Showing papers on "Partial differential equation published in 1983"

Shock Waves and Reaction-Diffusion Equations

Abstract: The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index For the second edition, typographical errors and other mistakes have been corrected and a new chapter on recent results has been added The new chapter contains discussion of the stability of travelling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for constructing travelling-wave solutions for systems of non-linear equations

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

Abstract: Integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels. Solutions are obtained for the screw dislocation and surface waves. Experimental observations and atomic lattice dynamics appear to support the theoretical results very nicely.

Viscosity solutions of Hamilton-Jacobi equations

Abstract: Publisher Summary This chapter examines viscosity solutions of Hamilton–Jacobi equations. The ability to formulate an existence and uniqueness result for generality requires the ability to discuss non differential solutions of the equation, and this has not been possible before. However, the existence assertions can be proved by expanding on the arguments in the introduction concerning the relationship of the vanishing viscosity method and the notion of viscosity solutions, so users can adapt known methods here. The uniqueness is then the main new point.

The Painlevé property for partial differential equations

Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems

Abstract: The purpose of this volume is to present the principles of the Augmented Lagrangian Method, together with numerous applications of this method to the numerical solution of boundary-value problems for partial differential equations or inequalities arising in Mathematical Physics, in the Mechanics of Continuous Media and in the Engineering Sciences.

Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

Abstract: The equations of motion of compressible viscous and heat-conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region. The global solution in time is proved to exist uniquely and approach the stationary state ast→∞, provided the prescribed initial data and the external force are sufficiently small.

A Taylor-Galerkin method for convective transport problems

Abstract: A method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second- and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank–Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov–Galerkin methods emerge and the new Taylor–Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.

The crack problem for a nonhomogeneous plane

Abstract: The plane elasticity problem for a nonhomogeneous medium containing a crack is considered. It is assumed that the Poisson's ratio of the medium is constant and the Young's modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy type kernel. Hence, its solution and the stresses around the crack tips have the conventional square root singularity. The solution is given for various loading conditions. The results show that the effect of the Poisson's ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible.

Solution of the Schrödinger equation by a spectral method II: Vibrational energy levels of triatomic molecules

Abstract: The spectral method utilizes numerical solutions to the time‐dependent Schrodinger equation to generate the energy eigenvalues and eigenfunctions of the time‐independent Schrodinger equation. Accurate time‐dependent wave functions ψ(r, t) are generated by the split operator FFT method, and the correlation function 〈ψ(r, 0) ‖ ψ(r, t)〉 is computed by numerical integration. Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ(r, t) with respect to time generate the eigenfunctions. Previous applications of the method were to two‐dimensional potentials. In this paper energy eigenvalues and wave functions obtained with the spectral method are presented for vibrational states of three‐dimensional Born–Oppenheimer potentials applicable to SO2, O3, and H2O. The energy eigenvalues are compared with results obtained with the variational method. It is concluded that the spectral method is an accurate tool for treating a variety of practical three‐dimensional potentials.

Partial differential equations of applied mathematics

Abstract: Preface. 1. Random Walks and Partial Differential Equations. 2. First Order Partial Differential Equations. 3. Classification of Equations and Characteristics. 4. Initial and Boundary Value Problems in Bounded Regions. 5. Integral Transforms. 6. Integral Relations. 7. Green's Functions. 8. variational and Other Methods. 9. Regular Perturbation Methods. 10. Asymptotic Methods. 11. Finite Difference Methods. 12. Finite Element Methods in Two Dimensions. Bibliography. Index.

Systems of Quasilinear Equations and Their Applications to Gas Dynamics

Abstract: This book is essentially a new edition, revised and augmented by results of the last decade, of the work of the same title published in 1968 by 'Nauka'. It is devoted to mathematical questions of gas dynamics. Topics covered include Foundations of the Theory of Systems of Quasilinear Equations of Hyperbolic Type in Two Independent Variables; Classical and Generalized Solutions of One-Dimensional Gas Dynamics; Difference Methods for Solving the Equations of Gas Dynamics; and Generalized Solutions of Systems of Quasilinear Equations of Hyperbolic Type.

