Nonhomogeneous Neumann Boundary Conditions Research Articles

A mathematical model for Alzheimer’s disease: An approach via stochastic homogenization of the Smoluchowski equation

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.

One dimensional phase transition problem modeling striped spin orbit coupled Bose-Einstein condensates

We study the behavior of a Modica-Mortola phase transition type problem with a non-homogeneous Neumann boundary condition. According to the parameters of the problem, this leads to the existence of either one component occupying most of the domain with an outer boundary layer containing the other component, or to many interfaces, on a periodic pattern. This is related to the striped behavior of a two component Bose-Einstein condensate with spin orbit coupling in one dimension. We prove that minimizers of the full Gross-Pitaevskii energy in 1D behave, in the Thomas-Fermi limit of strong intra-component interaction, like those of the simplified Modica-Mortola problem we have studied in the first part.

A high-order compact difference method for fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions

In a recent paper, Ren and Liu proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions.

A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation

In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency.

Steady-state voltage distribution in three-dimensional cusp-shaped funnels modeled by PNP.

We study here the bulk electro-diffusion properties of micro- and nanodomains containing a cusp-shaped structure in three-dimensions when the cation concentration dominates over the anions. To determine the consequences on the voltage distribution, we use the steady-state Poisson-Nernst-Planck equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain surface charge distribution that agree with numerical simulations. Finally, we analyze the consequences of several piecewise constant non-homogeneous surface charge densities, motivated by designing new nanopipettes. To conclude, when electro-neutrality is broken at the scale of hundreds of nanometers, the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. These formula provide a refined characterization of voltage distribution at steady-state in neuronal microdomains such as dendritic spines, but can also be used to design nanometric patch-pipettes.

Two-Dimensional Exact Subdomain Technique of Switched Reluctance Machines with Sinusoidal Current Excitation

This paper presents a two-dimensional (2D) exact subdomain technique in polar coordinates considering the iron relative permeability in 6/4 switched reluctance machines (SRM) supplied by sinusoidal waveform of current (aka, variable flux reluctance machines). In non-periodic regions (e.g., rotor and/or stator slots/teeth), magnetostatic Maxwell’s equations are solved considering non-homogeneous Neumann boundary conditions (BCs). The general solutions of magnetic vector potential in all subdomains are obtained by applying the interface conditions (ICs) in both directions (i.e., r- and θ-edges ICs). The global saturation effect is taken into account, with a constant magnetic permeability corresponding to the linear zone of the nonlinear B(H) curve. In this investigation, the magnetic flux density distribution inside the electrical machine, the static/dynamic electromagnetic torques, the magnetic flux linkage, the self-/mutual inductances, the magnetic pressures, and the unbalanced magnetic forces (UMFs) have been calculated for 6/4 SRM with two various non-overlapping (or concentrated) windings. One of the case studies is a M1 with a non-overlapping all teeth wound winding (double-layer winding with left and right layer) and the other is a M2 with a non-overlapping alternate teeth wound winding (single-layer winding). It is important to note that the developed semi-analytical model based on the 2D exact subdomain technique is also valid for any number of slot/pole combinations and for non-overlapping teeth wound windings with a single/double layer. Finally, the semi-analytical results have been performed for different values of iron core relative permeability (viz., 100 and 800), and compared with those obtained by the 2D finite-element method (FEM). The comparisons with FEM show good results for the proposed approach.

Improved receding horizon H∞ temperature spectrum tracking control for Debye media in microwave heating process

This paper considers the problem of tracking control for the coupled thermal-electromagnetic model with temperature-dependent permittivity in microwave heating process, where the incident electrical field is subject to the asymmetrical saturation. Due to the nonhomogeneous Neumann boundary condition and inherent infinite-dimensional characteristics for microwave heating process, it is difficult for us to readily design and implement tracking controller. With the help of improved spectral Galerkin method, the traditional partial differential equation (PDE) model is transformed into the linear finite-dimensional ordinary differential equation (ODE) model which can be subsequently used for controller design. Then, the receding horizon H∞ control strategy is improved to stabilize the derived uncertain ODE error model with asymmetrical input saturation and bounded external disturbance. The outcome of improved algorithm is formulated as a linear matrix inequalities (LMIs) optimization problem, which can be on-line updated by applying the finite-different time-domain (FDTD) method. Finally, the improved control strategy is successfully applied in one-dimensional cavity model for tracking temperature spectrum.

