Showing papers on "Partial differential equation published in 1980"

The Finite Difference Method in Partial Differential Equations

Abstract: Keywords: elements : finis ; equations : differentielles ; methodes de : calcul ; mathematiques Reference Record created on 2005-11-18, modified on 2016-08-08

A note on limiting cases of sobolev embeddings and convolution inequalities

Abstract: (1980). A note on limiting cases of sobolev embeddings and convolution inequalities. Communications in Partial Differential Equations: Vol. 5, No. 7, pp. 773-789.

Dynamics in nonglobally hyperbolic, static space-times

Abstract: Ordinary Cauchy evolution determines a solution of a partial differential equation only within the domain of dependence of the initial data surface. Hence, in a nonglobally hyperbolic space‐time, one does not have fully deterministic dynamics. We show here that for the case of a Klein–Gordon scalar field propagating in an arbitrary static space‐time, a physically sensible, fully deterministic dynamical evolution prescription can be given. If the cosmic censor hypothesis should be overthrown, a prescription of this sort could rescue deterministic physics.

Derivation of the Continuous-Time Random-Walk Equation

Abstract: The transport of electrons or excitations on a lattice randomly occupied by guests is considered. The equation governing the transport in any configuration is assumed to be the master equation. A projection operator technique is used to derive the exact equation governing the transport averaged over all configurations, which can be written as either a generalized master equation or the continuous-time random-walk equations (CTRW), establishing the correctness of the CTRW for these problems.

On some linear and nonlinear eigenvalue problems with an indefinite weight function

Abstract: (1980). On some linear and nonlinear eigenvalue problems with an indefinite weight function. Communications in Partial Differential Equations: Vol. 5, No. 10, pp. 999-1030.

A symmetry approach to exactly solvable evolution equations

Abstract: A method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest‐order derivative. The method, based on the symmetry structure of the equations, is applied to second‐order equations and then to third‐order equations which do not contain a second‐order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third‐order evolution equation which contains the Korteweg‐de Vries (KdV) equation and the modified KdV equation as particular cases.

Multilevel Adaptive Computations in Fluid Dynamics

Abstract: HE usual approach to solving partial differential boundary-value problems is first to discretize the problem in some preassigned manner (e.g., finite element or finite difference equations on a fixed grid), and then to submit the resulting discrete system to some numerical solver. In the multilevel adaptive technique (MLAT),1-23 discretization and solution processes are intermixed. A sequence of uniform grids (or "levels"), with geometrically decreasing mesh sizes, participates in the process. The cooperative solution process on these grids involves relaxation sweeps over each of them, coarse-grid-to-fine-grid interpolations of corrections and fineto-coarse transfers of residuals. This process has several important benefits. First, it acts as a very fast solver of the algebraic system of equations, since relaxation on each level is very efficient in liquidating those error components whose wavelength is comparable to that level's mesh size. General nonlinear boundary-value problems, such as Navier-Stokes equations in a general domain, are solved at computational work comparable to seven or so relaxation sweeps on the finest grid. The computer storage used may be much smaller than the number of discrete unknowns. Sections II-IX of this paper survey these fast solvers, emphasizing recent work8'10'11 on elliptic systems, such as Cauchy-Rieman n, Stokes, and Navier-Stokes equations. A new type of relaxation, called distributive Gauss Seidel (DGS), has been developed for such systems, based on the elliptic decomposition of the "symbol" of the finite-difference operators. 8 Also mentioned are fast solvers for compressible viscous flows, for transonic problems, and for unstable steady-state flows. Moreover, the fast multilevel solutions can actually represent better approximations than the full solutions of the difference equations. For example, one can combine the stability of upstream differencing with the accuracy of central differencing by using the first in relaxation and the latter in the residual transfers. One can obtain extrapolations to higher-order approximations by a trivial change in a lowerorder program. Or one can solve evolution problems very inexpensively by performing most time steps on coarse levels, in such a way that the finest-level accuracy is still maintained. These and related techniques (for ill-posed problems, bifurcation problems, and for parametric optimization) are surveyed in Sec. X. Finally, the multilevel structure provides, in a natural way, very flexible and adaptive discretization schemes, which can automatically and efficiently treat boundary layers and other singularities7 (Sec. XI). This is a survey paper. Naturally, not all the ideas expressed here can be rigorously supported at this time. They are described in more detail elsewhere. 2~8

Optimum experimental design for identification of distributed parameter systems

Abstract: A method to design optimal experiments for parameter estimation in distributed systems is given. The design variables considered are the boundary perturbation and the spatial location of measurement sensors. The design criterion used is the determinant of Fisher's information matrix. It is shown that suitable choice of these variables leads to improved parameter accuracy. Two examples are used to illustrate this method. The first example is concerned with sensor location for estimating the velocity of propagation and the damping coefficient of a vibrating string. The second example is concerned with the estimation of the thermal diffusivity and radiation constant for a heat diffusion process. It is also shown that the design philosophy can be applied to a wide class of systems described by partial differential equations.

Effect Analysis in Structural Equation Models Extensions and Simplified Methods of Computation

Abstract: One of the great virtues of structural equation models is that they permit the quantification of causal and noncausal sources of statistical relationship. The present article discusses efficient ma...

Factorization of operators I. Miura transformations

Abstract: The method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is here extended to some third‐order scattering operators, and transformations between several fifth‐order nonlinear evolution equations are derived. Further applications are discussed.