The painleve property for partial differential equations. ii. backlund transformation, lax pairs, and the schwarzian derivative

Abstract: In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.

Quantum linear problem for the sine-Gordon equation and higher representations

Abstract: A quantum linear problem is constructed which permits the investigation of the sine-Gordon equation within the framework of the inverse scattering method in an arbitrary representation of algebra . The corresponding R-matrix is found, satisfying the Yang-Baxter equation (the condition for the factorization of the multiparticle matrices of the scattering of particles on a straight line).

Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations.

Abstract: : Recent work by the authors and others has demonstrated the connections between the dynamic programming approach for two-person, zero-sum differential games and the new notion of viscosity solutions of Hamilton-Jacobi PDE, (Partial Differential Equations). The basic idea is that the dynamic programming optimality conditions imply that the values of a two-person, zero-sum differential game are viscosity solutions of appropriate PDE. This paper proves the above, when the values of the differential games are defined following Elliott-Kalton. This results in a great simplification in the statements and proofs, as the definitions are explicit and do not entail any kind of approximations. Moreover, as an application of the above results, the paper contains a representation formula for the solution of a fully nonlinear first-order PDE. This is then used to prove results about the level sets of solutions of Hamilton-Jacobi equations with homogeneous Hamiltonians. These results are also related to the theory of Huygen's principle and geometric optics.

Gyrokinetic approach in particle simulation

Abstract: A new scheme for particle simulation based on the gyrophase‐averaged Vlasov equation has been developed. It is suitable for studying linear and nonlinear low‐frequency microinstabilities and the associated anomalous transport in magnetically confined plasmas. The scheme retains the gyroradius effects but not the gyromotion; it is, therefore, far more efficient than conventional ones. Furthermore, the reduced Vlasov equation is also amenable to analytical studies.

Convergence of the Viscosity Method for Isentropic Gas Dynamics

Abstract: A convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.

On The Dirichletproblem for Quasilinear Equations

Abstract: (1983). On The Dirichletproblem for Quasilinear Equations. Communications in Partial Differential Equations: Vol. 8, No. 7, pp. 773-817.

The nonlinear development of Görtler vortices in growing boundary layers

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

The Compensated Compactness Method Applied to Systems of Conservation Laws

Abstract: One of the main difficulties in solving nonlinear partial differential equations lies in the following fact: after introducing a suitable sequence of approximations one needs enough a priori estimates to ensure the convergence of a subsequence to a solution; this argument is based on compactness results and in a nonlinear case one needs more estimates than in the linear case where weak continuity results can be used.

Numerical methods for semiconductor device simulation

Abstract: This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices. These methods have been encoded into our device simulation package which has successfully simulated complex devices in two and three space dimensions. We focus our discussion on nonlinear operator iteration, discretization and scaling procedures, and the efficient solution of the resulting nonlinear and linear algebraic equations. Our companion paper [13] discusses physical aspects of the model equations and presents results from several actual device simulations.

Superconductivity of networks. A percolation approach to the effects of disorder

Abstract: Solutions of the linearized Landau-Ginzburg equations on networks of thin wires are studied. We derive linear-difference equations for the value of the order parameter at the junctions of the net with the use of the explicit form of the solutions on the wires. The technique is shown to be applicable to the diffusion equation, to harmonic lattice vibrations, and to the Schroedinger equation and results in equations similar to tight-binding equations. The equations are solved and the upper critical field is determined for some simple finite nets, for the infinite square net, and for the triangular Sierpinski gasket. Dead-end side branches are shown to lead to a mass renormalization. On the square net the equations map on the Azbel-Hofstadter-Aubry model. When the coherence length is small, vortex cores can be accommodated in the holes of the net and there is no upper critical field. The equations on the Sierpinski gasket are solved by an iterative decimation process. The process determines a new length scale proportional to a power of the bare coherence length. The upper critical field is studied for a finite gasket and for a lattice of gaskets. With the use of scaling arguments the results are applied tomore » percolation clusters. Far from the percolation threshold the results are described by a renormalized correlation length of standard form. When this length becomes shorter than the correlation length for the percolation problem the critical field is shown to be constant or decreasing as the threshold is approached. Existing experiments are discussed and the importance of high-field-susceptibility measurements is emphasized.« less

