Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems

In this paper, we apply the Schwarz waveform relaxation (SWR) method to the one-dimensional Schrödinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schrödinger equation with time-independent linear potential, which is robust and scalable up to 500 subdomains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time-dependent linear potential or the nonlinear potential, we use a preprocessed linear operator for the zero potential case as a preconditioner which leads to a preconditioned algorithm. This ensures high scalability. In addition, some newly constructed absorbing boundary conditions are used as the transmission conditions and compared numerically.     

相关位函数和求解Maxwell方程的新方法

本文是文[1]的继续. 一、本文引入了一个新的位函数φ——相关位函数φ.它与经典的Helmboltz的标位Φ和矢位不同.由φ,我们得到了求解方程组?×=,?·=P的新公式.二、在交变电磁场中,我们引入了两个新的滞后位函数:滞后相关电位φ_e和滞后相关磁位φ_m.这两个位函数不同于经典的滞后位A和Φ,由φ_e和φ_m,我们得到了求解Maxwell方程组的新公式.三、指出了构建具有给定旋度函数(旋涡)的涡旋场的方法.     

On the scattering frequencies of time-dependent potentials

This paper discusses the scattering frequencies associated with the scalar wave equation and a time-periodic, real, potential. It is shown that the scattering frequencies form a discrete set in the complex plane and depend continuously on the potential. Existence of the scattering frequencies is proved for periodic potentials which are perturbations of a time independent, nonnegative, potential.     

Potential solutions of linear systems: The multi-criteria multiple constraint levels program

The realistic modeling of decision problems requires considerable flexibility in the model structure. Frequently one is faced with problems involving multiple criteria for which the constraint level is acceptable if a certain parameter (which may be a random variable) lies within a prescribed set. Furthermore, in formulating the problem, the criteria and constraints may be interchangeable. This requires a treatment which is more general than the nondominated solution in a multicriteria problem. We shall introduce the concept of a potential solution to cope with the above problem. To effectively locate these potential solutions, a generalization of the multicriteria (MC) simplex method, which handles multiple constraint levels (right hand sides) is developed. Geometric properties of adjacent potential solutions will be described together with a computational procedure which is based on the “connectedness” of the set of potential solutions. The natural duality relationship which exists in the double-MC simplex method and its consequences are also explored.     

Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearities modulated in space and time

This work deals with soliton solutions of the nonlinear Schrödinger equation with a diversity of nonlinearities. We solve the equation in a potential which oscillates in time between attractive and expulsive behavior, in the presence of nonlinearities which are modulated in space and time. Despite the presence of the periodically expulsive behavior of the potential, the results show that the nonlinear equation can support a diversity of localized excitations of the bright and dark types.     

Approximate continuation of harmonic functions in geodesy: A spline based least squares approach with regularization

This paper is concerned with the approximate reconstruction of the earth’s potential field from geometric and gravimetric data. This is an ill-posed problem involving typically large amounts of data which are to be continued by a harmonic function. The standard approach in geodesy is based on spherical harmonics which are globally supported. Thus, a least squares approach for the data fitting yields a linear system of equations with a fully populated system matrix. This becomes computationally prohibitive for large amounts of data and, therefore, presents the biggest bottleneck for fast and efficient computations.     

The application of the genetic algorithm to the minimization of potential energy functions

We adapted the genetic algorithm to minimize the AMBER potential energy function. We describe specific recombination and mutation operators for this task. Next we use our algorithm to locate low energy conformation of three polypeptides (AGAGAGAGA, A9, and [Met]-enkephalin) which are probably the global minimum conformations. Our potential energy minima are –94.71, –98.50, and –48.94 kcal/mol respectively. Next, we applied our algorithm to the 46 amino acid protein crambin and located a non-native conformation which had an AMBER potential energy 150 kcal/mol lower than the native conformation. This is not necessarily the global minimum conformation, but it does illustrate problems with the AMBER potential energy function. We believe this occurred because the AMBER potential energy function does not account for hydration.     

Global weak solution of the vlasov–poisson system for small electrons mass

We are studying the existence and weak stability of a Vlasov–Poisson syste with two typs of particles , in which the electrons are supposed to be at thermal equilibrium. This modifies the source term in the Poisson equaitonm\, and estimates in the Marcinkiewicz space M³ for the potential are used to get the strong compactness of approximations using a new regularized kernal which preservs an approriate energy inequality.     

A variational approach to transonic potential flow problems

This paper continues considerations of transonic potential flow problems by variational methods. A functional which is associated with a boundary value problem for the (full) potential equation and which possesses a real physical meaning is minimized over a class of admissible functions. These functions have to satisfy a non‐linear local entropy condition and a certain boundness constraint. Though this class is not a compact set of the underlying Hilbert space and though the functional need not be convex, the existence of a solution to the established variational problem can be proved by direct methods of the calculus of variations. Furthermore, some properties of minimizers concerning uniqueness, relation to the potential equation, and behaviour on supersonic regions are derived. Copyright © 2000 John Wiley & Sons, Ltd.     

