Showing papers on "Partial differential equation published in 1975"
TL;DR: In this article, conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time were investigated and sharp results were derived for certain classes of nonlinearities.
Abstract: A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in this paper. Extensions to parabolic and to abstract operator equations are also given.
700 citations
TL;DR: In this article, continuation and variational methods are developed to construct positive solutions for nonlinear elliptic eigenvalue problems, which contain models arising in chemical kinetics, nonlinear heat generation, and the gravitational equilibrium of polytropic stars.
Abstract: : Continuation and variational methods are developed to construct positive solutions for nonlinear elliptic eigenvalue problems. The class of equations studied contain in particular models arising in chemical kinetics, nonlinear heat generation, and the gravitational equilibrium of polytropic stars. (Author)
470 citations
TL;DR: In this paper, the prolongation structure of a closed ideal of exterior differential forms is discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation.
Abstract: The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open—i.e., does not terminate as a set of structure relations of a finite‐dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Backlund transformation, and derive the double Backlund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.
442 citations
Book•
01 Oct 1975
TL;DR: In this article, Copson gives a rigorous account of the theory of partial differential equations of the first order and of linear PDEs of the second order, using the methods of classical analysis.
Abstract: In this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first order and of linear partial differential equations of the second order, using the methods of classical analysis. In spite of the advent of computers and the applications of the methods of functional analysis to the theory of partial differential equations, the classical theory retains its relevance in several important respects. Many branches of classical analysing have their origins in the rigourous discussion of problems in applies mathematics and theoretical physics, and the classical treatment of the theory of partial differential equations still provides the best method of treating many physical problems. A knowledge of the classical theory is essential for pure mathematics who intend to undertake research in this field, whatever approach they ultimately adopt. The numerical analyst needs a knowledge of classical theory in order to decide whether a problem has a unique solution or not.
423 citations
TL;DR: In this paper, a two-dimensional finite-difference technique for irregular meshes is formulated for derivatives up to the second order, where the domain in the vicinity of a given central point is broken into eight 45 degree pie shaped segments and the closest finite difference point in each segment to the center point is noted.
257 citations
TL;DR: In this paper, a set of nonlinear partial differential equations that describe the movement of the saltwater front in a coastal aquifer is solved by the Galerkin-finite element method.
Abstract: The set of nonlinear partial differential equations that describe the movement of the saltwater front in a coastal aquifer is solved by the Galerkin-finite element method. Pressure and velocities are obtained simultaneously in order to guarantee continuity of velocities between elements. A layered aquifer is modeled either with a functional representation of permeability or by a constant value of permeability over each element.
236 citations
TL;DR: The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators.
Abstract: The numerical solution of physically realistic nonlinear partial differential equations (PDEs) is a complicated and highly problem-dependent process which usually requires the scientist to undertake the difficult and time-consuming task of developing his own computer program to solve his problem. This paper presents a software interface which can eliminate much of the expensive and time-consuming effort involved in the solution of nonlinear PDEs. The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators. Besides being portable, efficient, and easy to use, the software interface along with an ODE integrator will discretize the problem, select the time step and order, solve the nonlinear equations (checking for convergence, etc.), and maintain a user-specified time integration accuracy, all automatically and reliably. Physically realistic examples are given to illustrate the use and capability of the software.
218 citations
IBM1
196 citations
TL;DR: In this paper, the large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations and a perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear PDEs.
Abstract: The large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations. A perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear partial differential equations. The simply-supported boundary condition and the circumferential periodicity condition are satisfied. The resulting solution indicates that in addition to the fundamental modes, the response contains asymmetric modes as well as axisymmetric modes with the frequency twice that of the fundamental modes. In the previous investigations in which the Galerkins procedure was applied, only the additional axisymrnetric modes were assumed.
Vibrations involving a single driven mode response are investigated. The results indicate that the nonlinearity is either softening or hardening depending on the mode. The vibrations involving both a driven mode and a companion mode are also investigated. The region where the companion mode participates in the vibration is obtained and the effects due to the participation of the companion mode are studied.
An experimental investigation is also conducted. The
results are generally in agreement with the theory. "Non-stationary4 response is detected at some frequencies for large amplitude response where the amplitude drifts from one value to another. Various nonlinear phenomena are observed and quantitative comparisons with the theoretical results are made.
