Showing papers on "Partial differential equation published in 2002"

Dynamics of Evolutionary Equations

Abstract: Preface * 1 The Evolution of Evolutionary Systems * 2 Dynamical Systems: Basic Theory * 3 Linear Semigroups * 4 Basic Theory of Evolutionary Equations * 5 Nonlinear Partial Differential Equations * 6 Navier Stokes Dynamics * 7 Basic Principles of Dynamics * 8 Inertial Manifolds and the Reduction Principle * Appendices: Basics of Functional Analysis * Bibliography * Notation Index * Subject Index

Symmetry and Integration Methods for Differential Equations

Abstract: Introduction * Dimensional Analysis, Modeling, and Invariance * Lie Groups of Transformations and Infinitesimal Transformations * Ordinary Differential Equations (ODEs) * Partial Differential Equations (PDEs) * References * Author Index * Subject Index

A new integrable equation with peakon solutions

Abstract: We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact integrability of the new equation by constructing its Lax pair, and we explain its connection with a negative flow in the Kaup-Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions in the form of a superposition of multi-peakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa-Holm peakons.

Inverse Problems for Partial Differential Equations

Abstract: Auxiliary information from functional analysis and theory of differential equations the basic notions and notations inequalities some concepts and theorems of functional analysis linear partial differential equation of the first order the maximum principle for parabolic equations of second order the weak approximation method examples reducing to the concept of the weak approximation method general formulation of the weak approximation method two theorems - an example the linear partial differential equation identification problems for parabolic equations with Cauchy data the unknown source function the unknown lowest coefficient the unknown coefficient to the first order derivative an unknown coefficient to the time derivative inverse problem for a semilinear parabolic equation equations of Burgers type the splitting of one many-dimensional inverse problem into problems of lower dimension the identification of the source function for a system of composite type and parabolic equation the behaviour of the problem's solution under t-> the statement of the problem theorems of existence and uniqueness "on the whole" the behaviour of solution by t-> stationary problem convergence to the stationary problem solution unique solvability of the problem of identifying the source function for a parabolic equation the stabilization of the solution to the inverse problem the problem of determining the coefficient in a parabolic equation and some properties of its solution statement of the problem theorems of existence and uniqueness "on the whole" properties of solution under t-> two unknown coefficients of a parabolic type equation uniform boundary conditions inhomogeneous conditions of over-determination input data of the special form some inverse boundary problems unknown source function nonlinear heat equation hyperbolic equation with small parameter.

Functional equations and inequalities in several variables

Abstract: Functional Equations and Inequalities in Linear Spaces: Linear Spaces and Semilinear Topology Convex Functions Cauchy's Exponential Equation Polynomial Functions and Their Extensions Quadratic Mappings Quadratic Equation on an Interval Ulam-Hyers-Rassias Stability of Functional Equations: Additive Cauchy Equation Multiplicative Cauchy Equation Jensen's Functional Equation Gamma Functional Equation Stability of Homogeneous Mappings Stability of Functional Equations in Function Spaces Stability in the Lipschitz Norms Round-off Stability of Iterations Functional Equations in Set-Valued Functions: Cauchy's Set-Valued Functional Equation Pexider's Functional Equation Subadditive Set-Valued Functions Hahn-Banach Type Theorem and Applications Subquadratic Set-Valued Functions Iteration Semigroups of Set-Valued Functions.

Partial differential equations : methods and applications

Abstract: This text gathers, revises and explains the newly developed Adomian decomposition method along with its modification and some traditional techniques.

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Handbook of linear partial differential equations for engineers and scientists

Abstract: INTRODUCTION: SOME DEFINITIONS, FORMULAS, METHODS, AND SOLUTIONS Classification of Second Order Partial Differential Equations Basic Problems of Mathematical Physics Properties and Particular Solutions of Linear Equations Separation of Variables Method Integral Transforms Method Representation of the Solution of the Cauchy Problem via the Fundamental Solution Nonhomogeneous Boundary Value Problems with One Space Variable Nonhomogeneous Boundary Value Problems with Many Space Variables Construction of the Green's Functions: General Formulas and Relations Duhamel's Principles in Nonstationary Problems Transformation Simplifying Initial and boundary Conditions EQUATIONS OF PARABOLIC TYPE WITH ONE SPACE VARIABLE Constant Coefficient Equations Heat Equation with Axial or Central Symmetry and Related Equations Equations Containing Power Functions and Arbitrary Parameters Equations Containing Exponential Functions and Arbitrary Parameters Equations Containing Hyperbolic Functions and Arbitrary Parameters Equations Containing Logarithmic Functions and Arbitrary Parameters Equations Containing Trigonometric Functions and Arbitrary Parameters Equations Containing Arbitrary Functions Equations of Special Form PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Heat Equation Heat Equation with a Source Other Equations with Three Space Variables Equations with n Space Variables HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE Constant Coefficient Equations Wave Equation with Axial or Central Symmetry Equations Containing Power Functions and Arbitrary Parameters Equations Containing the First Time Derivative Equations Containing Arbitrary Functions HYPERBOLIC EQUATIONS WITH TWO SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equation with Two Space Variables HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Wave Equation Nonhomogeneous Wave Equation Telegraph Equation Other Equations with Three Space Variables Equations with n Space Variables ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Laplace Equation Poisson Equation Helmholtz Equation Other Equations Equations with n Space Variables HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS Third Order Partial Differential Equations Fourth Order One-Dimensional Nonstationary Equations Two-Dimensional Nonstationary Fourth Order Equations Fourth Order Stationary Equations Higher Order Linear Equations with Constant Coefficients Higher Order Linear Equations with Variable Coefficients SUPPLEMENT A: Special Functions and Their Properties SUPPLEMENT B: Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics REFERENCES INDEX

