Neumann problem

Begin by noting that C() satisfy the Neumann problem given by (5) and boundary conditions whose solution is Cj(x, y, t) = Cj(x, t). We now derive the overall macroscopic mass balance for the species. To this end we begin by deriving the closure problem for Cj. Given Cf(x,t), combine (6) with boundary conditions and neglect the advection induced by du/dt, to obtain the local Neumann problem... 

Example 5.4 Melting of a Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature The Stefan-Neumann Problem The previous example investigated the heat conduction problem in a semi-infinite solid with constant and variable thermophysical properties. The present Example analyzes the same conduction problem with a change in phase. 

Interest in such problems was first expressed in 1831 in the early work of G. Lame and B. P. Clapeyron on the freezing of moist soil, and in 1889 by J. Stefan on the thickness of polar ice and similar problems. The exact solution of the phase-transition problem in a semi-infinite medium is due to F. Neumann (who apparently dealt with this kind of problem even before Stefan), and thus, problems of this kind are called Stefan-Neumann problems. The interest in these problems has been growing ever since (7,8). 

There are three kinds of boundary conditions for elliptic equations. If the values of the unknown function are prescribed on the boundary, then the problem is called the Dirichlet problem. If the derivatives of the unknown function are prescribed on the boundary, then it is called the Neumann problem. If a linear combination of the function values and the derivatives is specified, then it is called the Robin problem. 

The Fourier transform can be used to solve the Dirichlet problem in the inhnite domain, and the Fourier sine transform can be used in the semi-inhnite domain. The Fourier cosine transform is appropriate for the Neumann problem in the semi-infinite domain. 

Numerical Experiments with the Qassical Difference Scheme Principles of Constructing Special Finite Difference Schemes for the Neumann Problem... 

We discuss the motivation for including the parameter e in the boundary condition (3.2b). For the Dirichlet problem (2.14), when the parameter tends to zero, boundary layers appear in a neighborhood of the lateral boundary. In this connection, the derivatives with respect to x increase unboundedly [see, e.g., estimates (2.70c)j. However, the product s(d/dx)u(x,t) is bounded uniformly with respect to the parameter e for all e (0, IJ. Therefore, when studying the Neumann problem (3.2) and considering the derivatives dldx)u(x, t) (e.g., when analyzing the... 

We say that the finite difference scheme solves the Neumann problem (3.5), if the grid solution converges to the solution of the boundary value problem, and for problem (3.6), if, in addition, the grid solution allows... 

Suppose that we want to find the solution of the Neumann problem for the singularly perturbed diffusion equation... 

Boundary value problems where the normal derivative 5p/5n is specified at the boundaries are known as Neumann problems. Their solutions are not unique, but only to the extent just described. If the flow rate, which is proportional to 5p/9n, is prescribed over part(s) of the boundary, and pressure itself is given over the remainder, the solution is again completely determined and unique. The reason is simple we have not unreasonably created mass. The required mass conservation will manifest itself at the boundaries where pressure was prescribed, and a net outflow or inflow will be obtained that is physically sound. Problems where both 9p/an and p are specified are referred to as mixed Dirichlet-Neumann problems or mixed problems. 

---