Shooting method

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

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. 

The three preceding equations may be solved simultaneously by the shooting method. A result for a first-order reaction is shown in Fig. 23-20, together with the case of uniform poisoning. 

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. 

Second order equations or a pair of first order equations require two conditions. When these are at the same value of x, say y0 and (dy/dx)0 at x0, a numerical integration can be started. When one condition is y0 at x0 and the other is (dy/dx)L at xL the shooting method can be applied. This consists of several steps ... 

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 ... 

The shooting method of solution is employed. Values of f at z = 1 are tried until one is found that results in satisfaction of the requirement at the other end after backward integration. 

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

The usual shooting method can be applied to the solution of the last three... 

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

Numerical solution of Eq (1) by the shooting method automatically gives the derivative at the external surface required in Eq (2). 

The effectiveness of an enzyme immobilized on a porous spherical pellet is to be calclulated with = 5 and (f> - 4. The shooting method for solving Eq (1) of problem P8.04.15 is described in problem P7.03.09. Several trial values of f0 at the center and the corresponding values at the surface, fx and [df/dp)x> are tabulated. ODE is applied to the equivalent pair of equations... 

The next level beyond the step-and-shoot method is the helical-scan approach. The conveyor belt of a heftcal-scan system moves continuously at a uniform speed and is synchronized with the rotating disk that holds the X-ray source and detectors. The synchronization ensures that by the time a bag has advanced by... 

Fig. A5.1. Construction of a Poincare section in the phase plane illustrating the shooting method of locating limit cycle solutions.

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... 

Discuss the pro s and con s of the shooting method compared to solving the equations as a coupled system wherein finite-difference forms of the momentum and energy equations are used (i.e., as discussed in Section 6.5). 

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). 

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