177 citations
TL;DR: In this article, the authors consider high order difference approximations of hyperbolic partial differential equations using Fast Fourier Transform (FFT) and discuss its accuracy, stability and speed on a computer.
Abstract: In this paper we consider high order difference approximations of ${\partial / {\partial x}}$ and especially one method based on the fast Fourier transform. It is used to approximate space derivatives for hyperbolic partial differential equations, and its accuracy, stability and speed on a computer are discussed.
TL;DR: In this paper, the existence of first integrals for non-convex systems with non-conservative forces is established. But the existence depends on the existence either of solutions of the generalized Noether-Bessel-Hagen equation or of the Killing system of partial differential equations.
Abstract: Noether's theorem and Noether's inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamilton's type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.
01 Oct 1975
TL;DR: In this article, a cubic spline approximation is presented which is suited for many fluid-mechanics problems and provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions.
Abstract: A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.
TL;DR: The equation of radiative transfer in the comoving frame makes possible an economical solution of the line formation problem in spherical atmospheres expanding with arbitrarily large velocities as mentioned in this paper, and a stable differencing scheme and a frequency-by-frequency elimination procedure have been developed to solve the partial differential equations that describe the radiation field.
Abstract: The equation of radiative transfer in the comoving frame makes possible an economical solution of the line formation problem in spherical atmospheres expanding with arbitrarily large velocities A stable differencing scheme and a frequency-by-frequency elimination procedure have been developed to solve the partial differential equations that describe the radiation field in the comoving frame Numerical results were obtained for a large number of illustrative models involving line formation by two-level atoms, electron scattering, and continuous absorption Selected results that simulate situations in the stellar winds of hot stars and similar objects are discussed In addition to P Cygni and other very broad profiles, extreme center-to-limb variations are obtained that show both limb darkening and limb brightening For very high velocity flows with very weak or nonexistent continuum and electron-scattering opacities, the flux profiles are very nearly symmetric about the laboratory wavelength and have shapes reminiscent of those observed in the nuclei of Seyfert galaxies Comparisons are presented between the results of Sobolev-type escape probability calculations and those obtained here The force of radiation on the gas is examined in a number of situations; the mechanism mentioned by Noerdlinger and Rybicki for the disruption of radiatively driven envelopes in planar geometriesmore » is shown to become inoperative for even slightly extended spherical systems« less
TL;DR: Some methods for the numerical solution of the regularized long-wave equation, u t + u x + u uu x − u xxt = 0, are described in this paper.
TL;DR: In this paper, a modified Wilson equation is derived explicitly based on an excess energy equation with Wilson''s local volume fractions and the Gibbs-Helmholtz correlation, expressed as the combination of the Wilson equation and a volume ratio term which has been introduced as a result of the derivation.
Abstract: A new equation is derived explicitly based on an excess energy equation with Wilson''s local volume fractions and the Gibbs-Helmholtz correlation. The equation is expressed as the combination of the Wilson equation and a volume ratio term which has been introduced as a result of the derivation. The new equation (modified Wilson equation) contains only two parameters for a binary system, and is applicable to both miscible and partially miscible systems. The equation is readily generalized to multicomponent systems without any additional parameters. The wide applicability of the new equation is shown in representing vapor-liquid and liquidliquid equilibria for binary and ternary systems. It is also shown that the original Wilson equation is obtained without obscurity by the same derivation as for the new equation.
TL;DR: In this paper, an efficient fourth order method for the multi-dimensional wave equation is presented, which is used to solve first order hyperbolic systems for application to inviscid flow calculations.
Abstract: : The results of this report are part of an effort to develop high accuracy efficient methods for solving hyperbolic equations. Here an efficient fourth order method for the multi-dimensional wave equation is presented. Similar techniques are being tried to solve first order hyperbolic systems for application to inviscid flow calculations.
TL;DR: In this paper, a sufficient condition for continuous dependence of a state of a system upon the observation, location of sensors for ensuring observability and continuity are discussed for several examples, and necessary and sufficient conditions for observability are presented for distributed-parameter systems described by linear partial differential equations of parabolic type.
Abstract: Questions regarding observability on the basis of the observed measurement data from a finite number of sensors are discussed for the distributed-parameter systems described by linear partial differential equations of parabolic type. Necessary and sufficient conditions for observability are presented. We obtain a sufficient condition for continuous dependence of a state of the system upon the observation, Location of sensors for ensuring observability and continuity are discussed for several examples.
