Elliptic Parabolic Partial Differential Equations Research Articles

Теорема існування для задачі масопереносу на графі

The mass transfer process in a porous medium is described using the Richards-Klute equation. This equation describes mass flows due to the actions of gravity and capillarity and allows modeling the mass transfer process with saturation limit. The Richards-Klute equation is a nonlinear elliptic-parabolic partial differential equation, so the main methods for solving it and modeling the mass transfer process are numerical methods. The article considers a model of a system of interconnected pipes, inside which the process of mass transfer takes place. Such systems are often found in agriculture and are actively used in the construction of irrigation systems. The article proposes to model pipe system using graphs, where pipes are represented by graph edges, and connection points or free ends of system pipes are represented by graph vertices. The article contains the definitions of the Richards-Klute equation on a graph in the usual and weak forms. On the edges of the graph, one-dimensional cases of the Richards-Klute equation are considered, while on the vertices either the boundary conditions are given or the equation that models the law of mass conservation is given. The definitions of the solution and weak solution of the Richards-Klute equation on the graph are also given. Also, the theorem of the existence of a weak solution of the Richards-Klute equation on a graph is proved. To prove the theorem of the existence of a weak solution of the Richards-Klute equation on a graph, the Kirchhoff transformation is used and conditions are given that are analogous to the conditions used in the proof of the existence of a weak solution of the Richards-Klute equation in regular domains in three-dimensional space, and which are defined in the classical work [1], which is devoted to the problems of existence and uniqness of weak solutions of elliptic-parabolic partial differential equations.

Derivation of a quasi-linear second-order elliptic-parabolic model for the efficiency of silicon solar cells

In this note, we propose a mathematical model for the efficiency of a silicon solar cell. The model is rigorously derived from continuity and transport assumptions, along with the Poisson equations to describe the behavior of charge carriers in a semiconductor. The photovoltaic phenomenon was thoroughly studied to develop the model, and some parameters that play a crucial role in its efficiency have been identified. Those parameters include the diffusion, the mobility and the absorption coefficients, the permittivity, the regions width and the minority carrier lifetime. The mathematical model obtained herein is a quasi-linear second-order elliptic-parabolic partial differential equation system with strong coupling and interface. The model is defined over a spatial domain composed of three regions with different electrical properties. The fundamental physical laws of the photovoltaic effect are employed here to develop the mathematical model and its boundary conditions.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations

This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multipliers, CPROP solves the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination. As a result, CPROP reduces the dimensionality of the optimization problem, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic and parabolic PDEs with changing parameters and nonhomogeneous terms.

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\mathbb{N}$, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615--646; Anal. Appl., 9 (2011), pp. 1--37], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\geq 1}$ of the random inputs, and we prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In a follow-up paper [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation], we explain two methods for how such best $N$-term truncations can be computed practically: by greedy-type algorithms as in [Ch. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291--467; C. J. Gittelson, Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland, 2011], or by multilevel Monte-Carlo methods as in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374]. The sparsity result allows, in conjunction with adaptive wavelet Galerkin schemes, for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [W. Dahmen and A. Kunoth, SIAM J. Control Optim., 43 (2005), pp. 1640--1675; M. D. Gunzburger and A. Kunoth, SIAM J. Control. Optim., 49 (2011), pp. 1150--1170; A. Kunoth, Numer. Algorithms, 39 (2005), pp. 199--220]; see [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation].

Meshless Method Based on Radial Basis Functions for Solving Parabolic Partial Differential Equations with Variable Coefficients

The method of approximate particular solutions is extended for solving initial-boundary-value problems for general parabolic partial differential equations (PDEs) with variable coefficients. The main idea is to reduce the parabolic PDEs into a series of elliptic PDEs and approximate the unknown solution by the closed-form particular solution using radial basis functions. Numerical experiments in two and three dimensions show that the proposed scheme is accurate and easy to implement.

Analytic Solution of the Anisotropic Bidomain Equations for Myocardial Tissue: The Effect of Adjoining Conductive Regions

The anisotropic bidomain model for the propagation of electrical activation in the human myocardium H consists of coupled elliptic-parabolic partial differential equations for the transmembrane potential Vm, intracellular potential phi(i), and extracellular potential phi(e) in H, together with quasi-static equations for the potential distribution phiB in the surrounding (passive) isotropic extracardiac regions B. Four local parameters sigma((i,e) (l,t)) specify the conductivities in the longitudinal (l) and transverse (t) directions with respect to cardiac muscle fibers. Continuous current flow is required at the interface S(H) between H and B. We derive analytic formulas for Vm, phi(e), phi(i), and phiB for plane wave propagation in a uniformly anisotropic slab surmounted by a homogeneous region of conductivity sigmaB. No assumptions are required regarding the anisotropy ratios of the conductivity coefficients. The properties of these solutions are examined with a view to providing insight into the effect of the passive region B on the propagation of Vm and phi(e) in H. We show that for a suitably chosen boundary condition, the problem can be reduced to solving the bidomain equations in H alone.

