Semi-infinite Domain

One nice feature of the finite element method is the use of natural boundaiy conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundaiy). The externally applied flux is still apphed at the shorter domain, and the solution inside the truncated domain is still vahd. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

In these variables there is only one parameter, r,. Therefore, for a given value of the parameter, the equation can be solved once and for all. Furthermore the entire family of solutions can be determined as a function of the single parameter. Consequently the nondimensional formulation has offered a potentially enormous reduction of the problem. In Chapter 6 we make extensive use of this form of nondimensionalization for problems on semi-infinite domains. 

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

The form of the solution for one-dimensional diffusion is illustrated in Fig. 5.3. The solution c(x,t) is symmetric about x = 0 (i.e., c(x,t) = c(—x,t)). Because the flux at this location always vanishes, no material passes from one side of the plane to the other and therefore the two sides of the solution are independent. Thus the general form of the solution for the infinite domain is also valid for the semi-infinite domain (0 < x < oo) with an initial thin source of diffusant at x = 0. However, in the semi-infinite case, the initial thin source diffuses into one side rather than two and the concentration is therefore larger by a factor of two, so that... 

The Laplace transform method is a powerful technique for solving a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain. After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved. The transformed solution is then back-transformed to obtain the desired solution. 

Solve the following boundary value problem on the semi-infinite domain with discontinuous initial conditions,... 

The coupled diffusion equations (6) together with the boundary conditions (9) and (10) can be solved in close form in the Laplace transform space, and numerically inverted to the time domain. At early time, the solution behaves according to the solution for a semi-infinite domain. At large time, the solution... 

One-dimensional (ID) heterogeneous non-reactive contaminant transport in a semi-infinite domain, described by the ADE is... 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

Figure 15. Numerical results for the first process time density process on the semi-infinite domain, for an Levy flight with Levy index a = 1.2. Note abscissa, is tp(t). For all initial conditions. to = 0.10 1.00, 10.0, and 100.0 the universal slope —3/2 in the log10-log10 plot is clearly reproduced, and it is significantly different from the two slopes predicted by the method of images and the direct definition of the first process time density.

Exponential Matrix Method for Linear BVPs with Semi-infinite Domains... 

The methodology developed in section 3.1.2 can be used for semi-infinite boundary conditions, also. The procedure for solving boundary value problems in semi-infinite domain is as follows ... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

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