Showing papers on "Partial differential equation published in 1990"

Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems

Abstract: Variational problems are part of our classical cultural heritage. The book gives an introduction to variational methods and presents on overview of areas of current research in this field. Particular topics included are the direct methods including lower semi-continuity results, the compensated compactness method, the concentration compactness method, Ekeland's variational principle, and duality methods or minimax methods, including the mountain pass theorems, index theory, perturbation theory, linking and extensions of these techniques to non-differentiable functionals and functionals defined on convex sets - and limit cases. All results are illustrated by specific examples, involving Hamiltonian systems, non-linear elliptic equations and systems, and non-linear evolution problems. These examples often represent the current state of the art in their fields and open perspective for further research. Special emphasis is laid on limit cases of the Palais-Smale condition.

Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates

Abstract: A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that

Output regulation of nonlinear systems

Abstract: The problem of controlling a fixed nonlinear plant in order to have its output track (or reject) a family of reference (or disturbance) signal produced by some external generator is discussed. It is shown that, under standard assumptions, this problem is solvable if and only if a certain nonlinear partial differential equation is solvable. Once a solution of this equation is available, a feedback law which solves the problem can easily be constructed. The theory developed incorporates previously published results established for linear systems. >

A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation

Abstract: Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, characterized by large mass balance errors and erroneous estimates of infiltration depth. Conversely, numerical solutions based on the mixed form of Richards equation can be shown to possess the conservative property, so that mass is perfectly conserved. This leads to significant improvement in numerical solution performance, while requiring no additional computational effort. However, use of the mass-conservative method does not guarantee good solutions. Accurate solution of the unsaturated flow equation also requires use of a diagonal time (or mass) matrix. Only when diagonal time matrices are used can the solution be shown to obey a maximum principle, which guarantees smooth, nonoscillatory infiltration profiles. This highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow.

Asymptotic-behavior for singularities of the mean-curvature flow

Abstract: is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time interval. Moreover, it was shown that for convex initial data Mo the surfaces Mt contract smoothly to a single point in finite time and become spherical at the end of the contraction. Here we want to study the singularities of (1) which can occur for nonconvex initial data. Our aim is to characterize the asymptotic behavior of Mt near a singularity using rescaling techniques. These methods have been used in the theory of minimal surfaces and more recently in the study of semilinear heat equations [3], [4]. An important tool of this approach is a monotonicity formula, which we establish in §3. Assuming then a natural upper bound for the growth rate of the curvature we show that after appropriate rescaling near the singularity the surfaces Mt approach a selfsimilar solution of (1). In §4 we consider surfaces Mt9 n > 2, of positive mean curvature and show that in this case the only compact selfsimilar solutions of (1) are spheres. Finally, in §5 we study the model-problem of a rotationally symmetric shrinking neck. We prove that the natural growth rate estimate is valid in this case and that the rescaled solution asymptotically approaches a cylinder.

Feature-oriented image enhancement using shock filters

Abstract: Shock filters for image enhancement are developed. The filters use new nonlinear time dependent partial differential equations and their discretizations. The evolution of the initial image $u_0 (x,y)$ as $t \to \infty $ into a steady state solution $u_\infty (x,y)$ through $u(x,y,t)$, $t > 0$, is the filtering process. The partial differential equations have solutions which satisfy a maximum principle. Moreover the total variation of the solution for any fixed $t > 0$ is the same as that of the initial data. The processed image is piecewise smooth, nonoscillatory, and the jumps occur across zeros of an elliptic operator (edge detector). The algorithm is relatively fast and easy to program.

Convergence of approximation schemes for fully nonlinear second order equations

Abstract: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied It is proved that any monotone, stable, and consistent scheme converges (to the correct solution), provided that there exists a comparison principle for the limiting equation Several examples are given where the result applies >

Liquid film on an unsteady stretching surface

Abstract: A fluid film lies on an accelerating stretching surface. A similarity transform reduces the unsteady Navier-Stokes equations to a nonlinear ordinary differential equation governed by a nondimensional unsteady parameter. Asymptotic and numerical solutions are found. The results represent rare exact similarity solutions of the unsteady Navier-Stokes equations

Differential Equations for Quantum Correlation Functions

Abstract: The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painleve transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrodinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.

H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations

Abstract: New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure

Abstract: The physical meaning of the constant {tau} in Cattaneo and Vernotte's equation for materials with a nonhomogeneous inner structure has been considered. An experimental determination of the constant {tau} has been proposed and some values for selected products have been given. The range of differences in the description of heat transfer by parabolic and hyperbolic heat conduction equations has been discussed. Penetration time, heat flux, and temperature profiles have been taken into account using data from the literature and the experimental and calculated results.

Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation

Abstract: In this work we adapt the techniques developed in [C1] to prove interior estimates for solutions of perturbations of the Monge-Ampere equation

Exponential decay for the semilinear wave equation with locally distributed damping

Abstract: (1990). Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping. Communications in Partial Differential Equations: Vol. 15, No. 2, pp. 205-235.

Rapid, stable fluid dynamics for computer graphics

Abstract: We present a new method for animating water based on a simple, rapid and stable solution of a set of partial differential equations resulting from an approximation to the shallow water equations. The approximation gives rise to a version of the wave equation on a height-field where the wave velocity is proportional to the square root of the depth of the water. The resulting wave equation is then solved with an alternating-direction implicit method on a uniform finite-difference grid. The computational work required for an iteration consists mainly of solving a simple tridiagonal linear system for each row and column of the height field. A single iteration per frame suffices in most cases for convincing animation.Like previous computer-graphics models of wave motion, the new method can generate the effects of wave refraction with depth. Unlike previous models, it also handles wave reflections, net transport of water and boundary conditions with changing topology. As a consequence, the model is suitable for animating phenomena such as flowing rivers, raindrops hitting surfaces and waves in a fish tank as well as the classic phenomenon of waves lapping on a beach. The height-field representation prevents it from easily simulating phenomena such as breaking waves, except perhaps in combination with particle-based fluid models. The water is rendered using a form of caustic shading which simulates the refraction of illuminating rays at the water surface. A wetness map is also used to compute the wetting and drying of sand as the water passes over it.

Finite Element Methods for Fluids

Abstract: Partial differential equations of fluid mechanics irrotational and weakly irrotational flows convention - diffusion phenomena the Stokes problem the Navier-Stokes equations Euler, compressible Navier-Stokes and the shallow water equations Appendix: a finite element program for fluids on the Macintosh

Classical and quantal nonrelativistic Chern-Simons theory

Abstract: We construct a nonrelativistic field theory for the second-quantized N-body system of point particles with Chern-Simons interactions. Various properties of this model are discussed: its obvious and hidden symmetries, its relation to a relativistic field theory, and its supersymmetric formulation. We present classical, static solutions--solitons--that satisfy a self-dual equation, which is equivalent to the Liouville equation; hence, it is completely solvable. The dynamical role of the Chern-Simons interaction is demonstrated: the interaction does not merely change statistics but also provides the forces that bind the classical solitons.

On the finite volume element method

Abstract: The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.

Soliton solutions to the gauged nonlinear Schrödinger equation on the plane.

Abstract: A gauged, nonlinear Schr\"odinger equation in two spatial dimensions is considered. This equation describes nonrelativistic matter interacting with Chern-Simons gauge fields. We find explicit static, self-dual solutions that satisfy the Liouville equation.

An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow

Abstract: In this paper we present a simple, general methodology for the generation of high-order operator decomposition (“splitting”) techniques for the solution of time-dependent problems arising in ordinary and partial differential equations. The new approach exploits operator integration factors to reduce multiple-operator equations to an associated series of single-operator initial-value subproblems. Two illustrations of the procedure are presented: the first, a second-order method in time applied to velocity-pressure decoupling in the incompressible Stokes problem; the second, a third-order method in time applied to convection-Stokes decoupling in the incompressible Navier-Stokes equations. Critical open questions are briefly described.

Tension-field theory

Abstract: A general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations. The principal contribution is a partial differential equation describing a geometrical property of tension trajectories. This is one of a system of two equations which describes the state of stress independently of the deformation. This system is strongly elliptic at any stable solution, whereas the deformation is described by a system of parabolic type. Controllable solutions, i.e. those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone, are obtained. The general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples. Existing small strain theories are shown to correspond to a singular limit of the general theory, at which the underlying system changes from elliptic to parabolic type.

A Nonlinear Theory of Generalized Functions

Abstract: Generalized functions on an open subset of En.- Generalized functions on an arbitrary subset of En.- Generalized solutions of nonlinear partial differential equations.

Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation

Abstract: A numerical and analytical study of the Kuramoto–Sivashinsky partial differential equation (PDE) in one spatial dimension with periodic boundary conditions is presented. The structure, stability, and bifurcation characteristics of steady state and time-dependent solutions of the PDE for values of the parameter $\alpha $ less than 40 are examined. The numerically observed primary and secondary bifurcations of steady states, as well as bifurcations to constant speed traveling waves (limit cycles), are analytically verified. Persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of $O( 2 )$. Their effect on the system dynamics is discussed, and several tertiary bifurcations, observed numerically, are presented.

