Shooting techniques

It may be noted that Eq. 5.45 could be solved as a third-order equation with a shooting technique. The equation could be also be transformed into a first-order and a second-order equation, which are solved as discussed in Section 5.2.2. In these cases the boundary conditions manifest themselves in somewhat different ways. 

Using a shooting technique, write a simulation to solve the nondimensional semiinfinite, stagnation-flow problem (i.e., Eq. 6.59). The first-order systems may be solved on a fixed mesh using an Euler method. 

Solve the coupled problem by a shooting technique. Take the Prandtl number to be Pr = 5. For constant-property, incompressible flow, note that the flow problem is not directly affected by the thermal solution. However, the thermal problem is affected by the flow through the velocity in the convective term. 

The model equations can be solved by a shooting technique start with an initial guess X for the conversion x(l), calculate the recycle flow rate/j from Eq. (4.37) and reactor-inlet concentration from Eq. (4.38), integrate the PFR Eqs. (4.34) and (4.35), check and update the guess X. This implies that it is theoretically possible to reduce the model to one equation with one variable ... 

Sometimes, instead of an initial value problem, the mathematical model of a chemical process is a boundary value problem in which values of the dependent variables are specified at different values of the independent variable t. The shooting technique consists of solving an initial value problem, but with an initial value vector a considered as a parameter to estimate (by optimization techniques) so that boundary conditions are satisfied. In this way, a boundary value problem is transformed into an initial value problem. 

Even with this simplification, we still must solve a split boundary value problem. A particularly convenient method of solution appears to be a "shooting" technique in which the... 

A number of one dimensional computer models have been developed to analyze thermionic converters. These numerical models solve the nonlinear differential equations for the thermionic plasma either by setting up a finite element mesh or by propagating across the plasma and iterating until the boundary conditions are matched on both sides. The second of these approaches is used in an analytical model developed at Rasor Associates. A highly refined "shooting technique" computer program, known as IMD-4 is used to calculate converter characteristics with the model ( ). 

No such closed-form solution exists for the more general case v / 0. The general form of Eq. 18 can be solved for small values of the deposition modulus, [3, though as we will see later, such solutions are not applicable to the problem of interest." Fortunately, very accurate numerical solutions to the boundary value problem posed by Eqs. 8 and 18 are readily obtained using a numerical shooting technique. 

The shooting technique involves converting the given boundary value problem to a system of initial value problems. The unknown initial conditions are guessed. These unknown conditions are then updated using the known boundary condition at X = 1. In this technique, the unknown initial condition at x = 0 is estimated using an optimization procedure. This is best illustrated using the next example. 

The catalyst pellet problem solved in example 3.2.2 is solved here using the shooting technique. The Maple program is given below ... 

Hence, we observe that the shooting technique can predict three multiple states in a catalyst pellet. The number of iterations required to obtain a converged solution depends on the initial guess and the scaling factor p. 

Note that with the shooting technique we obtain exact results. The finite difference solution is not as smooth as the shooting technique solution. 

In section 3.2.4, nonlinear boundary value problems were solved using shooting technique. The given boundary value problem was converted to a system of initial value problems. The unknown initial condition was obtained using an iteration and optimization procedure. This is a very robust technique and can be used to solve stiff boundary value problems. This technique is capable of predicting multiple steady states in a catalyst pellet. However, the number of iterations required for convergence can be prohibitively large for certain boundary value problems. 

Redo problem 6 using Maple s dsolve numeric command and shooting technique. 

Solve the Blasius equation (example 3.2.10) using the shooting technique. 

From problem 18, choose a value of O for which there are multiple steady states. For the chosen value of O, predict the multiple steady state concentration profiles using Maple s dsolve numeric command and shooting technique. 

Analyze problem 21 for multiple steady states. To do this, solve this problem using the shooting technique for the given set of parameters. Consider the behavior of a thin sheet of viscous liquid emerging from a thin slot at the base of a converging channel in connection with a method of lacquer application known as curtain coating. [6] The dimensionless governing equations and boundary conditions for the velocity are ... 

Solve this problem using Maple s dsolve numeric command, shooting technique and finite difference technique. Initially choose L = 5 and increase L to make sure that the solution has converged (i.e., change L = 6 dy... 

Obtain the first five eigenvalues and eigenfunctions using the shooting technique described in example 3.2.15. 

Should the calculated profiles in the enriching section be erroneous due to uncertainties in the boundary conditions the correct profiles can be found by employing a shooting technique. The input data for the simulation are again based on measurements subject to experimental error. By perturbing one or more values within the bounds of experimental error, the enriching section... 

Non-linear two point boundary value differential equations arise in fixed bed catalytic reactors mainly in connection with the diffusion and reaction in porous catalyst pellets. It may also arise in the modelling of axial and radial dispersion in the catalyst bed. In addition they also arise in cases of counter-current cooling or heating of the reactor. For this last case, the use of a shooting technique with an iterative procedure similar to the Newton method (Fox s method) seems to be the easiest and most straightforward technique (Kubicek and Hlavacek, 1983). 

Table 6.9 presents detailed simulation results for an industrial ammonia converter formed of three beds with interstage cooling between the beds. The simulation results are presented for both the empirical and the diffusion-reaction approaches for computing t]. For the diffusion-reaction approach two techniques are used for the solution of the two point boundary value differential equations, namely the shooting technique and the more efficient orthogonal collocation technique. 

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