Boundary shooting methods

With these two-point boundary conditions the dispersion equation, Eq. (23-50), may be integrated by the shooting method. Numerical solutions for first- and second-order reaciions are plotted in Fig. 23-15. 

Shooting methods attempt to convert a boundary value problem into an initial value problem. For example, given the preceding example restated as an initial value problem for which... 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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). 

Integration is started with known values of the dependent variables at one value of the independent variable, except when the "shooting method" is needed. Auxiliary algebraic equations can be entered to the program along with the differential equations and the boundary conditions. 

PI.05.02. SHOOTING METHOD FOR EQUATIONS WITH TWO-POINT BOUNDARY CONDITIONS... 

This two-point boundary condition problem is solved by the shooting method with this procedure ... 

Because the boundary conditions are at two points, the shooting method is applicable ... 

This is a two-point boundary value problem to which the "shooting method" is applicable, according to this procedure ... 

Solution by Shooting Solution of the boundary value problem described by Eq. 6.59 is usually accomplished numerically by a shooting method. To implement a shooting method, the third-order equations is transformed to a system of three first-order equations as... 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

Shooting method for bifurcation analysis of boundary value problems (with X. Song and L.D. Schmidt). Chem. Eng. Commun. 84,217-229 (1989). 

One of the simplest algorithms to solve a boundary value problem is the "shooting method." In this method, we assume initial values needed to make a boundary value problem into an initial value problem. We repeat this process until the solution of the initial value problem satisfies the boundary conditions. Therefore, proper initial conditions for the solution of the preceding problem are... 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

Instead of purely initial conditions, a mixture of initial and final conditions may be used [77—79]. However, such boundary value problems can, at least theoretically, be tranformed into initial value problems by having recourse, for example, to a shooting method (see Sect. 4.5). 

Shooting Methods The first method is one that utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Consider the nonlinear boundary value problem... 

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 —... 

The usual technique is to perform a numerical integration of Eq. (114) using boundary conditions at r = 0 and searching for the eigenvalue Er(X) by the shooting method applied iteratively until the boundary condition at r = R is obtained. For the potential equation (129), we will write down the exact transcendental equation for the energy for all values of R, 8, and X, and no numerical integration is needed. 

To convert a nonlinear boundary-value problem like (10 71), (10 72), and (10-75) to an initial-value form suitable for solution by means of standard numerical methods, the most common approach is to use a shooting method. In this method, we first guess a value of f" at rj = 0, say,... 

Similar equations can be written for the other components and for the enthalpy flux. An equation for the pressure drop is also required. The boundary conditions are of the Danckwerts type if the system is closed and are those of Example 9.2 for an open system. The shooting method of solution can be used for the closed system. The open system is more easily solved using the method of false transients described in Chapter 16. 

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