and molecular energy level, but a much stronger dipole moment and larger average electrostatic potential (ESP) compared with ITCC. After blending with the polymer donor PB2, the PB2:ITCC and PB2:FTCCBr blends exhibit favorable bulk-heterojunction morphologies and the same driving force, but the PB2:FTCCBr blend exhibits a large ESP difference. In OPV cells, the PB2:ITCC-based device produces a power conversion efficiency (PCE) of 11.0%, whereas the PB2:FTCCBr-based device gives an excellent PCE of 14.8% with an open-circuit voltage (VOC ) of 1.05V, which is the highest value among OPV cells with VOC values above 1.0V. When both acceptor-based devices work under a 1000 lux of 3000 K light-emitting diode, the PB2:ITCC-based 1 cm2 device yields a good PCE of 25.4%; in contrast, the PB2:FTCCBr-based 1 cm2 device outputs a record PCE of 30.2%. These results suggest that a large ESP offset in photovoltaic materials is important for achieving high-performance OPV cells.

Рассматривается математическая модель, описывающая стационарное ползущее течение неравномерно нагретой несжимаемой жидкости через ограниченную трехмерную область с локально-липшицевой границей. Выбранная модель представляет собой систему нелинейных дифференциальных уравнений в частных производных второго порядка со смешанными краевыми условиями: на участке протекания заданы давление, температура и касательная составляющая поля скоростей, а на твердых стенках сосуда используется условие прилипания и краевое условие типа Робена для температуры. Для данной краевой задачи вводится понятие слабого решения (пара «скорость-температура»), которое определяется как решение некоторой системы интегральных тождеств. Основной результат работы - теорема о существовании слабых решений в подпространстве декартова произведения двух соболевских пространств. Для доказательства этой теоремы дается операторная трактовка краевой задачи, выводятся априорные оценки решений и применяется теорема Лерэ-Шаудера о неподвижной точке вполне непрерывного отображения. Установлены энергетические равенства, которым удовлетворяют слабые решения.

In this paper, we introduce a model describing diffusion of species by a suitable regularization of a “forward–backward” parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g. as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via a maximal monotone graph acting on the time derivative of the concentration and presenting a strong coerciveness property.

Comparison of two ways of simulating a simple tubular heat exchanger

We study the existence of solutions of the general elliptic system of partial differential equations in the space where the principal part may be represented with the help of a CLIFFORDalgebra 𝔄. We construct an integral representation and discuss the properties of the kernels.

In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper Well-posedness and regularity for a generalized fractional Cahn-Hilliard system . More precisely, we study the ω-limit of the phase parameter y and characterize it completely. Our characterization depends on the first eigenvalues λ 1 ≥0 of one of the operators involved: if λ 1 >0, then the chemical potential μ vanishes at infinity and every element y ω of the ω-limit is a stationary solution to the phase equation; if instead λ 1 =0, then every element y ω of the ω-limit satisfies a problem containing a real function μ ∞ related to the chemical potential μ. Such a function μ ∞ is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions for μ ∞ to be uniquely determined and constant.

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

In this paper, we seek the exact and numerical solutions using Lie group analysis for one dimensional unsteady adiabatic flow in a self-gravitating ideal gas behind a cylindrical shock wave with axial magnetic field. With the help of Lie group analysis, the one-dimensional optimal system of sub-algebra is obtained for the system of equations of motion. With the help of optimal classes of infinitesimal generators, we constructed the similarity variable and transformation of the flow variables, which convert the system of partial differential equations into system of ordinary differential equations. In three particular cases, we have derived a general framework to solve the fundamental equations and exact feasible solutions are obtained. In two cases, the similarity solutions with exponential law and power law shock path are discussed. The similarity solution is obtained using numerical method in the case of power law shock path. It is obtained that the increase in the values of magnetic field strength and adiabatic index have the decaying effect on the shock wave. Also, increase in the strength of shock wave is witnessed with the rise in the gravitation parameter value. The effects of variation of magnetic field strength, adiabatic exponent and gravitational parameter on the flow variables are analyzed graphically.

A low dispersion, high accuracy finite element method for first order hyperbolic systems in several space variables