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain

Abstract: A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.

The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory

Abstract: We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, l ∈ , as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/l ∈ )3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-α equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-α model to the NSE by proving a convergence theorem, that as the length scale α 1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.

Stability of Travelling Waves

Abstract: An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of well-separated multi-bump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.

Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications

Abstract: An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

A New Test for Chaos

Abstract: We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. (This is an alternative to the usual approach of computing the largest Lyapunov exponent.) Our method is a 0-1 test for chaos (the output is a 0 signifying nonchaotic or a 1 signifying chaotic) and is independent of the dimension of the dynamical system. Moreover, the underlying equations need not be known. The test works equally well for continuous and discrete time. We give examples for an ordinary differential equation, a partial differential equation and for a map.

Non-convex potentials and microstructures in finite-strain plasticity

Abstract: A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.

Introduction to Symmetry Analysis

Abstract: Preface 1. Introduction to symmetry 2. Dimensional analysis 3. Systems of ODE's, first order PDE's, state-space analysis 4. Classical dynamics 5. Introduction to one-parameter Lie groups 6. First order ordinary differential equations 7. Differential functions and notation 8. Ordinary differential equations 9. Partial differential equations 10. Laminar boundary layers 11. Incompressible flow 12. Compressible flow 13. Similarity rules for turbulent shear flows 14. Lie-Backlund transformations 15. Invariance condition for integrals, variational symmetries 16. Backlund transformations and non-local groups Appendix 1. Review of calculus and the theory of contact Appendix 2. Invariance of the contact conditions under Lie point transformation groups Appendix 3. Infinite-order structure of Lie-Backlund transformations Appendix 4. Symmetry analysis software.

Convergence of Adaptive Finite Element Methods

Abstract: Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

A meshless, integration-free, and boundary-only RBF technique

Abstract: Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation techniques for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while non-singular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the only use of non-singular part of complete fundamental solution is also discussed.

A meshless, integration-free, and boundary-only RBF technique

Abstract: Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

Solution of the generalized Riemann problem for advection–reaction equations

Abstract: We present a method for solving the generalized Riemann problem for partial differential equations of the advection–reaction type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.

Local Discontinuous Galerkin Methods for the Stokes System

Abstract: In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For a class of shape regular meshes with hanging nodes we derive a priori estimates for the L2-norm of the errors in the velocities and the pressure. We show that optimal-order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k-1 for the pressure, for any $k\ge1$. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.

Weighted regularization of Maxwell equations in polyhedral domains

Abstract: We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient.

A Robust Finite Element Method for Darcy--Stokes Flow

Abstract: Finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed. The system is approximately a linear Stokes problem when the perturbation parameter is large, while it degenerates to a mixed formulation of Poisson's equation as the perturbation parameter tends to zero. It is established, basically by numerical experiments, that most of the proposed finite element methods for Stokes problem or the mixed Poisson's system are not well behaved uniformly in the perturbation parameter. This is used as the motivation for introducing a new "robust" finite element which exhibits this property.

A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative

Abstract: In this paper we prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in Hs for $s > \frac12$ for data small in L2 . To understand the strength of this result one should recall that for $s \frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schrodinger equation on the line is globally well-posed for large data in Hs for $s>\frac12$.

Evolution of Elastic Curves in $\Rn$: Existence and Computation

Abstract: We consider curves in ${\mathbb R}^n$ moving by the gradient flow for elastic energy, i.e., the L2 integral of curvature. Long-time existence is proved in the two cases when a multiple of length is added to the energy or the length is fixed as a constraint. Along these lines, a lower bound for the lifespan of solutions to the curve diffusion flow is observed. We derive algorithms for both the elastic flows and the curve diffusion equation. After a numerical test we compute several examples, including cases of curve diffusion in which a singularity develops.