One-dimensional Problem

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

Isoparametric mapping described in Section 1.7 for generating curved and distorted elements is not, in general, relevant to one-dimensional problems. However, the problem solved in this section provides a simple example for the illustration of important aspects of this procedure. Consider a master element as is shown in Figure 2.23. The shape functions associated with this element are... 

Note that in the one-dimensional problem illustrated here the Jacobian of coordinate transformation is simply expressed as dx7d and therefore... 

We begin the mathematical analysis of the model, by considering the forces acting on one of the beads. If the sample is subject to stress in only one direction, it is sufficient to set up a one-dimensional problem and examine the components of force, velocity, and displacement in the direction of the stress. We assume this to be the z direction. The subchains and their associated beads and springs are indexed from 1 to N we focus attention on the ith. The absolute coordinates of the beads do not concern us, only their displacements. 

These time steps are smaller than for one-dimensional problems. For three dimensions, the limit is... 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

The modeling of a groundwater chemical pollution problem may be one-, two-, or tlu-cc-dimcnsional. The proper approach is dependent on the problem context. For c.xamplc, tlie vertical migration of a chemical from a surface source to the water table is generally treated as a one-dimensional problem. Within an aquifer, this type of analysis may be valid if the chemical nipidly penetrates the aquifer so that concentrations are uniform vertically and laterally. This is likely to be the case when the vertical and latcrtil dimensions of the aquifer arc small relative to the longitudinal scale of the problem or when the source fully penetrates the aquifer and forms a strip source. 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

Here the constant C takes care of the relative importance of the second derivative influence. Instead of solving a front problem in the coordinates (x,t) (physical space) we perform the calculations in the computational space (C t). For one dimensional problems this adaptive grid transformation proved to be very successful. We can perform a transformation in a similar spirit for a two dimensional domain (x,y,t) -> A general sketch of this transformation... 

For the flat one-dimensional problem with the sole coordinate x, the Poisson equation can be written as... 

Thus, the three-dimensional problem has been reduced to three one-dimensional problems. 

The one-dimensional problem of the particle in a box was treateduxSectiQn S.4.1. Exact solutions were obtained, which were then restricted by the boundary conditions (0) — ty t) = 0. If the exact solutions were not known, the problem... 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

For the sake of mathematical simplicity wave motion was introduced before (1.3.2) as a one-dimensional problem. One of the most obvious features of waves, however, is their tendency to spread in all directions from the source of a disturbance. To represent such a wave in three dimensions requires two parameters, t and r, the distance from the source to any point in the medium, in a form like f(r — vt), in which v is the velocity at which the disturbance spreads, and the phase is r — vt. As the disturbance spreads it increases... 

We now get a fairly good feeling that handling boundary conditions in two dimensions is significantly more difficult than those of one-dimensional problems. Flux conditions can be applied to the boundaries using the method of fictitious points. 

Taking the one-dimensional problem with x as the space variable as an example, boundary conditions can be of three forms ... 

Instead of adapting the NewtonRaphson.m function we just use the Matlab function fzero which is a general routine for that kind of one-dimensional problem. ... 

One may compare this result with that of Section 1.2. The vibrational part of (1.13) is again identical to Eq. (1.68). The rotational part is, however, missing in the one-dimensional problem. It is worth commenting on this special feature of the vibrational problem. It arises from the fact that molecular potentials usually have a deep minimum at r = re. For small amplitude motion (i.e., for low vibrational states) one can therefore make the approximation discussed in the sentence following Eq. (1.13) of replacing r by re in the centrifugal term. In this most extreme limit of molecular rigidity, the vibrational motion is the same in one, two and three dimensions. 

Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U(2). There are two cases... 

One-dimensional problems present on one hand the simplest (and most studied) example of algebraic theory, and on the other hand involve some subtle problems that are worthwhile elucidating. 

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms... 

In view of Eq. (2.47), the states are doubly degenerate except for M=0. This is a peculiarity of one-dimensional problems, as it will be commented in the following. We choose in Eq. (2.55) only the positive branch of M,... 

The double degeneracy of the 0(2) case corresponds to the fact that the algebraic method describes in this case two Morse potentials related to each other by a reflection around x = 0. This is a peculiar feature of one-dimensional problems, and it does not appear in the general case of three dimensions. If one uses the 0(2) basis for calculations, this peculiarity can be simply dealt with by considering only the positive branch of M. 

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