Showing papers on "Partial differential equation published in 1988"
Book•
29 Jan 1988
TL;DR: In this article, the authors present an easily accessible introduction to one of the most important methods used to solve partial differential equations, which they call finite element methods for integral equations (FEME).
Abstract: Professor Johnson presents an easily accessible introduction to one of the most important methods used to solve partial differential equations. The bulk of the text focuses on linear problems, however a chapter extending the development of non-linear problems is also included, as is one on finite element methods for integral equations. Throughout the text the author has included applications to important problems in mathematics and physics, and has endeavored to keep the mathematics as simple as possible while still presenting significant results.
1,956 citations
TL;DR: On etudie la regularite limite des solutions faibles de l'equation divA(x,u,Du)+B(x andu, Du) = 0 avec des conditions aux limites conormales et de Dirichlet.
Abstract: On etudie la regularite limite des solutions faibles de l'equation divA(x,u,Du)+B(x,u,Du)=0 avec des conditions aux limites conormales et de Dirichlet
1,278 citations
Book•
01 Jan 1988
TL;DR: Navier-Stokes Equations as mentioned in this paper provide a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.
Abstract: Both an original contribution and a lucid introduction to mathematical aspects of fluid mechanics, Navier-Stokes Equations provides a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.
1,189 citations
TL;DR: Two stochastic processes that model the major modes of dispersal that are observed in nature are introduced, and explicit expressions for the mean squared displacement and other experimentally observable quantities are derived.
Abstract: In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of "runs" separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.
905 citations
TL;DR: In this article, an upwind differencing scheme of Roe-type for the MHD Riemann problem is constructed, in which the problem is first linearized around some averaged state which preserves the flux differences.
831 citations
TL;DR: In this article, the concept of an inertial manifold for nonlinear evolutionary equations, in particular for ordinary and partial differential equations, was introduced, which is an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations.
712 citations
Book•
25 Oct 1988
TL;DR: In this paper, the authors present an approach to the transport of finite-dimensional contact elements and the effect of the dimension of the Global Attractor on the acceleration of the contact elements.
Abstract: Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone Invariance Properties.- Consequences Regarding the Global Attractor.- Local Exponential Decay Toward Blocked Integral Surfaces.- Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- Choice of the Initial Manifold.- Construction of the Inertial Mainfold.- Lower Bound for the Exponential Rate of Convergence to the Attractor.- Asymptotic Completeness: Preparation.- Asymptotic Completeness: Proof of Theorem 12.1.- Stability with Respect to Perturbations.- Application: The Kuramoto-Sivashinsky Equation.- Application: A Nonlocal Burgers Equation.- Application: The Cahn-Hilliard Equation.- Application: A parabolic Equation in Two Space Variables.- Application: The Chaffee-Infante Reaction Diffusion Equation.- References.- Index.
523 citations
TL;DR: On etudie l'ensemble nul d'une solution u(t,x) de l'equation u t =a(x,t)u xx +b(x and t)u x +C(x,t) u t + c(x)t, t) as mentioned in this paper, sous des hypotheses tres generales sur les coefficients a, b, et c
Abstract: On etudie l'ensemble nul d'une solution u(t,x) de l'equation u t =a(x,t)u xx +b(x,t)u x +C(x,t)u, sous des hypotheses tres generales sur les coefficients a, b, et c
519 citations
TL;DR: A nonlinear electrostatic gyrokinetic Vlasov equation as well as a Poisson equation have been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport as mentioned in this paper.
Abstract: A nonlinear electrostatic gyrokinetic Vlasov equation as well as a Poisson equation have been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics In the derivation, the action‐variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov–Poisson system Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits
470 citations
TL;DR: In this article, the authors describe a general local smoothing effect for dispersive equations and systems, including the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations.
Abstract: Is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data? For the linear wave equation, for instance, the answer is no. However, in [10] T. Kato found a local smoothing property of the Korteweg-de Vries equation: the solution of the initial value problem is, locally, one derivative smoother than the initial datum. Kato's proof uses, in a curcial way, the algebraic properties of the symbol for the Korteweg-de Vries equation and the fact that the underlying spatial dimension is one. Actually, judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect, one would be inclined to believe this was a special property of the K-dV equation. This is not, however, the case. In this paper, we attempt to describe a general local smoothing effect for dispersive equations and systems. The smoothing effect is due to the dispersive nature of the linear part of the equation. All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations are included. We study, thus, equations and systems of the form
444 citations
TL;DR: This paper provides a preconditioned iterative technique for the solution of saddle point problems by reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration.
Abstract: This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration. Applications to the equations of elasticity and Stokes are discussed and the results of numerical experiments are given.
TL;DR: This paper provides a detailed exposition of model construction, structural stability of constructed models, stability of the scheme, etc, and considers the relationship between the CDS modeling and the conventional description in terms of partial differential equations, which leads to a new discretization scheme for semilinear parabolic equations.
