One-dimensional time-dependent

Self-similarity applies to one-dimensional, time-dependent problems in which dependence on one of two independent variables can be eliminated by nondimen-... 

Hasson, P. T., 1965, HYDRO—A Digital Model for One-Dimensional Time-Dependent Two-Phase Hydrodynamics, Part 2, RFR-492/RFN-210, A B Atomenergi, Nykoping, Sweden. (3)... 

Lamm I-L, C. Samuelsson, and H. Pettersson, Solution of the One-Dimensional Time-Dependent Diffusion Equation with Boundary Conditions Applicable to Radon Exhalation from Porous Materials, Coden Report LUNFD6(NFRA3042), Lund University, Lund (1983). 

Another model which includes interaction and for which partial results are available on the decay of initial correlations is that of the one dimensional time-dependent Ising model. This model was first suggested by Glauber,18 and analyzed by him for one-dimensional Ising lattices. Let us consider a one-dimensional lattice, each of whose sites contain a spin. The spin on site,/ will be denoted by s/t) where Sj(t) can take on values + 1, and transitions are made randomly between the two states due to interactions with an external heat reservoir. The state of the system is specified by the spin vector s(t) = (..., s- f), s0(t), Ji(0>---)- A- full description of the system is provided by the probability P(s t), but of more immediate interest are the reduced probabilities... 

Fig. 4.1. Schematic illustration of the evolution of a one-dimensional time-dependent wavepacket in the upper electronic state. The wavepacket is complex for t > 0 only its real part is shown here. Note that the upper horizontal axis does not correspond to a particular energyl The wavepacket is a superposition of stationary states corresponding to a broad range of energies, which are all simultaneously excited by the infinitely short light pulse indicated by the vertical arrow.

For one-dimensional, time dependent heat conduction with constant properties, we can select a slab for convenience of illustration. The typical problem is governed by the energy equation ... 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

The propagation of longitudinal acoustic waves in choked nozzles has been analyzed on the basis of the one-dimensional, time-dependent forms of equations (4-45) and (4-46) by introducing linearizations of the previously indicated type (for example, p = p(l + p )] for the stream wise velocity v as well—that is, v — v(l + i )—and by allowing the mean quantities p, p, and V to vary with the streamwise distance z through the nozzle, in a manner presumed known from a quasi-one-dimensional, steady-flow nozzle analysis [20]. The perturbation equations... 

A critique and more thorough review of the time-lag ideas has been published [7]. The principal value of time-lag concepts lies in the wide range of problems to which they can be applied with relative ease. Their principal deficiency lies in the difficulty of relating Xi to the fundamental processes occurring. If the one-dimensional, time-dependent conservation equations are linearized about any one of the steady-state solutions of Chapter 7 and p is calculated from a perturbation analysis, then it is found... 

In the theoretical analysis of shock instability, shock waves that are not too strong are presumed to propagate axially back and forth in a cylindrical chamber, bouncing off a planar combustion zone at one end and a short choked nozzle at the other [101], [102]. The one-dimensional, time-dependent conservation equations for an inviscid ideal gas with constant heat capacities are expanded about a uniform state having constant pressure p and constant velocity v in the axial (z) direction. Since nonlinear effects are addressed, the expansion is carried to second order in a small parameter e that measures the shock strength discontinuities are permitted across the normal shock, but the shock remains isentropic to this order of approximation. Boundary conditions at the propellant surface (z = 0) and at the... 

One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay bounded—as in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-Thien-Tanner constitutive equation [50]. 

Figure 17. Effective geometry of the model. The x axis corresponds to the quasi-one-dimensional time-dependent wavefunction.

Bjork, G. (1989) A one-dimensional time-dependent model of the vertical stratification of the upper Arctic Ocean. Journal of Physical Oceanography, 19, 52-67. 

Adjerid, S., Flaherty, J. E., A Moving Finite Element Method with Error Estimation and Refinement for One-Dimensional Time Dependent Partial Differential Equations, SIAM J. Numer. Anal., 23 (1986), 778-796... 

The dynamical behavior of strongly non-adiabatic multimode systems exhibits complex features that can perhaps be better understood by trying first to underline the aspects that they share with simple one-dimensional models basically derived from scattering theory. To this purpose we introduce a minimal model for the Cl and then discuss the limit of validity of an interpretation scheme grounded on the Landau-Zener formula as well as of a simple one-dimensional time-dependent model obtained by treating quantum-mechanically only the coupling mode. 

In this equation the ) (17) function assumes a similar role to the f(x,y, 17) function in the previous example. The reader is referred back to Section 12.5 for a discussion of this equation and the one dimensional time dependent solution. In the present chapter in Section 13.9 an example of linear diffusion into a two dimensional surface was presented. For that example, a triangular mesh array was developed and shown in Figure 13.15. The present example combines the nonlinear diffusion model of Section 12.5 with the FE mesh of Figure 13.15 to demonstrate a second nonlinear PDE solution using the FE approach. The reader should review this previous material as this seetion builds upon that material. 

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