SIMPLE, ACCURATE TREATMENT OF CURVED BOUNDARIES WITH DIRICHLET AND NEUMANN CONDITIONS

A simple and accurate numerical treatment of curved boundaries for the solution of elliptic or elliptic-parabolic partial differential equations typical of transport problems is described. The study is motivated by the importance of curved boundaries in applied and fundamental problems, especially those with Neumann conditions. Existing methods, including those employing unstructured or body-fitted meshes, are either clumsy, of low-order accuracy, or burdened with high computational overhead. The method employs elementary finitedifference meshes and places nodes on the curved boundaries to permit precise satisfaction of the boundary conditions. A key element of the method is the use of skew-tolerant algorithms to alleviate possible computational difficulties when boundary nodes are too close to interior nodes. Second-order equations and implementations are presented, easily extendable to third or higher orders. The higher-order space-time formulations and the implicit time integration's freedom from stability constraints render the approach potentially quite economical compared to conventional first-order approaches. The ability to satisfy Neumann boundary conditions accurately and economically can be a great asset in many problems, including free-boundary problems involving phase change and/or surface tension, as well as in pressure computations in Navier–Stokes solvers.

Cubature on Wiener space

It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.

Computer simulation of materials using parallel architectures

Algorithms are designed to implement molecular dynamics (MD) and quantum molecular dynamics (QMD) simulations on emerging concurrent architectures. For systems with finite-range interactions, a domain-decomposition algorithm is used to implement the multiple-time-step (MTS) approach to MD simulations on distributed-memory multiple instruction multiple data (MIMD) machines. Parallel algorithms are also designed for MD simulations of bulk Coulombic systems. The performances of these algorithms are tested on the Intel iPSC/860 system. The computational complexities of these algorithms are O( N) and parallel efficiencies close to 0.9. The core computational kernel of the QMD approach consists of solutions of parabolic partial differential equations (PDE) such as the time-dependent Schrödinger equation or time-dependent Kohn-Sham equation. This problem is coupled with another computationally intensive problem, i.e., solution of elliptic PDEs (the Poisson equation) for the long-range electron-electron interaction. We have designed parallel algorithms for both problems on SIMD (single instruction multiple data) machines. In the past two years, we have used the parallel computer architectures in our Concurrent Computing Laboratory for Materials Simulations (CCLMS) to carry out MD and QMD simulations on network glasses, ceramic composites, nanophase materials, solid C 60 and graphitic tubules, and quantum transport in nanoscale devices. Structural transformation, intermediate-range order and dynamic behavior of SiO 2 glass at high pressures are investigated with molecular dynamics. At high densities, the height of the first sharp diffraction peak is considerably diminished, its position changes from 1.6 to 2.2 Å −1, and a new peak appears at 2.85 Å -1. At twice the normal density, SiO bond length increases, SiO coordination changes from 4 to 6 and OSiO bond-angle changes from 109° to 90°. This is a tetrahedral to octahedral transformation, which was reported recently by Meade, Hem1ey, and Mao. Molecular dynamics simulations of porous silica, in the density range 2.2-0.1 g/cm 3, are carried out on a 41 472 particle system using a MIMD computer. The internal surface area, pore surface-to-volume ratio, pore-size distribution, fractal dimension, correlation length and mean-particle size are determined as a function of the density. Structural transition between a condensed amorphous phase and a low-density porous phase is characterized by these quantities. Various dissimilar porous structures with different fractal dimensions are obtained by controlling the preparation schedule and temperature.

On a Nonlinear Elliptic-Parabolic Partial Differential Equation System in a Two-Dimensional Groundwater Flow Problem

In this paper a nonlinear elliptic-parabolic system which arises in a two-dimensional groundwater flow problem is studied. Abstract results on evolution equations are employed to obtain existence and uniqueness results. Regularity and stability properties of the solution are also considered.

The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter

This paper studies the necessary and sufficient condition of uniformly convergent difference scheme for the elliptic-parabolic partial differential equation with a small parameter.

Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge.