Generalized shooting technique

Let us use the generalized shooting technique to solve for the backward integration of the tubular reactor with dispersion. The describing differential equations and boundary conditions are given by equations (7.1.9) to (7.1.12). The state functions are... 

Develop the generalized shooting technique for solving this problem. Write down the shooting system dynamics for this problem and the updating equations for unknown initial conditions. Discuss a computer implementation. 

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 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. 

Our definition of a committor in Eq. (1.107) is applicable to both stochastic and deterministic dynamics. In the case of deterministic dynamics, care must be taken that fleeting trajectories are initiated with momenta drawn from the appropriate distribution. As discussed in Section III.A.2, global constraints on the system may complicate this distribution considerably. The techniques described in Section III.A.2 and in the Appendix of [10] for shooting moves may be simply generalized to draw initial momenta at random from the proper equilibrium distribution. 

A popular method for the investigations of crystallization kinetics is to rapidly quench the sample at a temperature below the melting point and then perform an isothermal crystallization experiment. Apart from the scientific merits, there is obviously an interest in such temperature quench techniques from an industrial point of view. The temperature control in a fast quench does not in general exist and only the start and final temperature can be controlled (109). An elegant way to perform a temperature quench experiment is to keep the sample above the beamline in a furnace at an elevated temperature and then, shoot, it into a second fiimace which is mounted on the beamline (101). A time resolution of 0.1 s/frame is feasible at present (109). It should be stressed that the limitations here are more in the temperature control than in the SR beamlines. 

Example 5.4 Flow of a Non-Newtonian Fluid. Write a general MATLAB function for solution of a boundary value problem by the shooting method using the Newton s technique. Apply this function to find the velocity profile of a non-Newtonian fluid that is flowing through a circular tube as shown in Fig. E5.4a. Also calculate the volumetric flow rate of the fluid. The viscosity of this fluid can be described by the Carreau model [5] ... 

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