Boundary value problems computational parameters

If we consider the limiting case where p=0 and q O, i.e., the case where there are no unknown parameters and only some of the initial states are to be estimated, the previously outlined procedure represents a quadratically convergent method for the solution of two-point boundary value problems. Obviously in this case, we need to compute only the sensitivity matrix P(t). It can be shown that under these conditions the Gauss-Newton method is a typical quadratically convergent "shooting method." As such it can be used to solve optimal control problems using the Boundary Condition Iteration approach (Kalogerakis, 1983). 

The two-point boundary value problem for the effective potential (11)—(16) was solved numerically, by using the shooting methods [32], The computations were performed for the following range of parameters r = T / I], = 0.08 —... 

As the next step up in complexity, we consider the case of multiple reactions. Some analytical solutions are available for simple cases with multiple reactions, and Axis provides a comprehensive list [2], but the scope of these is limited. We focus on numerical computation as a general method for these problems. Indeed, we find that even numerical solution of some of these problems is challenging for two reasons. First, steep concentration profiles often occur for realistic parameter values, and we wish to compute these profiles accurately. It is not unusual for species concentrations to change by 10 orders of magnitude within the pellet for realistic reaction and diffusion rates. Second, we are solving boundary-value problems because the boundary conditions are provided at the center and exterior surface of the pellet. Boundary-value problems (BVPs) are generally much more difficult to solve than initial-value problems (IVPs). 

We emphasize that z(x), which is the solution of the difference problem on uniform grids, depends essentially on the two parameters e and N, where the parameter e characterizes the physical essence of the boundary value problem, while N (or h) characterizes the applied numerical method (or the computational resources of our computer). 

Thus, in the case of the boundary value problem (1.16), (1.24), the use of the scheme (1.17), (1.18), (1.25) leads, for small values of the parameter s, to sharp underestimation of the computed normalized flux (and also of the solution gradient) on the boundary. Therefore, even qualitatively, the normalized flux cannot be approximated by the computed flux e-uniformly. 

It is known that, in the case of singularly perturbed elliptic equations for which (as the parameter s equals zero) the equation does not contain any derivatives with respect to the space variable, the principal term in the singular part of the solution is described by an ordinary differential equation similar to Eq. (1.16a) (see, e.g., [3-6]). Thus, it can be expected that, when solving singularly perturbed elliptic and parabolic equations using classical difference schemes, one faces computational problems similar to the computational problems for the boundary value problem (1.16). 

Recall that we are interested in the behavior of the error in the approximate solution for various values of the parameter e. To compute the solution of the boundary value problem (2.16), (2.18), we use a classical finite difference scheme. We now describe this scheme. On the set G the uniform rectangular grid... 

In this section, we consider singularly perturbed diffusion equations when the diffusion flux is given on the domain boundary. We show (see Section III.B) that the error in the approximate solution obtained by a classical finite difference scheme, depending on the parameter value, can be many times greater than the magnitude of the exact solution. For the boundary value problems under study we construct special finite difference schemes (see Sections III.C and III.D), which allow us to find the solution and diffusion flux. The errors in the approximate solution for these schemes and the computed diffusion flux are independent of the parameter value and depend only on the number of nodes in the grid. 

If, for a fixed value of the parameter e, the solution of the boundary value problem has the continuous derivatives (4.24), where K = 4, the computed normalized diffusion flux P (x, t) converges to the actual flux P x, t) e-uniformly on G, ... 

The finite element method (FEM) has become the dominant computational method in structural engineering. In general, the input parameters in the standard FEM assume deterministic values. In earthquake engineering, at least the excitation is often random. However, considerable uncertainties might be involved not only in the excitation of a structure but also in its material and geometric properties. A rational treatment of these uncertainties needs a mathematical concept similar to that underlying the standard FEM. Thus, FEM as a numerical method for solving boundary value problems has to be extended to stochastic boundary value problems. The extension of the FEM to stochastic boundary value problems is called stochastic finite element method (SEEM). 

An algorithm for computing the decision boundary thus requires the choice of the kernel function frequently chosen are radial basis functions (RBFs). A further input parameter is the priority of the size constraint for used in the optimization problem (Equation 5.38). This constraint is controlled by a parameter that is often denoted by y. A large value of y forces the size of to be small, which can lead to an overht and to a wiggly... 

Therefore, we use POLYMATH to solve some examples in this textbook. With POLYMATH one simply enters all equations and the corresponding parameter values into the computer with the initial (rather, boundary) conditions and they are solved and displayed on the screen. It is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation to obtain an analytical solution of the problem. The basic procedure for reactor modeling and simulation is shown in Figure 13-26. 

---