Finely continuously differentiable functions

This paper establishes an explicit characterization of those real-valued functions on a finely open set in Euclidean space which are continuously differentiable with respect to the fine topology of classical potential theory. It improves and extends previous work of several authors. Differentiability of all orders is considered, and some consequences of the characterization are deduced.     

求解非线性规划问题的一类对偶算法

本文提出了一类求解不等式约束非线性规划问题的构造性对偶算法,我们证明在适当的条件下,势函数的罚参数存在一个阀值,当罚参数小于这个阀值时,由这一方法所产生的序列局部收敛于问题的一个Kuhn-Tucker解,我们也建立了解的依赖于罚参数的误差上界,最后,我们给出了一个特残势函数的数值结果。     

Formulation variationnelle pour le calcul de la diffraction d'une onde acoustique par une surface rigide

This work is a continuation of [1]. We give a space-time variational formula to the problem of the scattered acoutic wave by a hard body, using the double layer retarded potential technique. New schemes are constructed from this variationnal formula, for which we prove the stability and errors estimates.     

On the spectral instability of the Sturm-Liouville operator with a complex potential

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector.     

Inverse scattering for the magnetic Schrödinger operator

We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n?3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston (1995) [5] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument.     

Nonlinear Schrödinger equation containing the time-derivative of the probability density: A numerical study

A nonlinear Schrödinger equation that contains the time-derivative of the probability density is investigated, which is motivated by the attempt to include the recoil effect of radiation. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigen-states of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the normalization of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential (1D ISWP) and for several time-dependent potentials that tend to the former as time increases. Results show that for various initial states, the wave function always evolves into some eigen-state of the corresponding linear Hamiltonian of the 1D ISWP. For a small time-dependent perturbation potential, solutions present the process similar to the spontaneous transition between stationary states. For a periodical potential with an appropriate frequency, solutions present the process similar to the stimulated transition. This nonlinear Schrödinger equation thus presents the state evolution similar to the wave-function reduction.     

On numerical schemes for phase-field models for electrowetting with electrolyte solutions

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase-field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Excessiveness of harmonic functions for certain diffusions

In an earlier paper⁽⁴⁾ the author has shown that a diffusion process whose potential kernel satisfies certain analytic conditions has all of its excessive harmonic functions, which are not identically infinite, continuous. This paper shows that under these conditions (concerning its potential kernel), the excessiveness of its nonnegative harmonic functions isautomatic.     

A Fourier method for computing the perturbation to a two‐dimensional electric potential by a conductor of arbitrary shape

Introducing an electric conductor into a region pervaded by an initial electric potential perturbs that potential by inducing a charge distribution on the conductor's surface, necessary to guarantee that the surface is an equipotential of the total potential. Some numerical method is required to compute the perturbation potential, when the conductor's shape does not admit a standard analytic solution. For two‐dimensional situations, a method is proposed for solving for the perturbation potential that involves expansion of the boundary perturbation potential and its normal derivative as truncated Fourier series. This boundary potential is known to within an additive constant from the requirement that its sum with the initial potential must be a constant. The standard representation theorem for the Dirichlet problem gives a consistency relation between the boundary function and its normal derivative, which here becomes a set of linear algebraic relations between Fourier series coefficients, with matrix entries found by appropriate applications of the fast Fourier transform. These are solved for the boundary derivative coefficients; at any point exterior to the conductor, the perturbation potential can then be evaluated from the two sets of Fourier coefficients, using further application of the fast Fourier transform. Examples are shown for two conductor shapes, with several initial potentials. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 673–683, 2001     

Potential transients for an electrochemical corrosion reaction under constant current conditions

Due to the electrochemical nature of almost all corrosion reactions, electrochemical methods are commonly used to measure the corrosion rate of a metal in the laboratory or in the field. In particular, steady state methods are the most widely used for corrosion rate measurements. Transient methods, which can be much more efficient, traditionally rely on an equivalent linear circuit representing the surface kinetics, with negligible mass transport effects. This has been reported to predict transients which are not observed experimentally in many practical situations. In this paper, we consider the galvanostatic method, wherein a constant current is applied across a corroding metal surface and the transient potential response is recorded. The resulting boundary value problems incorporating mixed kinetic and diffusion control involve highly nonlinear, coupled boundary conditions. We present numerical and approximate analytical solutions which can be incorporated into corrosion analysis routines in order to calculate corrosion parameters. The analytical expressions open the possibility of measuring corrosion parameters by merely fitting a class of elementary functions to experimental potential transients. This leads to a significant reduction in the number of computations required for the curve fitting, and hence increasing the overall efficiency of the measurement process compared to the conventional steady state methods.