Efficient compact finite difference method for variable coefficient fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions in conservative form

In this paper, a high-order compact finite difference scheme is derived for the time fractional sub-diffusion equation with nonhomogeneous Neumann boundary conditions in conservative form. Based on the L2- $$1_{\sigma }$$ approximation formula of the time fractional derivative, a high-order efficient compact finite difference method is developed. The unconditional stability of the resulting scheme and its convergence of second order in time and fourth order in space are rigourously proved by a discrete energy analysis method. The theoretical results are illustrated by examples.

Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation–fragmentation–diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng–Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.

New Subdomain Technique for Electromagnetic Performances Calculation in Radial-Flux Electrical Machines Considering Finite Soft-Magnetic Material Permeability

The most significant assumption in the subdomain technique (i.e., based on the formal resolution of Maxwell’s equations applied in subdomain) is that the iron parts are considered to be infinitely permeable, so that the saturation effect is neglected. In this paper, we present a novel contribution on improving this 2-D technique in polar coordinates by focusing on the consideration of iron relative permeability. In non-periodic regions, magnetostatic Maxwell’s equations are solved considering non-homogeneous Neumann boundary conditions, and the general solution is obtained by applying the interfaces conditions in both directions (i.e., $r$ - and $\theta $ -edges ICs). The proposed model is relevant for different types of radial-flux electrical machines with(out) permanent magnets (PMs) supplied by a direct or alternate current (with any waveforms). For example, the semi-analytical model has been implemented for spoke-type PM machines. The magnetic field calculations have been performed for two different values of iron core relative permeability (viz, 100 and 600), and compared with those obtained by the 2-D finite-element method. The semi-analytical results are in a very good agreement with those obtained numerically, considering both amplitude and waveform.

Receding horizon H∞ guaranteed cost tracking control for microwave heating medium with temperature-dependent permittivity

This paper considers the temperature spectrum tracking control of microwave heating model, in the presence of asymmetrical input saturation, nonhomogeneous Neumann boundary condition and temperature-dependent permittivity. The sufficient condition for the existence of receding horizon H∞ guaranteed cost control is proposed based on the derived finite-dimensional ordinary differential equation (ODE) error model. Furthermore, by on-line updating and solving linear matrix inequalities (LMIs) optimization problem, the constrained tracking controller can be obtained in the sense of minimizing H∞ norm and satisfying the quadratic cost performance. The proposed control strategy is implemented on a one-dimensional cavity heating model and its performance is evaluated through the simulation.

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted L2- and L∞-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order min{2−α,1+α}, where α∈(0,1) is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.

Infinitely many solutions for a class of hemivariational inequalities involving p(x)-Laplacian

Abstract In this paper hemivariational inequality with nonhomogeneous Neumann boundary condition is investigated. The existence of infinitely many small solutions involving a class of p(x) - Laplacian equation in a smooth bounded domain is established. Our main tool is based on a version of the symmetric mountain pass lemma due to Kajikiya and the principle of symmetric criticality for a locally Lipschitz functional.

