Dimensional variables boundary value problem

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

When solving difference boundary-value problems for Poisson s equation in rectangular, angular, cylindrical and spherical systems of coordinates direct economical methods are widely used that are known to us as the decomposition method and the method of separation of variables. The calculations in both methods for two-dimensional problems require Q arithmetic operations, Q — 0 N og. N), where N is the number of the grid nodes along one of the directions. 

Obviously, the above transformed governing equations for a linear viscoelastic material (Eqs. 9.33- 9.36) are of the same form as the governing equations for a linear elastic material (Eqs. 9.25 - 9.28) except they are in the transform domain. This observation leads to the correspondence principle for three dimensional stress analysis For a given a viscoelastic boundary value problem, replace all time dependent variables in all the governing equations by their Laplace transform and replace all material properties by s times their Laplace transform (recall, e.g., G (s) = sG(s)),... 

On the basis of their initial and boundary conditions, partial differential equations may be further classified into initial-value or boundary-value problems. In the first case, at least one of the independent variables has an open region. In the unsteady-state heat conduction problem, the time variable has the range 0 r >, where no condition has been specified at r = eo therefore, this is an initial-value problem. When the region is closed for all independent variables and conditions are specified at all boundaries, then the problem is of the boundary-value type. An example of this is the three-dimensional steady-state heat conduction problem described by the equation... 

The term is square brackets is the solution value at some time t and is assumed to be some known function of the spatial variables. For a given time and time step h, the form of this equation is identical to the general form of the two dimensional boundary value problem previously considered. As discussed in the previous section this is also very similar to the use of the w parameter in the ADI technique to stabilize the alternating direction iterative method. Thus at any given time and time step, the code routines aheady developed for the two dimensional BVP can... 

The reference and the scale have the same units as x. A nonzero reference indicates that the difference between x and xref is important to the specific problem at hand rather than the absolute value of x itself. xref is determined from the problem and is typically some value of x at the boundary or some initial time. The scale is some combination of other dimensional quantities that are relevant to the physical problem, such as boundary and initial conditions and physical constants. If chosen properly, the scale provides a measure of the range of values that the variable x — xref will take for the particular physical problem. The ideal choice of the reference and scale quantities for a given problem will result in order-unity values for the nondimensional quantities. 

A principal assumption for similarity is that there exists a viscous boundary layer in which the temperature and species composition depend on only one independent variable. The velocity distribution, however, may be two- or even three-dimensional, although in a very special way that requires some scaled velocities to have only one-dimensional content. The fact that there is only one independent variable implies an infinite domain in directions orthogonal to the remaining independent variable. Of course, no real problems have infinite extent. Therefore to be of practical value, it is important that there be real situations for which the assumptions are sufficiently valid. Essentially the assumptions are valid in situations where the viscous boundary-layer thickness is small relative to the lateral extent of the problem. There will always be regions where edge effects interrupt the similarity. The following section provides some physical evidence that supports the notion that there are situations in which the stagnation-flow assumptions are valid. 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

Program Description The MATLAB funct onparabolic]D.m is developed to solve the parabolic partial differential equation in an unsteady-state one-dimensional problem. The boundary conditions arc passed to the function in the same format as that of Example 6.1, with the exception that they are given in only the. xi-direction. The function also needs the initial condition, u, which is a vector containing the values of the dependent variable for all x at time f = 0. 

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