Abstract: We present a computationally efficient scheme of modeling the phase-ordering dynamics of thermodynamically unstable phases. The scheme utilizes space-time discrete dynamical systems, viz., cell dynamical systems (CDS). Our proposal is tantamount to proposing new Ansa$iuml---tze for the kinetic-level description of the dynamics. Our present exposition consists of two parts: part I (this paper) deals mainly with methodology and part II [S. Puri and Y. Oono, Phys. Rev. A (to be published)] gives detailed demonstrations. In this paper we provide a detailed exposition of model construction, structural stability of constructed models (i.e., insensitivity to details), stability of the scheme, etc. We also consider the relationship between the CDS modeling and the conventional description in terms of partial differential equations. This leads to a new discretization scheme for semilinear parabolic equations and suggests the necessity of a branch of applied mathematics which could be called ``qualitative numerical analysis.''
TL;DR: Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.
Abstract: A procedure is given for constructing basis sets for the radial Dirac equation from B splines. The resulting basis sets, which include negative-energy states in a natural way, permit the accurate evaluation of the multiple sums over intermediate states occurring in relativistic many-body calculations. Illustrations are given for the Coulomb-field Dirac equation and tests of the resulting basis sets are described. As an application, relativistic corrections to the second-order correlation energy in helium are calculated. Another application is given to determine the spectrum of thallium (where finite--nuclear-size effects are important) in a model potential. Construction of B-spline basis sets for the Dirac-Hartree-Fock equations is described and the resulting basis sets are applied to study the cesium spectrum.
TL;DR: In this article, the existence of Semiclassical Bound States of Nonlinear Schrodinger Equations with Potentials of the Class (V)a is investigated and discussed.
Abstract: (1988). Existence of Semiclassical Bound States of Nonlinear Schrodinger Equations with Potentials of the Class (V)a. Communications in Partial Differential Equations: Vol. 13, No. 12, pp. 1499-1519.
Book•
31 Dec 1988
TL;DR: In this paper, the authors present a decomposition method for differential and partial differential Equations of the An Polynomials for composite nonlinearity and linearization problems.
Abstract: I: A Summary of the Decomposition Method.- 1: The Decomposition Method.- 1.1 Introduction.- 1.2 Summary of the Decomposition Method.- 1.3 Generation of the An Polynomials.- 1.4 The An for Differential Nonlinear Operators.- 1.5 Convenient Computational Forms for the An Polynomials.- 1.6 Calculation of the An Polynomials for Composite Nonlinearities.- 1.7 New Generating Schemes - the Accelerated Polynomials.- 1.8 Convergence of the An Polynomials.- 1.9 Euler's Transformation.- 1.9.1 Solution of a Differential Equation by Decomposition.- 1.9.2 Application of Euler Transform to Decomposition Solution.- 1.9.3 Numerical Comparison.- 1.9.4 Solution of Linearized Equation.- 1.10 On the Validity of the Decomposition Solution.- 2: Effects of Nonlinearity and Linearization.- 2.1 Introduction.- 2.2 Effects on Simple Systems.- 2.3 Effects on SOlution for the General Case.- 3: Research on Initial and Boundary Conditions for Differential and Partial Differential Equations.- II: Applications to the Equations of Physics.- 4: The Burger's Equation.- 5: Heat Flow and Diffusion.- 5.1 One-Dimensional Case.- 5.2 Two-Dimensional Case.- 5.3 Three-Dimensional Case.- 5.4 Some Examples.- 5.5 Heat Conduction in an Inhomogeneous Rod.- 5.6 Nonlinear Heat Conduction.- 5.7 Heat Conduction Equation with Discontinuous Coefficients.- 5.8 Nonlinear Boundary Conditions.- 5.9 Comparisons.- 5.10 Uncoupled Equations with Coupled Conditions.- 6: Nonlinear Oscillations in Physical Systems.- 6.1 Oscillatory Motion.- 6.2 Pendulum Problem.- 6.3 The Duffing and Van der Pol Oscillators.- 7: The KdV Equation.- 8: The Benjamin-Ono Equation.- 9: The Sine-Gordon Equation.- 10: The Nonlinear Schrodinger Equation and the Generalized Schrodinger Equation.- 10.1 Nonlinear Schrodinger Equation.- 10.2 Generalized Schrodinger Equation.- 10.3 Schrodinger's Equation with a Quartic Potential.- 11: Nonlinear Plasmas.- 12: The Tricomi Problem.- 13: The Initial-Value Problem for the Wave Equation.- ChaDter 14: Nonlinear Dispersive or Dissipative Waves.- 14.1 Wave Propagation in Nonlinear Media.- 14.2 Dissipative Wave Equations.- 15: The Nonlinear Klein-Gordon Equation.- 16: Analysis of Model Equations of Gas Dynamics.- 17: A New Approach to the Efinger Model for a Nonlinear Quantum Theory for Gravitating Particles.- 18: The Navier-Stokes Equations.- Epilogue.
TL;DR: In this paper, it was shown that viscosity solutions in W 1,∞ of the second order, fully nonlinear, equation F(D2u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) f is uniformly elliptic, and nonincreasing in u. This method was completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that was used by Crandall & Lions [3] and Cr
Abstract: We prove that viscosity solutions in W1,∞ of the second order, fully nonlinear, equation F(D2u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.