Matrix methods applied to acoustic waves in multilayers

Abstract: Matrix methods for analyzing the electroacoustic characteristics of anisotropic piezoelectric multilayers are described. The conceptual usefulness of the methods is demonstrated in a tutorial fashion by examples showing how formal statements of propagation, transduction, and boundary-value problems in complicated acoustic layered geometries such as those which occur in surface acoustic wave (SAW) devices, in multicomponent laminates, and in bulk-wave composite transducers are simplified. The formulation given reduces the electroacoustic equations to a set of first-order matrix differential equations, one for each layer, in the variables that must be continuous across interfaces. The solution to these equations is a transfer matrix that maps the variables from one layer face to the other. Interface boundary conditions for a planar multilayer are automatically satisfied by multiplying the individual transfer matrices in the appropriate order, thus reducing the problem to just having to impose boundary conditions appropriate to the remaining two surfaces. The computational advantages of the matrix method result from the fact that the problem rank is independent of the number of layers, and from the availability of personal computer software that makes interactive numerical experimentation with complex layered structures practical. >

The Poincare-Bendixson theorem for monotone cyclic feedback systems

Abstract: We prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inR n of the form $$x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).$$ We apply our results to a variety of models of biological systems.

Distribution theory and Fourier analysis

Abstract: I. Test Functions.- Summary.- 1.1. A review of Differential Calculus.- 1.2. Existence of Test Functions.- 1.3. Convolution.- 1.4. Cutoff Functions and Partitions of Unity.- Notes.- II. Definition and Basic Properties of Distributions.- Summary.- 2.1. Basic Definitions.- 2.2. Localization.- 2.3. Distributions with Compact Support.- Notes.- III. Differentiation and Multiplication by Functions.- Summary.- 3.1. Definition and Examples.- 3.2. Homogeneous Distributions.- 3.3. Some Fundamental Solutions.- 3.4. Evaluation of Some Integrals.- Notes.- IV. Convolution.- Summary.- 4.1. Convolution with a Smooth Function.- 4.2. Convolution of Distributions.- 4.3. The Theorem of Supports.- 4.4. The Role of Fundamental Solutions.- 4.5. Basic Lp Estimates for Convolutions.- Notes.- V. Distributions in Product Spaces.- Summary.- 5.1. Tensor Products.- 5.2. The Kernel Theorem.- Notes.- VI. Composition with Smooth Maps.- Summary.- 6.1. Definitions.- 6.2. Some Fundamental Solutions.- 6.3. Distributions on a Manifold.- 6.4. The Tangent and Cotangent Bundles.- Notes.- VII. The Fourier Transformation.- Summary.- 7.1. The Fourier Transformation in ? and in ?'.- 7.2. Poisson's Summation Formula and Periodic Distributions.- 7.3. The Fourier-Laplace Transformation in ?'.- 7.4. More General Fourier-Laplace Transforms.- 7.5. The Malgrange Preparation Theorem.- 7.6. Fourier Transforms of Gaussian Functions.- 7.7. The Method of Stationary Phase.- 7.8. Oscillatory Integrals.- 7.9. H(s), Lp and Holder Estimates.- Notes.- VIII. Spectral Analysis of Singularities.- Summary.- 8.1. The Wave Front Set.- 8.2. A Review of Operations with Distributions.- 8.3. The Wave Front Set of Solutions of Partial Differential Equations.- 8.4. The Wave Front Set with Respect to CL.- 8.5. Rules of Computation for WFL.- 8.6. WFL for Solutions of Partial Differential Equations.- 8.7. Microhyperbolicity.- Notes.- IX. Hyperfunctions.- Summary.- 9.1. Analytic Functionals.- 9.2. General Hyperfunctions.- 9.3. The Analytic Wave Front Set of a Hyperfunction.- 9.4. The Analytic Cauchy Problem.- 9.5. Hyperfunction Solutions of Partial Differential Equations.- 9.6. The Analytic Wave Front Set and the Support.- Notes.- Exercises.- Answers and Hints to All the Exercises.- Index of Notation.

Partial Differential Equations: Analytical Solution Techniques

Abstract: 1. The Diffusion Equation.- 2. Laplace's Equation.- 3. The Wave Equation.- 4. Linear Second-Order Equations with Two Independent Variables.- 5. The Scalar Quasilinear First-Order Equation.- 6. Nonlinear First-Order Equations.- 7. Quasilinear Hyperbolic Systems.- 8. Approximate Solutions by Perturbation Methods.- A.1. Review of Green's Function for ODEs Using the Dirac Delta Function.- A.2. Review of Fourier and Laplace Transforms.- A.3. Review of Asymptotic Expansions.- References.