Boundary Value Problems dimension

Diffusion problems in one dimension lead to boundaiy value problems. The boundaiy conditions are applied at two different spatial locations at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Example 5 Consider now the third boundary-value problem (9). As in Example 2 of Section 1 it will be convenient to introduce the space = 0 , of the dimension A +1 consisting of all grid functions defined on the uniform ... 

Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial... 

Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

In this section we consider problems in which there is convective and diffusive transport in one spatial dimension, as well as elementary chemical reaction. The computational solution of such problems requires attention to discretization on a mesh network and solution algorithms. For steady-state situations the computational problem is one of solving a boundary-value problem. In chemically reacting flow problems it is not uncommon to have steep reaction fronts, such as in a flame. In such a case it is important to provide adequate mesh resolution within the front. Adaptive mesh schemes are used to accomplish this objective. 

For boundary value problems of ordinary differential equations the first step is to reduce the problem to a first-order system of DEs of dimension n > 1 as described in Section 1.2.2. Once this is done, one needs to solve the system of DEs... 

This transformation allows for equal distribution in the y-space while concentrating the lines close to the x = L boundary. Parameter a sets the spacing of the lines. This technique is called MOL1D (Method Of Lines in 1 Dimension) and is suitable for solving parabolic and hyperbolic initial boundary value problems in one dimension. 

Fig. 4. Schematic representation of the problem geometry corresponding to a pattern involving a step distribution of the active-area density. The domain Q can be viewed as a two-dimensional, symmetric section of a larger, three-dimensional configuration, as indicated. The relative dimensions are shown, and the boundary segments of the corresponding boundary-value problem are indicated. (Reprinted by permission of the publisher, The Electrochemical Society, Inc. [25]).

Fig. 25. The coupled boundary-value problem involving the potential, the metal-ion concentration, and the leveUng-agent concentration. The geometry of the problem is schematically shown, and the base-case dimensions are given. (Copyright 1992 by International Business Machines Corporation reprinted with permission [63]).

The differential equation, (1.2), is an ordinary differential equation because there is only one independent variable, x In this case, equations in one space dimension are boundary value problems, because the conditions are provided at two different locations. While it is also possible to solve this problem using Excel and MATLAB, it is much simpler to use FEMLAB. Transient heat transfer in one space dimension is governed by... 

A body is called thin if one (or more) of its characteristic dimensions is much smaller than the others. A thin rod and a thin plate are examples of such bodies. We consider a boundary value problem describing a heat conduction process in a thin rod, where the ratio e of the thickness of the rod to its length is a small parameter. To simplify the presentation, we consider the problem for a planar rod, that is, in the thin rectangle (OsjfSl) X (0[Pg.166]

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections. 

Full development of the methods to solve elasticity boundary value problems in either two or three dimensions is beyond the scope of this text. Here we outline the two major approaches. 

Consider the following boundary-value problem. A plate of dimensions 2H 2L 2b is indented by two opposite punches of dimensions 2b 2a, Fig.la. 

As the dimension increases, the fill-in problem becomes more acute, and we may not have enough memory to store even the contents of, 4. For these reasons, in our later discussion of boundary value problems (Chapter 6), we consider iterative methods for solving linear systems that are not susceptible to fill-in and that do not require us even to store in memory the components of. ... 

Before considering some additional examples of the finite difference method for two dimensional boundary value problems, it is useful to consider the possibility of adding another dimension to the problem and this is usually a time dimension. The prototypical example of such an equation is the time dependent diffusion equation ... 

Many physical two dimensional boundary value problems also involve a time dimension. Just as with the FD technique, solving such problems with the FE technique represent an important application of the FE approach. The approach here is very similar to that used in Section 12.7 for the FD technique and the reader is referred to that section for a more detailed explanation of the approach. Again the diffusion equation provides a prototypical example of such a two dimensional boundary value problem with an additional time dimension ... 

In these equations the primes indicate time derivatives at some time step t+h and the un and unn functions are to be evaluated at time step t where good approximations to the time derivatives are assumed to be known. In this manner the partial derivatives with respect to time can be eliminated from the equation and the solution can be bootstrapped along from one time increment to an additional time increment. This of course assumes that the problem is that of an initial value problem in the time dimension where a solution is known for some initial time at all points in the x-y space. If the problem involves a second time derivative, then the initial first time derivative must be known at all points in the x-y space. Such PDEs involving a boundary value problem in the spatial dimension and an initial value problem in time cover a broad range of practical engineering problems. 

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