An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows

The primary objective of this short work is the identification of alternate routes for the determination of exact and numerical solutions of the Navier-Stokes equations in the specific case of surface-tension driven thermal convection. We aim to elaborate a theoretical approach in which the typical kinematic boundary conditions required at the free surface by this kind of flows can be replaced by a homogeneous Neumann condition using a class of ‘continuous’ distribution functions by which no discontinuities or abrupt variations are introduced in the model. The rationale for such a line of inquiry can be found (1) in the potential to overcome the typical bottlenecks created by the need to account for a shear stress balance at the free surface in the context of analytic models for viscoelastic and other non-Newtonian fluids and/or (2) in the express intention to support existing numerical (commercial or open-source) tools where the possibility to impose non-homogeneous Neumann boundary conditions is not an option. Both analytic solutions and (two-dimensional and three-dimensional) numerical “experiments” (concerned with the application of the proposed strategy to thermocapillary and Marangoni-Bénard flows) are presented. The implications of the proposed approach in terms of the well-known existence and uniqueness problem for the Navier-Stokes equations are also discussed to a certain extent, indicating possible directions of future research and extension.

A non-smooth three critical points theorem for general hemivariational inequality on bounded domains

In this paper we are concerned with the study of a hemivariationalinequality with nonhomogeneous Neumann boundary condition. Weestablish the existence of at least three solutions of the problem by usingthe nonsmooth three critical points theorem and the principle of symmetriccriticality for Motreanu-Panagiotopoulos type functionals.

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d‐dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017

Mixed elliptic problems involving the <inline-formula><tex-math id="M1">\begin{document}$p-$\end{document}</tex-math></inline-formula>Laplacian with nonhomogeneous boundary conditions

In this paper, mixed elliptic problems involving the p-Laplacian and with nonhomogeneous boundary conditions are investigated. At first, the existence of one non-trivial solution, under a suitable behaviour on the nonlinearity and without requiring neither conditions at zero nor conditions at infinity, is established. Then, by adding a condition at infinity on the nonlinearity, also a second non-trivial solution is guaranteed. Some special cases are pointed out as, in particular, the existence of one non-trivial solution when the datum is \begin{document} $(p-1)-$ \end{document} sublinear at zero and the existence of two non-trivial solutions when the nonlinear term is again \begin{document} $(p-1)-$ \end{document} sublinear at zero and, in addition, more than \begin{document} $(p-1)-$ \end{document} superlinear at infinity. As a consequence, the existence of two non-trivial solutions for concave-convex nonlinearities is emphasized. Finally, the case of a simple \begin{document} $(p-1)-$ \end{document} superlinearity at infinity is considered and it is also observed that the same results hold when the nonlinear behaviour, described before for the datum, is instead assumed by the nonhomogeneous Neumann boundary conditions. Concrete examples of applications are also given. The approach is based on variational methods and critical point theory. Precisely, a non-zero local minimum theorem and a two non-zero critical points theorem are applied.

From a Microscopic to a Macroscopic Model for Alzheimer Disease: Two-Scale Homogenization of the Smoluchowski Equation in Perforated Domains

In this paper, we study the homogenization of a set of Smoluchowski’s discrete diffusion–coagulation equations modeling the aggregation and diffusion of \(\beta \)-amyloid peptide (A\(\beta \)), a process associated with the development of Alzheimer’s disease. In particular, we define a periodically perforated domain \(\Omega _{\epsilon }\), obtained by removing from the fixed domain \(\Omega \) (the cerebral tissue) infinitely many small holes of size \(\epsilon \) (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of A\(\beta \) by the neuron membranes. Then, we prove that, when \(\epsilon \rightarrow 0\), the solution of this micromodel two-scale converges to the solution of a macromodel asymptotically consistent with the original one. Indeed, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in that the scalar diffusion coefficients defined at the microscale are replaced by tensorial quantities.

非斉次Neumann境界条件を含むPoisson方程式に対する粒子法の誤差評価とその応用

非斉次Neumann境界条件を含むPoisson方程式に対する粒子法の誤差評価とその応用

Quenching for a Diffusion System with Coupled Boundary Fluxes

In this paper, we investigate a diffusion system of two parabolic equations with more general singular coupled boundary fluxes. Within proper conditions, we prove that the finite quenching phenomenon happens to the system. And we also obtain that the quenching is non-simultaneous and the corresponding quenching rate of solutions. This extends the original work by previous authors for a heat system with coupled boundary fluxes subject to non-homogeneous Neumann boundary conditions.