TL;DR: In this paper, it was shown that among a family of supersymmetric extensions of the Kortewegde Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation.
Abstract: It is shown that among a one‐parameter family of supersymmetric extensions of the Korteweg–de Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation. Its modified version is also discussed.
Book•
01 Jan 1988
TL;DR: In this paper, the procedures to perform nonlinear soil-structure-interaction analysis in the time domain are summarized, where the nonlinearity is restricted to the structure and possibly an adjacent irregular soil region.
Abstract: The procedures to perform nonlinear soil-structure-interaction analysis in the time domain are summarized. The nonlinearity is restricted to the structure and possibly an adjacent irregular soil region. The unbounded soil (far field) must remain linear in this formulation. Besides the direct method where local frequency-independent boundary conditions are enforced on the artificial boundary, various formulations based on the substructure method are addressed, ranging from a discrete model with springs, dashpots and masses to boundary-element methods with convolution integrals involving either the dynamic-stiffness coefficients or the Green's functions in the time domain via the iterative hybrid-frequency-time-domain analysis procedure with the nonlinearities affecting only the right-hand side of the equations of motion.
TL;DR: In this paper, the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear Stochastic Differential Equations of the generalized Ito type (containing both jump and diffusion components) is considered.
Abstract: This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Ito type (containing both jump and diffusion components). We derive a particular functional partial differential equation which characterizes the exact likelihood function of a discretely sampled Ito process. In addition, we show by a simple counterexample that the common approach of estimating parameters of an Ito process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consis
TL;DR: In this article, the authors introduced new classes of symmetries for partial differential equations, which are neither point nor Lie-Backlund symmetric, and they are determined by a completely algorithmic procedure.
Abstract: New classes of symmetries for partial differential equations are introduced. By writing a given partial differential equation S in a conserved form, a related system T with potentials as additional dependent variables is obtained. The Lie group of point transformations admitted by T induces a symmetry group of S. New symmetries may be obtained for S that are neither point nor Lie–Backlund symmetries. They are determined by a completely algorithmic procedure. Significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation.
TL;DR: In this paper, the authors consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains and demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids.
Abstract: We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.
TL;DR: In this paper, the equivalence between semi-explicit and implicit differential algebraic equations was shown and it was shown that the theory for the former applies to the latter of one lower index.
Abstract: In this paper we show an equivalence between semi-explicit and implicit differential-algebraic equations so that the theory for the former applies to the latter of one lower index. We also discuss ...
TL;DR: In this article, it was shown that the initial value problem for Boussinesq-type equations is always locally well posed and that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds.
Abstract: Certain generalizations of one of the classical Boussinesq-type equations,
$$u_{tt} = u_{xx} - (u^2 + u_{xx} )_{xx} $$
are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.
TL;DR: In this article, a new mixed finite element formulation for the equations of linear elasticity is considered, where the variables approximated are the displacement, the unsymmetric stress tensor and the rotation.
Abstract: A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.
TL;DR: On demontre un theoreme d'existence general pour des solutions distribution positives, a comportement singulier prescript, de l'equation scalaire semilineaire provenant de la deformation conforme des metriques de Riemann as discussed by the authors.
Abstract: On demontre un theoreme d'existence general pour des solutions distribution positives, a comportement singulier prescript, de l'equation scalaire semilineaire provenant de la deformation conforme des metriques de Riemann
TL;DR: In this paper, a selfconsistent and energy-conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell's equations for finite-beta plasmas, is derived.
Abstract: A self‐consistent and energy‐conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite‐beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.
TL;DR: In this paper, the boundary conditions for nonlinear hyperbolic systems of conservation laws were formulated based on the vanishing viscosity method and the Riemann problem, and the equivalence between these two conditions was studied.
TL;DR: The solution of the Stokes problem is approximated by three stabilized mixed methods, one due to Hughes, Balestra, and Franca and the other two being variants of this procedure.
Abstract: The solution of the Stokes problem is approximated by three stabilized mixed methods, one due to Hughes, Balestra, and Franca and the other two being variants of this procedure. In each case the bilinear form associated with the saddle-point problem of the standard mixed formulation is modified to become coercive over the finite element space. Error estimates are derived for each procedure.
TL;DR: In this paper, the Sato equation is introduced, and it is shown that the generalized Lax equation, the Zakharov-Shabat equation and the IST scheme can be derived from the sato equation.
Abstract: An .elementary introduction to Sato theory is given. Starting with an ordinary differential equation, introducing an infinite number of time variables, and imposing a certain time dependence on the solutions, we obtain the Sato equation which governs the time development of the variable coefficients. It is shown that the generalized Lax equation, the Zakharov-Shabat equation and the IST scheme are generated from the Sato equation. It is revealed that the r-function becomes the key function to express the solutions of the Sato equation. By using the results of the representation theory of groups, it is shown that the r-function is governed by the partial differential equations in the bilinear forms which are closely related to the PlUcker relations.