Difference problem well-posed

On the concept of well-posedness for a difference problem. There is another matter which is one of some interest. In conformity with statements of problems of mathematical physics, it is fairly common to call a problem well-posed if the following conditions are satisfied ... 

We say that a difference problem (scheme) is well-posed if for all sufficiently small < /ly... 

When the parameters differ by more than one order of magnitude, matrix A may appear to be ill-conditioned even if the parameter estimation problem is well-posed. The best way to overcome this problem is by introducing the reduced sensitivity coefficients, defined as... 

When the parameters differ by several orders of magnitude between them, the joint confidence region will have a long and narrow shape even if the parameter estimation problem is well-posed. To avoid unnecessary use of the shape criterion, instead of investigating the properties of matrix A given by Equation 12.2, it is better to use the normalized form of matrix A given below (Kalogerakis and Luus, 1984) as AR. 

Some gum users now take gum in a pre-prepared form. Spray dried gum acacia has been used in pharmaceutical products for some time. The spray dried gum offers the pharmaceutical manufacturer a clean ready to use product. Instant forms of gum acacia have been offered by suppliers for some time. The instant products can be rapidly made into solution and used. Obviously the instant gum is more expensive. A manufacturer that uses gum as a minor ingredient may well find that the capital and labour cost of purifying raw gum is not cost effective. A company that uses gum acacia as a major ingredient might come to a different conclusion. Instantised gums pose different problems to the analytical chemist.. One approach that can be used is to have an optical rotation specification for the product. Even this approach is not entirely proof against a material that contains a blend of gums of different optical rotation. 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

The introduction of biological and chemical threat agents into bodies of water poses several different problems. The nature of the problems posed will depend on the biological and physical properties of the threat agent used and the nature of the water body. To a large extent, the chemical characteristics of the biological and chemical agents (see Table 4.1) and the water quality parameters of a water body control solubility of the material as well as the sorption of the material to soil particles (suspended or bed sediments). Many of the transport phenomena are discussed in Chapter 3. 

Enzymes have many potential advantages when used as catalysts for chemical synthesis. The unique properties offered by these biocatalysts are, first of all, their often outstanding chemo-, regio-, and, in particular, stereoselectivity. Furthermore, enzymes are highly efEcient catalysts working under very mild conditions. However, enzymes do also have some drawbacks that may limit their potential use, such as ability to accept a limited substrate pool only, and a moderate operational stability. Ways of overcoming most of these potential limitations exist and they pose in most cases more of a perceived than a real problem. Well over 100 different biocatalytic processes have been implemented on an industrial scale [1]. A few processes are... 

Unfortunately, for a given domain of interest boundary conditions can be chosen that over- or underspecify the problem. An example of an over-specified problem is a constant area duct, with a fixed fluid velocity at the inlet and but different at the outlet. Naturally, for an incompressible fluid both conditions cannot be physically satisfied in the absence of a mass source. In the same sense, a closed box with only heat flux boundary conditions is under specified since the temperature level is not constrained, and therefore unpredictable. It becomes clear that defining well posed boundary conditions is quite important for the proper solution of a flow problem. An easy way to check for well posed boundary conditions is to ask yourself Could the chosen configuration be physically recreated in the laboratory . 

There may be several different vrays to model a well-posed physical problem mathematically. Some of these formulations may be sensitive to small perturbations, whereas others may not. 

In the table the second, third, and fourth problems each result from a permutation of the known and unknown quantities that occur in the bubble-T calculation. We refer to these as P-problems, because each problem is well-posed when values are specified for P independent intensive properties, where the value of T is given by the phase rule (9.1.14). However, the flash problem in Table 11.1 differs from the others in that it is an P -problem it is well-posed when values are specified for T independent intensive properties, with the value of T given by (9.1.12). Flash calculations pertain to separations by flash distillation in which a known amount N of one-phase fluid, having known composition z, is fed to a flash chamber. When T and P of the chamber are properly set, the feed partially flashes, producing a vapor phase of composition xP in equilibrium with a liquid of composition x ). The problem is to determine these compositions, as well as the fraction of feed that flashes NP/N. Unlike the other problems in Table 11.1, the flash problem involves the relative amounts in the phases and therefore a solution procedure must invoke not only the equilibrium conditions (11.1.1) but also material balances. 

It is commonly believed that a correct mathematical presentation of physical situations ought to result in properly posed problems. In two-phase flow problems, however, the existence of an assumed physical situation, e.g., stratified wavy flow configuration, is not certain under all operational conditions. Therefore, ill-posedness in some domains of the parameters space does not necessarily imply that the formulation is globally incorrect. Moreover, the boundary of the well-posed domain may have physical significance since it signals the existence of additional physical features which the original model neglects. When these features become consequential, one expects a different physical behavior, such as transition to a different flow pattern, and a different model is required to simulate this transition. 

The equations (4.5.2a) with (4.5.1) are bilinear in the variables yi and mj. The whole set of constraints generally does not determine a unique solution, but only what is called an (analytic) solution manifold, a (mathematically smooth, but curved ) subset of the variables space. Only if a sufficiently great number of variables is measured and adjusted so as to make the system solvable, the remaining variables can be computed. We shall not attempt to analyze the problem in the same (mathematically rigorous) manner as in Chapter 3. An example in Section 5.5 will show that such problems may sometimes also be not well-posed . For a different approach, see further Chapter 8. 

The methodological and design issues are still of relevance today, as differences in design, or methodological variations or shortcomings, may be responsible for the differences in results between different studies. The methodological and design issues are discussed below, as well as some relevant theoretical points, while the problems they pose in interpretation are considered later. 

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