Showing papers on "Hyperbolic partial differential equation published in 1984"
TL;DR: On considere le probleme de Dirichlet as discussed by the authors for des equations elliptiques non lineaires for a fonction reelle u definie dans la fermeture d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞
Abstract: On considere le probleme de Dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture Ω d'un domaine borne Ω dans R n avec une frontiere ∂Ω C ∞
936 citations
TL;DR: A survey of finite element methods for convectiondiffusion problems and first-order linear hyperbolic problems can be found in this article, where the authors give a survey of some recent work by the authors.
706 citations
Book•
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TL;DR: The abstract cauchy problem for time-dependent equations was introduced in this paper and applied to second-order parabolic equations in functional analysis, where the abstract problem can be expressed as a vector-valued distribution.
Abstract: Editor's statement Foreword Preface 1. Elements of functional analysis 2. The caucy problem for some equations of mathematical physics: the abstract cauchy problem 3. Properly posed cauchy problems: general theory 4. Dissipative operators and applications 5. Abstract parabolic equations: applications to second order parabolic equations 6. Perturbation and approximation of abstract differential equations 7. Some improperly posed cauchy problems 8. The abstract cauchy problem for time-dependent equations 9. The cauchy problem in the sense of vector-valued distributions References Index.
461 citations
Book•
11 Aug 1984
TL;DR: Inverse Methods for Reflector Imaging as discussed by the authors, the Dirac Delta Function, Fourier Transforms, and Asymptotics are used for direct scattering problems, and the Wave Equation in Two and Three dimensions.
Abstract: First-Order Partial Differential Equations. The Dirac Delta Function, Fourier Transforms, and Asymptotics. Second-Order Partial Differential Equations. The Wave Equation in One Space Dimension. The Wave Equation in Two and Three Dimensions. The Helmholtz Equation and Other Elliptic Equations. More on Asymptotic Techniques for Direct Scattering Problems. Inverse Methods for Reflector Imaging. Each chapter includes references. Index.
439 citations
TL;DR: On considere le probleme de Cauchy pour l'equation d'onde semi lineaire (ο 2 /∂t 2 -Δ) as discussed by the authors,
294 citations
TL;DR: In this article, a finite element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated.
Abstract: A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.
259 citations
TL;DR: In this paper, it is shown that the finite difference technique can be used to transform some important linear distributed processes described by partial differential equations into so-called Roesser discrete state-space models.
192 citations
TL;DR: In this article, the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures are explained and the cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of PDE.
Abstract: Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations Training examples are also given
185 citations
TL;DR: The Taylor-Galerkin method is employed to derive accurate and efficient numerical schemes for the solution of time-dependent advection-diffusion problems and is successively extended to deal with nonlinear and multi-dimensional problems.
157 citations
TL;DR: In this article, a symmetric version of the regularized long-wave equation is shown to describe weakly nonlinear ion acoustic and space-charge waves, which possesses hyperbolic secant squared solitary waves and has four known invariants.
Abstract: A symmetric version of the regularized‐long‐wave equation is shown to describe weakly nonlinear ion acoustic and space‐charge waves. The equation possesses hyperbolic secant squared solitary waves and has four known invariants. Numerical solutions are compared with previous results on the regularized‐long‐wave equation.
114 citations
TL;DR: In this article, the total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to a system of linear partial differential equations with constant coefficients, and complete integrability of the KP equation is proved by using this linear system.
TL;DR: On etudie des systemes de lois de conservation hyperboliques et d'equations d'ondes completement non lineaires as mentioned in this paper, et al.
Abstract: On etudie des systemes de lois de conservation hyperboliques et d'equations d'ondes completement non lineaires
TL;DR: In this paper, a theory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary, where instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0.
Abstract: A th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundstrom. In particular we investigate l2-instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.
TL;DR: In this paper, the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)−dimensional Minkowski space M(n,1).
Abstract: Symmetry reduction is studied for the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)‐dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ1, ... ,ξk as invariants of a subgroup G of the Poincare group P(n,1), having generic orbits of codimension k≤n in M(n,1), reduces the equation to a PDE in k variables. All codimension‐1 symmetry variables in M(n,1) (n arbitrary), reducing the equation studied to an ODE are found, as well as all codimension‐2 and ‐3 variables for the low‐dimensional cases n=2,3. The type of equation studied includes many cases of physical interest, in particular nonlinear Klein–Gordon equations (such as the sine–Gordon equation) and Hamilton–Jacobi equations.
TL;DR: In this article, new oscillation criteria are established for the first order functional differential equation y'(t)+p(t)y(g(t))=0 and its nonlinear analogue.
Abstract: New oscillation criteria are established for the first order functional differential equation (*) y'(t)+p(t)y(g(t))=0and its nonlinear analogue. The results are presented so that a remarkable duality existing between the case where (*) is retarded (g(t) t) is apparent. Possible extension of the results for (*) to equations with several deviating arguments is attempted. Finally, it is shown that there exists a class of autonomous equations for which the oscillation situation can be completely characterized.
TL;DR: The partial differential equation treated here is the formal limit of the p-harmonic equation in R2 for p→∞ as mentioned in this paper, and the singular solutions constructed here bring new light on these questions.
