Boundary value problems concentrated sources

It is not obvious, but other studies (Ryan etal., 1981) have shown that the reaction source in equations 1.124 and 1.125 makes a negligible contribution to c y. In addition, one can demonstrate (Whitaker, 1999) that the heterogeneous reaction, k CAy, can be neglected for all practical problems of diffusion and reaction in porous catalysts. Furthermore, the non-local diffusion term is negligible for traditional systems, and under these circumstances the boundary value problem for the spatial deviation concentration takes the form... 

In the case when the sutetance concentration at both ends of the segment (0,1] differs from zero, boundary layers appear in the neighborhood of both ends of the interval. The qualitative behavior of the solution for the stationary boundary value problem with distributed sources... 

We consider the simplest meaningful example leading to singularly perturbed boundary value problems. Suppose that we want to find the distributions of concentration C of a substance in a homogeneous material or in a solid material layer with thickness L. Suppose that the quantity C depends only on the variable y, which is the distance to one side of the material, and that generally speaking, the quantity C varies in time T, that is, C — C y,r). Assume also that inside the material the distributed sources of the substance have a density F y, t). Suppose that the diffusion coefficient D is constant. In this case, the distribution of the substance in a material layer is described by the diffusion equation... 

In this section, we consider the singularly perturbed diffusion equation when linear combinations of the solution and its diffusion flux are given on the domain boundary. Such boundary conditions make it possible to realize any of the boundary conditions considered in Sections II and III. Moreover, concentrated sources act inside the domain. These sources lead to the appearance of interior layers. Thus, in addition to the computational problems accompanying the solution of the boundary value problems in Sections II and III, there arise new problems due to the presence of these interior layers. 

We begin our consideration with a simple example that brings us to the singularly perturbed boundary value problems considered in this section. Suppose that it is required to find the function C(y, t), which is the distribution of temperature in a homogeneous material, or in a layer of solid material with a thickness 2L. As in Section II.A, we assume that the distributed heat sources of density F y, t) act inside the material. Besides these sources, in the middle part of the material a concentrated source of the strength Q(t) is situated at y = 0. In a simplified variant, the temperature distribution in a layer of material is described by the heat equation... 

On the set G we consider the boundary value problem for the singularly perturbed equation of parabolic type with a concentrated source on 5 ... 

Note, in particular, one feature in the behavior of the approximate solutions of boundary value problems with a concentrated source. It follows from the results of Section II that, in the case of the Dirichlet problem, the solution of the classical finite difference scheme is bounded 6-uniformly, and even though the grid solution does not converge s-uniformly, it approximates qualitatively the exact solution e-uniformly. But now, in the case of a Dirichlet boundary value problem with a concentrated source, the behavior of the approximate solution differs sharply from what was said above. For example, in the case of a Dirichlet boundary value problem with a concentrated source acting in the middle of the segment D = [-1,1], when the equation coefficients are constant, the right-hand side and the boundary function are equal to zero, the solution is equivalent to the solution of the problem on [0,1] with a Neumann condition at x = 0. It follows that the solution of the classical finite difference scheme for the Dirichlet problem with a concentrated source is not bounded e-uniformly, and that it does not approximate the exact solution uniformly in e, even qualitatively. 

The analysis of heat exchange processes, in the case of the plastic shear of a material, leads us to singularly perturbed boundary value problems with a concentrated source. Problems such as these were considered in Section IV, where it was shown that classical difference schemes give rise to errors, which exceed the exact solution by many orders of magnitude if the perturbation parameter is sufficiently small. Besides, a special finite difference scheme, which allows us to approximate both the solution and... 

Thus, problem (5.3) is a boundary value problem for a singularly perturbed equation with a concentrated source, which was considered in Section IV. 

The special finite difference schemes constructed here allow one to approximate solutions of boundary value problems and also normalized di sion fluxes. They can be used to solve effectively applied problems with boundary and interior layers, in particular, equations with discontinuous coefficients and concentrated factors (heat capacity, sources, and so on). Methods for the construction of the special schemes developed here can be used to construct and investigate special schemes for more general singularly perturbed boundary value problems (see, e.g., [4, 17, 18, 24, 35-39]). 

In the ensuring we assume that the interaction between species a and is reciprocal, i.e. = DijSap where S p is Kronecker s delta. For simplicity we treat only one species, and set the concentration as c = c , the source as p/ = ya where the superscript implies that this value is rapidly changed in a scale of the micro-domain as mentioned previously. Then, if we assume incompressibility of fluid (p = constant, 9vf/9v , = 0), we have the following initial-boundary value problem governing the advective-diffusive movement of the species ... 

Hence, the two sources of nonlinearity in Eq. (54) are moisture concentration c and dilatational strain Consequently, Eqs. (48) and (54) are coupled. The diffusion boundary-value problem must be solved in conjunction with the nonlinear viscoelasticity boundary-value problem by using an iterative procedure. Finite-element formulation of Eq. (54) is standard (see Reddy(47)) and is given by... 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

---