Neumann conditions

In a similar fashion, taking a homogeneous Neumann condition in the z direction... 

Here C is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form... 

With LSV, the quasireversible and irreversible cases might also be interesting models, both of which have mixed boundary conditions, lying somewhere between the extremes of Dirichlet and Neumann conditions, because here we have fluxes at the electrode, determined by heterogeneous rate constants (depending on potential) and concentrations at the electrode. Also,... 

Here (1), (2), (7), (8) are the diffusion equations with reversible hydrogen capture by the traps at the membrane layers (3), (9) are initial conditions (4), (10) are non-linear boundary conditions of the third kind (Neumann conditions) (5), (6) are mating of layers condition c(t, x), u(t,x) are concentrations of dissolved... 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

Uniform flow is assumed at inlet. The gas pressure is set at outlet. The particles are not allowed to leave the reactor. For the scalar variables, except pressure, Dirichlet boundary conditions are used at inlet, whereas Neumann conditions are employed at the other boundaries [92]. 

At boundary nodes where the variable values are given by Dirichlet conditions, no model equations are solved. When the boundary condition involve derivatives as defined by Neumann conditions, the boundary condition must be discretized to provide the required equation. The governing equation is thus solved on internal points only, not on the boundaries. Mixed or Robin conditions can also be used. These conditions consist of linear combinations of the variable value and its gradient at the boundary. A common problem does arise when higher order approximations of the derivatives are used at... 

Flux specified (called a Neumann condition or boundary condition of the second kind), with specified Nq or q. ... 

Conversely, if a PEC wall is located on a tangential magnetic field node, then due to the homogeneous Neumann condition, 9i/tan/9 = 0, the above procedure should be pertinently... 

Introduction. Transient one-dimensional conduction external to long circular cylinders is considered in this section. The conduction equation, the boundary and initial conditions, and the solutions for the Dirichlet and Neumann conditions are presented. The conduction equation for the instantaneous temperature rise 0(r, t) - T, in the region external to a long circular cylinder of radius a is... 

Neumann Condition. The instantaneous surface temperature is given by... 

S. S. Sadhal, Exact Solutions for the Steady and Unsteady Diffusion Problems for a Rectangular Prism Cases of Complex Neumann Conditions, ASME 84-HT-83, 22nd Heat Transfer Conference, Niagara Falls, NY, Aug. 6-8,1984. 

We note that T Th as x oo where the subscripts u and b refer to unbiuned and burned respectively, however, we employ the Neumann condition in our model. The solution is periodic in ip with period 27t. We observe that % is derivable from the time-independent solution of the problem asTf, = Tu+(3Yu-We nondimensionaUze as in [21] by introducing... 

Neumann condition Dlrlchlet condition No heat flux... 

The first is the Dirichlet condition, which says that one has a set of discrete measurements of the voltage of a subset of the outer surface. The second is the natural Neumann condition. While it does not look much different from the formulation of the direct problem, the inverse formulations are ill-posed. The bioelectric inverse problem in terms of primary current sources does not have a unique solution, and the solution does not depend continuously on the data. Thus, to obtain useful solutions, one must try to... 

For the FD method, it turns out that the Dirichlet boundary condition is easy to apply while the Neumann condition takes a little extra effort. For the FE method it is just the opposite. The Neumann boundary condition... 

A different boundary condition is the Neumann condition or derivative boundary condition. An example is seen with chronopotentiometry. Equation sets (4 and 10). The procedure here is that a value of Co is computed such that it fits with the concentration profile (set of points C. .. Cn), so as to satisfy the gradient specification. Using the simple two-point approximation (33) and given a G value, this yields an expKcit expression for Co,... 

Here, the boundary condition at the surface of the aggregate is a Neumann condition since it is considered perfectly insulating. 

Furthermore, homogeneous Neumann conditions are imposed at the whole tube wall except a special part. This part is A = dO y =... 

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. 

When using exphcit methods, time-step constraints are natural small time steps are needed to represent the mixture in a stream. In the case of compressible flows, the Fourier-Neumann condition gives for the wave equation... 

---