Associated systems, initial value problem

In [202] the authors studied the numerical solution of systems of differential equations with oscillatory solutions which is associated with initial value problems which can be written as ... 

Due to these inner iterations via IVP solvers and due to the need to solve an associated nonlinear systems of equations to match the local solutions globally, boundary value problems are generally much harder to solve and take considerably more time than initial value problems. Typically there are between 30 and 120 I VPs to solve numerous times in each successful run of a numerical BVP solver. 

In Section 9.4.1, selected numerical methods are examined for solving the initial value problems associated with first-order differential equations. Those methods are also applicable to higher-order differential equations following the reduction to a system of first-order equations. For example, the second-order differential equation... 

The basis for spectral association is the role played by eigenvalues in describing the dynamics of a linear system. Consider the initial value problem given by ... 

The clogging effect can be considered as a reduction in the value of the surface filtration constant for practical purposes. Indeed, when clogging takes place, the surface filtration constant can be given by its initial value ko multiplied by a decreasing time function. This assumption is frequently used when the function is obtained from experiments [3.19, 3.20]. In our example, if we do not consider the friction (and heat transfer) we can note that only a concrete mass transfer problem can be associated with the membrane separation process. The first step before starting to build the general mathematical model, concerns the division of the system into different elementary sections. Indeed, we have a model for the filtration device (i.e. the membrane and its envelope), for the pump (P) and for the reservoir of concentrated suspension (RZ) (Fig. 3.7). 

These mathematical representations are complex and it is necessary to use numerical techniques for the solution of the initial-boundary value problems associated with the descriptions of fluidized bed gasification. The numerical model is based on finite difference techniques. A detailed description of this model is presented in (11-14). With this model there is a degree of flexibility in the representation of geometric surfaces and hence the code can be used to model rather arbitrary reactor geometries appropriate to the systems of interest. [The model includes both two-dimensional planar and... 

Association of Textile Chemists and Colorists recently [3] described the incorporation of small amoimts of fluorocarbon derivative in a polymeric material normally used to treat textiles for water repellency. They observed that the fluorocarbon preferentially adsorbed at the interfaces and decreased the values to 16 to 18 dynes per cm. Their films clearly showed the ability to self-heal, for when the initially adsorbed layer was deliberately scraped off, additional molecules quickly adsorbed at the interface when the polymer matrix was recured at an elevated temperature. The usefulness of adsorbed films of surface active molecules is thus apparent, and one may expect wide application of this technique to specific problems. The present study, in combination with previous investigations of wettability and surface activity in organic liquids, forms an excellent guide for the design and synthesis of further surface active agents for polymeric systems. 

Now let us return to the integration problem of a system of conjugate differential equations. As mentioned in the preceding Section, when solving the problems with the use of the calculus of variations, usually there is a need to overcome the difficulties associated with finding the initial y/ddo) values. 

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