Showing papers on "Partial differential equation published in 1989"

Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Abstract: On etudie les solutions regulieres non negatives de l'equation conformement invariante −Δu=u (n+2)/(n−2) , u>0 dans une boule perforee, B 1 (0)\{0}⊂R n , n≥3, avec une singularite isolee a l'origine

Solitons: An introduction

Abstract: This textbook is an introduction to the theory of solitons and its diverse applications to nonlinear systems that arise in the physical sciences. The authors explain the generation and properties of solitons, introducing the mathematical technique known as the Inverse Scattering Transform. Their aim is to present the essence of inverse scattering clearly, rather than rigorously or completely. Thus, the prerequisites (i.e., partial differential equations, calculus of variations, Fourier integrals, linear waves and Sturm–Liouville theory), and more advanced material is explained in the text with useful references to further reading given at the end of each chapter. Worked examples are frequently used to help the reader follow the various ideas, and the exercises at the end of each chapter not only contain applications but also test understanding. Answers, or hints to the solution, are given at the end of the book. Sections and exercises that contain more difficult material are indicated by asterisks.

New similarity reductions of the Boussinesq equation

Abstract: Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

Interior a priori estimates for solutions of fully non-linear equations

Abstract: On etend une theorie de perturbations aux solutions d'equations uniformement elliptiques d'ordre 2 totalement non lineaires

Convergence of Adomian's Method

Abstract: Adomian has developed a numerical technique using special kinds of polynomials for solving non‐linear functional equations. General conditions and a new formulation are proposed for proving the convergence of Adomian's method for the numerical resolution of non‐linear functional equations depending on one or several variables. The methods proposed are applicable to a very wide class of problems.

Quasilinear elliptic equations involving critical Sobolev exponents

Abstract: Soit G un sous ensemble ouvert borne de R N et 1 0 dans G, u=0 sur ∂G ou a∈L ∞ (G) et p*=Np/(N−p)

Spectral element methods for the incompressible Navier-Stokes equations

Abstract: Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.

On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's

Abstract: We prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second-order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.-L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions.

A posteriori error estimators for the Stokes equations

Abstract: We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.

The perturbed test function method for viscosity solutions of nonlinear PDE

Abstract: The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

Nonlinear Galerkin methods

Abstract: This article presents a new method of integrating evolution differential equations—the non-linear Galerkin method—that is well adapted to the long-term integration of such equations.While the usual Galerkin method can be interpreted as a projection of the considered equation on a linear space, the methods considered here are related to the projection of the equation on a nonlinear manifold. From the practical point of view some terms have been identified as small, and sometimes.(but not always) disregarded.

An absolutely stabilized finite element method for the stokes problem

Abstract: On presente une formulation a elements finis absolument stabilisee pour le probleme de Stokes On etablit des estimations d'erreurs optimales dans la norme L 2 pour l'approximation des champs de vitesses et de pressions

Asymptotic methods in equations of mathematical physics

Abstract: Part 1 The stationary phase method: on asymptotic expansions the stationary phase method the stationary phase method, the multidimensional case the problem of waves on the surface of a liquid the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation the scattering problem the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order general partial differential equations of the first order the Hamilton-Jacobi equation example propagation of light waves in an inhomogenous medium characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula problems with rapidly oscillating initial data discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium the Lagrange maniford the precanonical operator the canonical operator, construction of a formal asymptotic solution a field in an isotropic medium with parabolic wave front more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces elliptic problems elliptic problems with a parameter inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial equations and systems of the type of the Helmholtz equation radiation conditions the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients equations with variable coefficients and problems in the interior of a bounded domain the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t introduction short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering proof of theorems 1 and 2. Part contents...

Elements of Green's Functions and Propagation: Potentials, Diffusion, and Waves

Abstract: Part 1 Introduction: the dirac delta function ordinary differential equations partial differential equations - a preview. Part 2 Potentials: Poisson's equation - introduction - dirichlet problems - Neumann problems - some points of principle. Part 3 Diffusion: the diffusion equation - unbounded space - general theory. Part 4 Waves: the wave equation - general theory - unbounded space - examples. Part 5 The Helmholtz equation and diffraction.

Navier-stokes flow in r3 with measures as initial vorticity and morrey spaces

Abstract: (1989). Navier-stokes flow in r3 with measures as initial vorticity and morrey spaces. Communications in Partial Differential Equations: Vol. 14, No. 5, pp. 577-618.

Stationary stokes and Navier-Stokes systems on two-or three-dimensional domains with corners. Part I: linearized equations

Abstract: The $H^s $-regularity (s being real and nonnegative) of solutions of the Stokes system in domains with corners is studied. In particular, a $H^2 $-regularity result on a convex polyhedron that generalizes Kellogg and Osborn’s result on a convex polygon to three-dimensional domains is stated. Sharper regularity on a cube and on other domains with corners is attained. Conditions for the problem to be Fredholm are also given, and its singular functions along with those of the nonlinear problem are studied in the second part of this paper.

A second order splitting method for the Cahn-Hilliard equation

Abstract: A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.

Finite element-galerkin approximation of the eigenvalues and Eigenvectors of selfadjoint problems

Abstract: : This paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or, more generally, Galerkin methods. Suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let M(lambda) denote the space of eigenvectors corresponding to lambda. Keywords: Hilbert space.

Stress analysis by boundary element methods

Abstract: 1. Solution of Partial Differential Equations by the Boundary Integral Equation Method (BIEM). Ordinary differential equations. Partial differential equations. 2. Elastostatics. Governing equations and fundamental solutions. The Somigliana identity. Boundary integral equations (BIE). Integral representation of stresses. Boundary integro-differential equations (BIDE). Stresses on the boundary. BIE and BIDE for an anisotropic medium. Axisymmetric problems. Semi-infinite problems. Numerical implementation. Numerical examples. 3. Elastodynamics. Equations of motion. Fundamental solutions. Integral representations of displacements and stresses. Boundary integral and integro-differential equations. Numerical solution. Alternative formulation of boundary element method. Numerical examples. 4. Thermoelasticity. Governing equations. Fundamental solutions. Integral representations of displacements and temperature. Integral representations of temperature gradients and stresses. Boundary integral and integro-differential equations. Numerical implementation. Alternative BEM formulation. Application to fracture mechanics. Numerical examples. 5. Micropolar Thermoelasticity. Equations of motion. Fundamental solutions in three dimensions. Fundamental solutions for plane problems. Fundamental solutions for antiplane problems. Integral representations. BIE and BIDE. Applications to fracture mechanics. 6. Elastoplasticity. Governing equations. Boundary integral formulations. Elastoplastic stress-strain relations. Incremental computations for elastoplasticity. 7. Viscoelasticity. Rheological models and the correspondence principle. Boundary integral formulation. Schapery's inversion algorithm. Application to fracture mechanics. 8. Thin Elastic Plates in Bending. Governing equations in classical plate theory. Integral formulation. Numerical solution. Large deflections. Berger equation. Plates on elastic foundations. 9. Stress Analysis by Hybrid Methods. Computation of stresses at internal points from stresses on boundary. Combination of BIE with holographic interferometry measurements. Hybrid method in fracture mechanics. Appendices. References. Subject Index.

Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations

Abstract: The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Backlund transformations, prolongation structures, etc. A notion of a nonlocal cobweb is introduced which seems quite useful for dealing with nonlocal objects.