Showing papers on "Partial differential equation published in 1994"

A New Integrable Shallow Water Equation

Abstract: Publisher Summary This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation in shallow water theory. Such equations possess soliton solutions-coherent (spatially localized) structures that interact nonlinearly among themselves and then re-emerge, retaining their identity and showing particle-like scattering behavior. This chapter discusses a newly discovered, completely integrable dispersive shallow-water equation found by Camassa and Holm in 1993. This equation is obtained by using a small-wave-amplitude asymptotic expansion directly in the Hamiltonian for the vertically averaged incompressible Euler's equations, after substituting a solution ansatz of columnar fluid motion and restricting to an invariant manifold for unidirectional motion of waves at the free surface under the influence of gravity. Section II of the chapter derives the one-dimensional Green–Naghdi equations. Section III uses Hamiltonian methods to newly discovered equation for unidirectional waves. Section IV analyzes the behavior of the solutions of the equation and shows that certain initial conditions develop a vertical slope in finite time. It is also shown that there exist stable multisoliton solutions. Section V demonstrates the existence of an infinite number of conservation laws for the equation that follow from its bi-Hamiltonian property.

Ginzburg-Landau Vortices

Abstract: The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research. The text is concerned with the study in two dimensions of stationary solutions uE of a complex valued Ginzburg-Landau equation involving a small parameter E. Such problems are related to questions occuring in physics, such as phase transistion phenomena in superconductors and superfluids. The parameter E has a dimension of a length, which is usually small. Thus, it should be of interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u* of minimizers uE exists. Moreover, u* is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree - or winding number - of the boundary condition. Each singularity has degree one - or, as physicists would say, vortices are quantized. The singularities have infinite energy, but after removing the core energy we are led to a concept of finite renormalized energy. The location of the singularities is completely determined by minimizing the renormalized energy among all possible configurations of defects. The limit u* can also be viewed as a geometrical object. It is a minimizing harmonic map into S1 with prescribed boundary condition g. Topological obstructions imply that every map u into S1 with u=g on the boundary must have infinite energy. Even though u* has infinite energy one can think of u* as having "less" infinite energy than any other map u with u=g on the boundary. The material presented in this book covers mostly recent and original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations and complex functions. It is designed for researchers and graduate students alike and can be used as a one-semester text.

Fracture and crack growth by element free Galerkin methods

Abstract: Element free Galerkin (EFG) methods are methods for solving partial differential equations that require only nodal data and a description of the geometry; no element connectivity data are needed. This makes the method very attractive for the modeling of the propagation of cracks, as the number of data changes required is small and easily developed. The method is based on the use of moving least-squares interpolants with a Galerkin method, and it provides highly accurate solutions for elliptic problems. The implementation of the EFG method for problems of fracture and static crack growth is described. Numerical examples show that accurate stress intensity factors can be obtained Without any enrichment of the displacement field by a near-crack-tip singularity and that crack growth can be easily modeled since it requires hardly any remeshing.

Asymptotic stability of solitary waves

Abstract: We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation $$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$ is asymptotically stable Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, $$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 $$ In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 3 0, withf∈C4) The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation) We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part) The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

Solving Schrödinger’s equation around a desired energy: Application to silicon quantum dots

Abstract: We present a simple, linear‐in‐size method that enables calculation of the eigensolutions of a Schrodinger equation in a desired energy window. We illustrate this method by studying the near‐gap electronic structure of Si quantum dots with size up to Si1315H460(≊37 A in diameter) using a plane wave pseudopotential representation.

Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data

Abstract: (1994). Semilinear heat equations and the navier-stokes equation with distributions in new function spaces as initial data. Communications in Partial Differential Equations: Vol. 19, No. 5-6, pp. 959-1014.

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

Abstract: We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.

First-order system least squares for second-order partial differential equations: part I

Abstract: This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in $n=2$ or $3$ dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the $H(\divv) \times H^1$ norm and to yield optimal convergence for finite element subspaces of $H(\divv) \times H^1$. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the $(H^1)^{n+1}$ norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of $(H^1)^{n+1}$. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection--diffusion--reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.

Analysis of the K-epsilon turbulence model

Abstract: This book is aimed at applied mathematicians interested in numerical simulation of turbulent flows. The book is centered around the k - {epsilon} model but it also deals with other models such as subgrid scale models, one equation models and Reynolds Stress models. The reader is expected to have some knowledge of numerical methods for fluids and, if possible, some understanding of fluid mechanics, the partial differential equations used and their variational formulations. This book presents the k - {epsilon} method for turbulence in a language familiar to applied mathematicians, stripped bare of all the technicalities of turbulence theory. The model is justified from a mathematical standpoint rather than from a physical one. The numerical algorithms are investigated and some theoretical and numerical results presented. This book should prove an invaluable tool for those studying a subject that is still controversial but very useful for industrial applications. (authors). 71 figs., 200 refs.

