Boundary-value problems continuation

We concentrate primarily on the first boundary-value problem associated with equation- (3) in the rectangle D = Q < x < l,0[Pg.300]

Here the problem is given as an initial value problem, although the concepts can easily be generalized to boundary value problems and even partial differential equations. Note also that both continuous variables, x (parameters), and functions of time, U(t) (control profiles), are included as decision variables. Constraints can also be enforced over the entire time domain and at final time. 

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

The original problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it, is required to find a continuous in the rectangle >t = 0 < < 1, 0 [Pg.459]

Overall the system of equations (continuity and momentum) is third order, nonlinear, ordinary-differential equation, boundary-value problem. The boundary conditions require no-slip at the plates and specified wall-injection velocities,... 

Stagnation flows represent a very important class of flow configurations wherein the steady-state Navier-Stokes equations, together with thermal-energy and species-continuity equations, reduce to systems of ordinary-differential-equation boundary-value problems. Some of these flows have great practical value in applications, such as chemical-vapor-deposition reactors for electronic thin-film growth. They are also widely used in combustion research to study the effects of fluid-mechanical strain on flame behavior. 

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

Although the specific boundary conditions depend on the details of the flow situation and the domain, there are common elements in the boundary conditions that are considered here. Equations 6.159 through 6.163 represent a ninth-order boundary-value problem, requiring nine boundary conditions. The continuity equation is a first-order equation that requires only a boundary condition on u at the stagnation surface. The second-order transport equations demand boundary conditions at each end of the domain, 0 < z < zend-... 

The composition boundary values entering into Eqs. (All) represent external values for Eqs. (A10). With some further assumptions concerning the diffusion and reaction terms, this allows an analytical solution of the boundary-value problem [Eqs. (A10) and (All)] in a closed matrix form (see Refs. 58 and 135). On the other hand, the boundary values need to be determined from the total system of equations describing the process. The bulk values in both phases are found from the balance relations, Eqs. (Al) and (A2). The interfacial liquid-phase concentrations xj are related to the relevant concentrations of the second fluid phase, y , by the thermodynamic equilibrium relationships and by the continuity condition for the molar fluxes at the interface (57,135). 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

A typical result of numerical analysis is an estimate of the error U — Uh between the solution U of the continuous problem (. e., the solution of the initial boundary value problem) and the solution Uh of the discrete problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.)... 

Either of the boundary-value problems can be uniquely solved numerically if any positive value of the slip velocity U = Unix, 1) would be prescribed on the common boundary z = 1. However, this value is physically suited if it secures the continuity of the transversal velocity and the shear stress. The conjugation boundary condition... 

The principle of well-posedness states that the description of the motion should be such that a solution to the initial boundary value problem exists and depends continuously on the initial and boundary conditions. 

A wide range of situations, where statements (6.5) and (6.10) may be used can be found for the free problems (i.e. R = oo) with some types of external potentials V(r, rf). The same methods may be used for external potentials in Dirichlet problems and other boundary value problems of the type Equation (1.1), when R does not depend on p. For example, let us suppose that the integral G(r, rj) (see Equation (6.3)) exists for any r e [0, / ], G is a continuous... 

Various formulations of hnite element methods have been proposed. For an exhaustive account on hnite element methods, the reader is referred to Chen (2005), Donea (2003), Reddy (2005), etc. We present here one of the popular formulations known as the weak formulation of the governing differential equation that, instead of requiring the solution to be twice continuously differentiable, requires that the derivative of the solution be square integrable. We illustrate the weak formulation of the boundary-value problem. Equation (2.157). 

For this formulation, the solution to the inverse problem is unique [7] however, there still exists the problem of continuity of the solution on the data. The linear algebraic counterpart to the elliptic boundary value problem is often useful in discussing this problem of noncontinuity. The numerical solution to all elliptic boundary value problems (such as the Poisson and Laplace problems) can be formulated in terms of a set of linear equations, = b. For the solution of Laplace s equation, the system can be reformulated as ... 

Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving boundary value problems require that the continuous domain be broken up into discrete elements, the so-called mesh or grid, which one can use to approximate the governing equation (s) using the particular numerical technique (finite element, boundary element, finite difference, or multigrid) best suited to the problem. 

COLMOD An automatic continuation code for solving stiff boundary value problems. 

The discrete version of the hedonic equilibrium model is the basis of the second numerical technique we used to generate the relationship between wages and injury risk. Because the system of nonlinear equations in (3.6) and (3.8) theoretically represents equilibrium completely the technique does not encounter boundary value problems. In addition, as the number of submarkets increases the model approaches the continuous case so numerical differences between the two approaches... 

The theory for cyclic voltametric methods has not been worked out quantitatively. In his discussion, Delahay indicated that the initial concentration of reacting species will vary from one cycle to another and the initial conditions for the boundary value problem cannot be easily stated. As a result the experimental peak currents should not necessarily be in agreement with derived theoretical values. Delahay concluded that compared with conventional voltametry, voltametry with continuously changing potential has several disadvantages which limit its usefulness. ... 

We give some estimates of the solution and its derivatives for problem (4.12), (4.11). Suppose that, for a fixed value of the parameter, the solution of the boundary value problem is sufficiently smooth on each set G and has the continuous derivatives... 

The solution of the boundary value problem is continuous on G and sufiiciently smooth on every subdomain Gf The solution can be represented in the form... 

---