The extended Graetz problem with prescribed wall flux

Abstract: An analytical solution is obtained to the extended Graetz problem with prescribed wall flux, based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first-order partial differential equations. The solution obtained is simple, computationally efficient and in striking contrast with incomplete numerical efforts in the past.

Decay of solution to parabolic conservation laws

Abstract: (1980). Decay of solution to parabolic conservation laws. Communications in Partial Differential Equations: Vol. 5, No. 4, pp. 449-473.

Space‐time finite elements incorporating characteristics for the burgers' equation

Abstract: A new finite element method is presented for the numerical solution of the Burgers' equation. The method is based on a weighted residual formulation in which the method of characteristics is incorporated by employing isoparametric space–time elements with the sides joining the nodes at subsequent time levels oriented along the characteristic lines of the hyperbolic partial differential equation associated with the Burgers' equation. This method is capable of solving the Burgers' equation accurately for values of viscosity ranging from very small to large. The utility and accuracy of the method are demonstrated by a number of examples and a comparison of the numerical results with the exact solution is presented in one case.

Simple Method for Predicting Drainage from Field Plots

Abstract: When the one-dimensional moisture flow equation is simplified by applying the unit gradient approximation, a first-order partial differential equation results. The first-order equation is hyperbolic and easily solved by the method of P. 0. Lax. Three published K(6) relationships were used to generate three analytical solutions for the drainage phase following infiltration. All three solutions produced straight lines or nearly straight lines when log of total water above a depth was plotted versus log of time. Several suggestions for obtaining the required parameters are presented and two example problems are included to demonstrate the accuracy and applicability of the method. Additional Index Words: Cauchy problem, redistribution, characteristic value problem, hydraulic conductivity, infiltration. Sisson, J. B., A. H. Ferguson, and M. Th. van Genuchten. 1980. Simple method for predicting drainage from field plots. Soil Sci. Soc. Am. J. 44:1147-1152. W DRAINING from soil profiles is an important factor in many contemporary and environmental problems. In the Northern Great Plains this water is responsible for the annual destruction of thousands of hectares of cropland by contributing to the formation and growth of saline seeps (Brown and Ferguson, 1973; Ferguson et al., 1972). In some irrigated areas, subsurface drainage water contributes to river pollution (Wierenga and Patterson, 1972). Estimation of the rate and quantity of drainage water contributing to these problems is essential to finding feasible solutions. But estimating these variables requires predicting the hydrologic behavior of large areas and frequently the characterization of many soils under field conditions. This necessitates the use of simple, yet accurate, models which contain parameters that can be obtained on site as quickly as possible. This paper considers a special class of models based on the assumption of a unit gradient of the total potential head. Several studies (cf. Black et al., 1969, and Davidson et al., 1969, among others) have shown that a unit gradient often exists during the redistribution and drainage phases when a uniform profile is draining freely in the absence of a shallow water table. Three solutions are presented, each based upon a different conductivity equation. Two example problems are further included to demonstrate the use of the present approach. THEORETICAL CONSIDERATIONS The equation for predicting the one-dimensional flow of water in porous materials is (Taylor and Ashcroft, 1972): 80 3T 9 37 [1] 9 — 0(z,t) is the volumetric moisture content, t = time, z = depth (positive downward), K = K(0) is the hydraulic conductivity, and H H(6,z) = h(6)-z; i.e., H(hydraulic head) = h (pressure head) — ^(gravitational head). When a unit gradient in the total head H is assumed, 9H/3z = -1, Eq. [1] becomes

The Inverse Scattering Transform

Abstract: A detailed description of the inverse scattering transform associated with the generalized Zakharov-Shabat and Schrodinger eigenvalue problems is given. The close analogy with the ideas of the Fourier transform is emphasized and the general expansions for the unknown functions in terms of the squared eigenfunctions and their derivatives are developed for both eigenvalue problems. The results for the Schrodinger equation are new.1 The partial differential equations which are solvable by the inverse scattering transforms associated with these eigenvalue problems are identified, almost by inspection, and classified according to the nature of the dispersion relation. Of particular interest are those classes which are integrable but which do not possess conserved quantities, and also those equations which are integrable but for which the spectrum is not invariant. Several examples, the coherent pulse propagation problem, the nonlinear Schrodinger equation and the sine-Gordon equation, are used to illustrate some of the important points. Finally, a singular perturbation theory for examining the effects of perturbations over long times is given. Last-minute revisions and additions have been made to Sects.6.12 and 13 to reflect some recent developments.

The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic

Abstract: : The results of Part I are extended to include linear spatially periodic problems - solutions of the initial value are shown to exist and decay like exp(- lambda (t to the beta power)). Then the full non-linear Boltzmann equation with a soft potential is solved for initial data close to equilibrium. The non-linearity is treated as a perturbation of the linear problem, and the equation is solved by iteration. (Author)

On solving certain nonlinear partial differential equations by accretive operator methods

Abstract: We use similar functional analytic methods to solve (a) a fully nonlinear second order elliptic equation, (b) a Hamilton-Jacobi equation, and (c) a functional/partial differential equation from plasma physics The technique in each case is to approximate by the solutions of simpler problems, and then to pass to limits using a modification of G Minty’s device to the spaceL ∞

On the Hamiltonian structure of evolution equations

Abstract: The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikii on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.