Moving mesh partial differential equations (MMPDES) based on the equidistribution principle

Abstract: This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh equations that have been used by others. Their stability is analyzed and it is seen that a key term for most of these moving mesh PDEs is a source-like term that measures the level of equidistribution. It is shown that under weak assumptions mesh crossing cannot occur for most of them. Finally, numerical experiments for these various moving mesh PDEs are performed to study their relative properties.

The Couette-Taylor problem

Abstract: This monograph presents a systematic and unified approach to the non-linear stability problem and transitions in the Couette-Taylor problem, by the means of analytic and constructive methods. The most "elementary" one-parameter theory is first presented. More complex situations are then analyzed (mode interactions, imperfections, non-spatially periodic patterns). The whole analysis is based on the mathematically rigorous theory of centre manifold and normal forms, and symmetries are fully taken into account. These methods are very general and can be applied to other hydrodynamical instabilities, or more generally to physical problems modelled by partial differential equations.

An Introduction to Nonlinear Partial Differential Equations

Abstract: Preface. 1. Partial Differential Equations. 1.1 Partial Differential Equations. 1.1.1 PDEs and Solutions. 1.1.2 Classification. 1.1.3 Linear vs. Nonlinear. 1.1.4 Linear Equations. 1.2 Conservation Laws. 1.2.1 One Dimension. 1.2.2 Higher Dimensions. 1.3 Constitutive Relations. 1.4 Initial and Boundary Value Problems. 1.5 Waves. 1.5.1 Traveling Waves. 1.5.2 Plane Waves. 1.5.3 Plane Waves and Transforms. 1.5.4 Nonlinear Dispersion. 2. First-Order Equations and Characteristics. 2.1 Linear First-Order Equations. 2.1.1 Advection Equation. 2.1.2 Variable Coefficients. 2.2 Nonlinear Equations. 2.3 Quasi-linear Equations. 2.3.1 The general solution. 2.4 Propagation of Singularities. 2.5 General First-Order Equation. 2.5.1 Complete Integral. 2.6 Uniqueness Result. 2.7 Models in Biology. 2.7.1 Age-Structure. 2.7.2 Structured predator-prey model. 2.7.3 Chemotherapy. 2.7.4 Mass structure. 2.7.5 Size-dependent predation. 3. Weak Solutions To Hyperbolic Equations. 3.1 Discontinuous Solutions. 3.2 Jump Conditions. 3.2.1 Rarefaction Waves. 3.2.2 Shock Propagation. 3.3 Shock Formation. 3.4 Applications. 3.4.1 Traffic Flow. 3.4.2 Plug Flow Chemical Reactors. 3.5 Weak Solutions: A Formal Approach. 3.6 Asymptotic Behavior of Shocks. 3.6.1 Equal-Area Principle. 3.6.2 Shock Fitting. 3.6.3 Asymptotic Behavior. 4. Hyperbolic Systems. 4.1 Shallow Water Waves Gas Dynamics. 4.1.1 Shallow Water Waves. 4.1.2 Small-Amplitude Approximation. 4.1.3 Gas Dynamics. 4.2 Hyperbolic Systems and Characteristics. 4.2.1 Classification. 4.3 The Riemann Method. 4.3.1 Jump Conditions for Systems. 4.3.2 Breaking Dam Problem. 4.3.3 Receding Wall Problem. 4.3.4 Formation of a Bore. 4.3.5 Gas Dynamics. 4.4 Hodographs and Wavefronts. 4.4.1 Hodograph Transformation. 4.4.2 Wavefront Expansions. 4.5 Weakly Nonlinear Approximations. 4.5.1 Derivation of Burgers' Equation. 5. Diffusion Processes. 5.1 Diffusion and Random Motion. 5.2 Similarity Methods. 5.3 Nonlinear Diffusion Models. 5.4 Reaction-Diffusion Fisher's Equation. 5.4.1 Traveling Wave Solutions. 5.4.2 Perturbation Solution. 5.4.3 Stability of Traveling Waves. 5.4.4 Nagumo's Equation. 5.5 Advection-Diffusion Burgers' Equation. 5.5.1 Traveling Wave Solution. 5.5.2 Initial Value Problem. 5.6 Asymptotic Solution to Burgers' Equation. 5.6.1 Evolution of a Point Source. 6. Reaction-Diffusion Systems. 6.1 Reaction-Diffusion Models. 6.1.1 Predator-Prey Model. 6.1.2 Combustion. 6.1.3 Chemotaxis. 6.2 Traveling Wave Solutions. 6.2.1 Model for the Spread of a Disease. 6.2.2 Contaminant transport in groundwater. 6.3 Existence of Solutions. 6.3.1 Fixed-Point Iteration. 6.3.2 Semi-Linear Equations. 6.3.3 Normed Linear Spaces. 6.3.4 General Existence Theorem. 6.4 Maximum Principles. 6.4.1 Maximum Principles. 6.4.2 Comparison Theorems. 6.5 Energy Estimates and Asymptotic Behavior. 6.5.1 Calculus Inequalities. 6.5.2 Energy Estimates. 6.5.3 Invariant Sets. 6.6 Pattern Formation. 7. Equilibrium Models. 7.1 Elliptic Models. 7.2 Theoretical Results. 7.2.1 Maximum Principle. 7.2.2 Existence Theorem. 7.3 Eigenvalue Problems. 7.3.1 Linear Eigenvalue Problems. 7.3.2 Nonlinear Eigenvalue Problems. 7.4 Stability and Bifurcation. 7.4.1 Ordinary Differential Equations. 7.4.2 Partial Differential Equations. References. Index.

