Mathematical modeling weighting

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

It cannot be stated generally how accurate the predicted results are. Due to the limitations of geometric, physical, and mathematical modeling, not all of the produced numbers (e.g., air velocity vectors) are at a high level of accuracy, and the results are therefore subjected to experienced weighting. In some cases, the values can be as accurate as within 5% of the real values in other cases, they are not as accurate as could be wished. But results can be still very strong and helpful in a comparative judgment, i.e., if a number of similar case.s are compared with observed tendencies. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

The algebraic/iterative and the brute force methods are numerical respectively computational techniques that operate on the chosen mathematical model. Raw residuals r are weighted to reflect the relative reliabilities of the measurements. 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

A mathematical model has been developed which allows the calculation of the degradation of polymeric drug delivery systems. The model has been shown to accurately simulate both the drug release and molecular weight changes in such systems. The concentration of anhydride levels affect the erosion characteristics of... 

Detailed studies on the lipase-catalyzed polymerization of divinyl adipate and 1,4-butanediol were performed [41-44]. Bulk polymerization increased the reaction rate and molecular weight of the polymer however, the hydrolysis of the terminal vinyl ester significantly limited the formation of the polyester with high molecular weight. A mathematical model describing the kinetics of this polymerization was proposed, which effectively predicts the composition (terminal structure) of the polyester. 

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

No explicit mathematical model of the method was presented. However, a short descriptive model was outlined For each run, the average ignition and combustion rate (expressed as weight of fuel ignited or burnt per unit bed area and unit time) were calculated by determining the time taken for the ignition front to pass down through the bed and the completion of burn-out, respectively. No discussion is presented about limitations and assumptions of the method. 

For concentrated solutions of amorphous polymers, Bueche s mathematical model shows the ratio of zero shear viscosities of branched and linear polymer above the critical molecular weight in the entanglement region to be (28) ... 

The objective function of the mathematical model involves the minimization of the total annualized cost which consists of appropriately weighted investment and operating cost. It takes the form ... 

A complete description of the process must consist of kinetic equations for all components of the reactive mass, including all fractions of different molecular weights and intermediate and byproducts as well. Such an exact approach is usually superfluous for modelling any real process and should not be applied, because excessive detail actually prevents achievement of the final goal due to overcomplicating of the analysis. Therefore, why correct (necessary and sufficient) choice of the parameters for quantitative estimation is of primary importance in mathematical models of a technological process. 

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