Numerical methods shooting techniques

The solution of the concentration profile ( >(z) should be specified for given temperature T, film thickness D, and the average blend composition in this film <(( . The parameters T and <( , important in experiments, might be translated [60] into interaction parameter % and the chemical potential difference Ap.more convenient in calculations. Thus, for say D, %, and Ap known and kept constant, the profile ( >(z) may be obtained (Eq. 50) by varying the reservoir concentration (]>b until the boundary conditions (Eq. 51) are met. If a few solutions exist, the relevant ones are those with minimal overall free energy F (Eq. 49). Such a shooting procedure was developed by Flebbe et al. [60]. A numerical method which starts from an arbitrary assumed profile and modifies its discretized form until conditions equivalent to Eqs. (50), (51) and (53) are met has also been proposed recently by Eggleton [222]. The solutions yielded by this technique may however correspond to metastable states. Concentration profiles in thin films were also evaluated by other theoretical treatments [93,118,177,219,221]. 

Three basic numerical techniques are introduced. These are shooting techniques, quasilinearization with the use of the principle of superposition, and the method of adjoints. 

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. 

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. 

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