Geometric statistics in turbulence

Abstract: The author presents results regarding certain average properties of incompressible fluids derived from the equations of motion. The author estimates the average dissipation rate, the average dimension of level sets. The role played by the field of direction of vorticity in the three-dimensional Euler and Navier-Stokes equations is discussed and a class of two-dimensional equations that are useful models of incompressible dynamics is described. The author presents results concerning scaling exponents in turbulence.

Exact and approximate controllability for distributed parameter systems

Abstract: We consider a system whose state is given by the solution y to a Partial Differential Equation (PDE) of evolution, and which contains control functions, denoted by v.

Solution of the Redfield equation for the dissipative quantum dynamics of multilevel systems

Abstract: We present a new method for solving the Redfield equation, which describes the evolution of the reduced density matrix of a multilevel quantum‐mechanical system interacting with a thermal bath. The method is based on a new decomposition of the Redfield relaxation tensor that makes possible its direct application to the density matrix without explicit construction of the full tensor. In the resulting expressions, only ordinary matrices are involved and so any quantum system whose Hamiltonian can be diagonalized can be treated with the full Redfield theory. To efficiently solve the equation of motion for the density matrix, we introduce a generalization of the short‐iterative‐Lanczos propagator. Together, these contributions allow the complete Redfield theory to be applied to significantly larger systems than was previously possible. Several model calculations are presented to illustrate the methodology, including one example with 172 quantum states.

Moving finite elements

Abstract: This book is mainly concerned with finite element methods for time-dependent partial differential equations when the grids are allowed to move in time, but also describes grid generation techniques which include grid adjustment. The mechanism for grid movement derives from a generalization of the residual minimization technique which is familiar from the Galerkin finite element method. The book brings together most of the work done over the last decade or so which has been stimulated by Miller's original idea, and discusses the interrelationships between the techniques of the method and the established ideas of the method of characteristics, Hamilton's equations, the Legendre transformation, and grid equidistribution. The book highlights the issues involved and should provide the reader with a clear view of the current state of the subject and prompt further research.

Wavelet–Galerkin solutions for one‐dimensional partial differential equations

Abstract: In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies1) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund2 and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.

Spectral approach to solving three-dimensional Maxwell's diffusion equations in the time and frequency domains

Abstract: We describe a new explicit three-dimensional solver for the diffusion of electromagnetic fields in arbitrarily heterogeneous conductive media The proposed method is based on a global Krylov subspace (Lanczos) approximation of the solution in the time and frequency domains We derive solutions stable to spurious curl-free modes and provide estimates of the computer complexity involved in the calculations Such estimates together with numerical experiments attest to a computationally efficient method suitable for large-scale problems Also included are modeling examples drawn from practical geophysical applications

Methods for Analysis of Nonlinear Elliptic Boundary Value Problems

Abstract: The theory of nonlinear elliptic equations is currently one of the most actively developing branches of the theory of partial differential equations. This book investigates boundary value problems for nonlinear elliptic equations of arbitrary order. In addition to monotone operator methods, a broad range of applications of topological methods to nonlinear differential equations is presented: solvability, estimation of the number of solutions, and the branching of solutions of nonlinear equations. Skrypnik establishes, by various procedures, a priori estimates and the regularity of solutions of nonlinear elliptic equations of arbitrary order. Also covered are methods of homogenization of nonlinear elliptic problems in perforated domains. The book is suitable for use in graduate courses in differential equations and nonlinear functional analysis.