Abstract: The partial differential equation treated here is the formal limit of the p-harmonic equation in R2, for p→∞. Questions related to the smoothness of solutions and the possible existence of stationary points are central for the theory. The “singular” solutions constructed here bring new light on these questions.
TL;DR: In this paper, the first order stochastic partial differential equations of the parabolic type were studied and the Cauchy problem of the first-order SDPs was studied.
Abstract: Publisher Summary This chapter discusses the first order stochastic partial differential equations. The chapter studies the Cauchy problem of the first order stochastic partial differential equations of the parabolic type. The method of stochastic characteristic curve has been proposed to construct a solution of a suitable linear stochastic partial differential equation of first order. The chapter defines the first order stochastic partial differential equation rigorously and then introduces the associated stochastic characteristic equation. The proof of the existence and uniqueness of local solutions associated with a given initial condition is given. The chapter discusses quasi-linear, semi-linear and linear equation as special cases. The existence of the maximal or global solution is shown. The chapter also focuses on the regularity of Ito integral and the Stratonovich integral.
TL;DR: In this article, a Harnack inequality for degenerate parabolic equations is proposed for partial differential equations, which is a generalization of the one we consider in this paper. pp. 719-749.
Abstract: (1984). A Harnack inequality for degenerate parabolic equations. Communications in Partial Differential Equations: Vol. 9, No. 8, pp. 719-749.
Book•
01 Jan 1984
TL;DR: In this article, it was pointed out that the nonlinear wave equation can be solved by quadratures, where a and c are constants, A(y) and B (y) are arbitrary functions; a t-dependence of all these quantities can also be accommodated.
Abstract: It is pointed out that the nonlinear wave equation
can be solved by quadratures. Here a and c are constants, A(y) and B(y) (arbitrary) functions; a t-dependence of all these quantities can also be accommodated. This wave equation can also be rewritten in the (purely differential) form
via the substitutions .
01 Jan 1984
TL;DR: An implicit, finite-difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional, curvilinear coordinate system based on the pseudo compressibility approach, which employs an implicit, approximate factorization scheme.
Abstract: An implicit, finite-difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional, curvilinear coordinate system. The pressure-field solution is based on the pseudo compressibility approach in which the time derivative pressure term is introduced into the mass conservation equation to form a set of hyperbolic equations. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds stresses, that are uncoupled from the implicit scheme, are lagged by one time-step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computed, and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.
01 Mar 1984
TL;DR: The main thrust of this AFOSR supported research has been to extend the hyperbolic partial differential equation procedure to three dimensional applications and to study ways of applying the procedure to body shapes that have discontinuous derivatives.
Abstract: An efficient numerical mesh generation scheme capable of creating orthogonal or nearly orthogonal grids about moderately complex three dimensional configurations is described. The mesh is obtained by marching outward from a user specified grid on the body surface. Using spherical grid topology, grids have been generated about full span rectangular wings and a simplified space shuttle orbiter.
TL;DR: In this paper, a totally one-sided first-order and second-order scheme is presented employing a numerically calculated characteristic speed direction and are combined into a transportive, monotonicity-preserving hybrid scheme using the method of flux correction.
TL;DR: In this article, the authors demonstrate that a time-dependent Kadomtsev-Petviashvili equation has the Painleve property for partial differential equations, and show that removing the painleve expansion yields an auto Backlund transformation and a representation in Zakharov-Shabat form.
TL;DR: The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered in this paper, where the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian is addressed.
01 Jan 1984
TL;DR: In this paper, a fonction a valeurs reelles C 1 (0,∞) satisfaisant a(s)≥0 pour s∈(0, ∞).
Abstract: Soit a(•) une fonction a valeurs reelles C 1 (0,∞) satisfaisant a(s)≥0 pour s∈(0,∞). On considere le probleme aux valeurs limites et initiales pour des equations hyperboliques quasi lineaires degenerees. On montre l'existence, l'unicite, la regularite et le comportement asymptotique des solutions
TL;DR: In this paper, it is shown that by taking into account all integrability conditions for a given system of partial differential equations, a more systematic treatment of several problems in the theory of nonlinear evolution equations may be performed.
01 Jul 1984
TL;DR: In this article, the authors describe the procedures used to create 0 and C-type solution grids around arbitrary two-dimensional bodies, which are a modification of the 0-type hyperbolic method of Steger and Chaussee.
Abstract: : This report describes the procedures used to create 0 and C-type solution grids around arbitrary two-dimensional bodies. The grid generation procedure is a modification of the 0-type hyperbolic method of Steger and Chaussee. The procedure described in Reference (1) was modified by including additional dissipation terms and by changing the form of the dissipation described. The modifications necessary to produce a C-type grid are discussed and examples are provided. A brief description of the theory and development of the governing hyperbolic equations is provided. Example results for both an airfoil section and a complex body (X-24C) are shown. A discussion of the user definable input variables and their effects on the resulting grid is included. For many solution procedures it is desirable that the distribution of points around the airfoil be 'second order smooth.' Since most airfoils are defined with a somewhat random distribution of points, a routine was used to redefine the coordinates in a smooth distribution. The routine also allows selective clustering of grid points in regions of interest. A brief description of this routine and the effects of the different input parameters are included as Appendix A.