TL;DR: In this paper, a Bremmer type series solution of the three dimensional reduced wave equation is obtained by iterating generalizations of the Bellman-Kalaba integral equations, which provides systematic corrections to the parabolic approximation.
TL;DR: In this article, the authors present a technique for the analysis of unsteady, two-dimensional diffusive heat-or mass-transfer problems characterized by moving irregular boundaries, including an immobilization transformation and a numerical scheme.
TL;DR: In this article, the authors considered the unsteady laminar compressible boundary-layer flow in the immediate vicinity of a two-dimensional stagnation point due to an incident stream whose velocity varies arbitrarily with time, and the governing partial differential equations involving both time and the independent similarity variable were transformed into new co-ordinates with finite ranges by means of a transformation which maps an infinite interval into a finite one.
Abstract: The unsteady laminar compressible boundary-layer flow in the immediate vicinity of a two-dimensional stagnation point due to an incident stream whose velocity varies arbitrarily with time is considered. The governing partial differential equations, involving both time and the independent similarity variable, are transformed into new co-ordinates with finite ranges by means of a transformation which maps an infinite interval into a finite one. The resulting equations are solved by converting them into a matrix equation through the application of implicit finite-difference formulae. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) a constantly accelerating stream and (ii) a fluctuating stream. The results show that in the former case both the skin-friction and the heat-transfer parameter increase steadily with time after a certain instant, while in the latter they oscillate, thus responding to the fluctuations in the free-stream velocity.
TL;DR: In this paper, the evolution of the power spectrum of surface gravity waves is described by means of a transport equation and the effects of a slowly varying, prescribed ocean current and nonlinear wave-wave interactions are included.
Abstract: : The evolution of the power spectrum of surface gravity waves is described by means of a transport equation. The effects of a slowly varying, prescribed ocean current and nonlinear wave-wave interactions are included. A definition due to Wigner of a localized power spectrum is used to derive the transport equation from the dynamical equations describing surface wave motion.
TL;DR: In this paper, the existence and uniqueness of the Cauchy problem for the evolution equation with respect to weak global solutions of boundary value problems with quasilinear partial differential equations of mixed Sobolev-parabolic-elliptic type, boundary conditions with mixed space-time derivatives, and those of the fourth or fifth type were studied.
Abstract: The Cauchy problem for the evolution equation $Mu'(t) + N(t,u(t)) = 0$ is studied, where M and $N(t, \cdot )$ are, respectively, possibly degenerate and nonlinear monotone operators from a vector space to its dual. Sufficient conditions for existence and for uniqueness of solutions are obtained by reducing the problem to an equivalent one in which M is the identity but each $N(t, \cdot )$ is multivalued and accretive in a Hilbert space. Applications include weak global solutions of boundary value problems with quasilinear partial differential equations of mixed Sobolev-parabolic-elliptic type, boundary conditions with mixed space-time derivatives, and those of the fourth or fifth type. Similar existence and uniqueness results are given for the semilinear and degenerate wave equation $Bu''(t) + F(t,u'(t)) + Au(t) = 0$, where each nonlinear $F(t, \cdot )$ is monotone and the nonnegative B and positive A are self-adjoint operators from a reflexive Banach space to its dual.
TL;DR: In this article, a theoretical analysis of transport through charged membranes is presented using a cell model, where three sets of partial differential equations describe the system: the generalized Nernst-Planck flux equations, the Navier-Stokes equation, and the Poisson-Boltzmann equation.
Abstract: A theoretical analysis of transport through charged membranes is presented using a cell model. In an attempt to improve the capillary tube model, we have modeled the membrane by an array of charged spheres. Three sets of partial differential equations describe this system: the generalized Nernst–Planck flux equations, the Navier–Stokes equation, and the Poisson–Boltzmann equation. These equations are averaged over a cell volume to yield a set of ordinary differential equations on the gross scale, that is, a length scale of the order of the membrane thickness. It is shown for hyperfiltration that the cell model will reject salt more efficiently than the tube model, and in the limit of Stokes’ flow the present analysis reproduces the rejection coefficient previously obtained for the capillary tube model. Results for the electrodialysis mode of membrane operation are also presented. Comparison is made with the capillary tube model for the same total charge and equal volume to surface ratio. The effect of a s...