Boundary value problems Neumann problem

Fig. 1.8. (a) Dirichlet s problem, (b) Neumann s problem, (c) the third boundary value problem. 

Example 2.11 The Neumann Boundary-Value Problem Find the harmonic function in the hrst quadrant x,y > 0 satisfying... 

Using the example of a boundary value problem for a singularly perturbed ordinary differential equation with Neumann boundary conditions, we discuss some principles of constructing special finite difference schemes. These principles will be used in Section III.D to construct special finite difference schemes for singularly perturbed equations of the parabolic type. 

Thus, for the Neumann boundary value problem (3.23), a special finite difference scheme has been constructed. Its solution z(x),xE.D,, and the function xED allow us to approximate the solution of the... 

Now we study the efficiency of the special finite difference scheme in the case of the Neumann boundary value problem (3.7), (3.9). On the set G(3 g) we introduce the special grid... 

Thus, we see that the newly constructed finite difference schemes are indeed effective and that they allow us to approximate the solution and the normalized diffusion fluxes g-uniformly for both Dirichlet and Neumann boundary value problems with singular perturbations. 

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. 

Since the boundary value problem is of the Neumann type, the solution will not be unique without loss of generality, we fix the value of to zero, say, at... 

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. 

This chapter has been used to illustrate the method discussed in Chap. 2, which may be used to solve thermoviscoelastic boundary value problems involving temperature fields simultaneously varying with position and time. The method relies upon the solution of integral equations in terms of Neumann series expansions. 

Basic equations of the theory of elastic diatomic media, each particle of which includes two different atoms, are given. By means of the semi-inverse method of Saint-Venant, the stresses and displacements in a bar subjected to a terminal load are derived. Satisfaction of the balance and of the generalized Beltrami-Michell compatability equations leads to four (as compared with three classical) Neumann type boundary value problems of the potential theory. A numerical example is solved and illustrated by a graph. 

Once the random field involved in the stochastic boundary value problem has been discretized, a solution method has to be adopted in order to solve the boundary value problem numerically. The choice of the solution method depends on the required statistical information of the solution. If only the first two statistical moments of the solution are of interest second moment analysis), the perturbafion method can be applied. However, if a. full probabilistic analysis is necessary, Galerkin schemes can be utilized or one has to resort to Monte Carlo simulations eventually in combination with a von Neumann series expansion. 

It remains to describe how to handle the boundary value Cq. Clearly, for the RL variant, there is no problem because the last concentration value calculated is C(, and Cq can then be computed from all the other C values, now known, according to the boundary condition. This leaves the LR problem. If the boundary concentration is determined as such (the Dirichlet condition, for example the Cottrell experiment), then this is simply applied. It is with derivative (Neumann) boundary conditions that there is a (small) problem. Here, we know an expression for the gradient G at the electrode. For simplicity, assume a two-point gradient approximation at time t + ST... 

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. 

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... 

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... 

This important relation holds for all the states for the Dirichlet problem (see e.g. [5], Sect. XIII.15). However, the analogous relation is wrong for the other types of boundaries [29]. For example, the constant function in a sphere (that is, the wavefunction r/r(r) = const) satisfies the Neumann boundary conditions and has a zero value for the kinetic energy. Hence the energy functional is the mean value for the potential, that is, it equals —3/(2R) for the hydrogen atom. When the sphere radius R goes to zero, the energy value decreases, in contrast to Equation (2.4). 

When solving how problems numerically, a Neumann boundary is described as an insulated boundary (or impermeable boundary), which means that there is no fiux at the boundary, while a Dirichlet boundary indicates that the value of head (potential, concentration, etc.) is constant at the boundary. Constant boundary conditions are not able to describe the nature of the electrokinetic transport realistically due to the existence of fiux boundaries caused by the electrode reactions and advection of fluid. In Cao s model, the boundary conditions apphed at the inlet and outlet of the soil column maintained the equahty between the flux of solute at the inside of the column and the flux of solute immediately outside of the column. The following boundary condition was used at the inlet (Lafolie and Hayot, 1993) ... 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

The points (-1, j) are located outside the object therefore, u., have fictitious values. Their calculation, however, is necessary for the evaluation of the Neumann boundary condition. Eq. (6.37), written for all y (/ = 0,1,..., q), provides ( + 1) additional equations but at the same time introduces ( + 1) additional variables. To counter this, Eq. (6.34) is also written for (q + 1) points along this boundary (at x = 0), thus providing the necessary number of independent equations for the solution of the problem. 

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