Review of elastic and electromagnetic wave propagation in horizontally layered media

Abstract: The objective of this paper is to provide a unified treatment of elastic and electromagnetic (EM) wave propagation in horizontally layered media for which the parameters in the partial differential equations are piece‐wise continuous functions of only one spatial variable. By applying a combination of Fourier, Laplace, and Bessel transforms to the partial differential equations describing the elastic or EM wave propagation I obtain a system of 2n linear ordinary differential equations. The 2n×2n coefficient matrix is partitioned into 4n×n submatrices. By a proper choice of variables, the diagonal submatrices are zero and the off‐diagonal submatrices are symmetric. All the results in the paper are derived from the symmetry properties of this general equation. In the appendices it is shown that three‐dimensional elastic waves, cylindrical P‐SV waves, acoustic waves, and electromagnetic waves in isotropic layered media can all be represented by an equation with the same properties. The symmetry properties of...

Elementary Applied Partial Differential Equations With Fourier Series and Boundary Value Problems

Abstract: 1. Heat Equation. 2. Method of Separation of Variables. 3. Fourier Series. 4. Vibrating Strings and Membranes. 5. Sturm-Liouville Eigenvalue Problems. 6. An Elementary Discussion of Finite Difference Numerical Methods for Partial Differential Equations. 7. Partial Differential Equations with at Least Three Independent Variables. 8. Nonhomogeneous Problems. 9. Green's Functions for Time-Independent Problems. 10. Infinite Domain Problems--Fourier Transform Solutions of Partial Differential Equations. 11. Green's Functions for Time-Dependent Problems. 12. The Method of Characteristics for Linear and Quasi-Linear Wave Equations. 13. A Brief Introduction to Laplace Transform Solution of Partial Differential Equations. 14. Topics: Dispersive Waves, Stability, Nonlinearity, and Perturbation Methods. Bibliography. Selected Answers to Starred Exercises. Index.

Regularity of the derivatives of solutions to certain degenerate elliptic equations

Abstract: On montre que pour p fixe, 1

Position-dependent effective masses in semiconductor theory. II.

Abstract: A compound semiconductor possessing a slowly varying position-dependent chemical composition is considered. An effective-mass equation governing the dynamics of electron (or hole) motion using the Kohn-Luttinger representation and canonical transformations is derived. It is shown that, as long as the variation in chemical composition may be treated as a perturbation, the effective masses become constant, position-independent quantities. The effective-mass equation derived here is identical to the effective-mass equation derived previously by von Roos (1983), using a Wannier representation.

The small dispersion limit of the Korteweg‐de Vries equation. III

Abstract: In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of uxxx tends to zero. This formula contains n parameters β1, …, βn whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t

Boundary-value problems for partial differential equations in non-smooth domains

Abstract: CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References

On the inhomogeneous two‐plasmon instability

Abstract: The two‐plasmon instability in warm inhomogeneous plasma for a normally incident pump is considered. The complex eigenfrequencies of the absolute instability are obtained by reducing the linearized fluid equations to a Schrodinger equation in wavenumber space. These eigenvalues are obtained in several ways. One is by combining a perturbation expansion in powers of the reciprocal scale length with WKB theory. The resulting algebraic equations are solved by three analytical approximations and by direct numerical solution. A second way is by analysis of the Schrodinger equation using an interactive WKB computer code. A third way is by the use of a shooting code. These methods are all used and compared for threshold curves and growth rates above threshold. Some eigenfunction forms are also obtained. The threshold is near (v0/ve)2k0 L =3, and varies weakly with β≂v4e/v20c2, rising from near 2 to about 4 over six decades of variation of β. The corresponding critical value of (ky/k0)2 is near 0.2/β over this ran...