Polynomials mathematical modeling

Therefore, research focused on the development of new membrane materials and up-scaling for production is still important. In that direction, and in order to understand better the membrane behaviour of PEO-PBT copolymers, the CO2 permeability (Table 12.1) were fitted to polynomial mathematical models (Equation 12.2), and plotted versus the molecular weight (M ) of the PEO block, as shown in Figure 12.2. 

In the basis of Placket-Burman, using the steepest ascent experiment determined the Central Composite Design s experiment design center, and the 3 significant factors of inulin added amount, peptone content and liquid medium volume were optimized through response surface, therefore established the Endoinulinase activity s Quadratic polynomial mathematical model and anal) ed... 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

Lack of fit of the obtained model has to be statistically checked, so that, if needed, the polynomial degree may be augmented. Knowing the mathematical model of the research subject for several responses is a prerequisite in solving optimization with multiple responses. The computation of this is solved geometrically or by use of computers and the method of linear algebra. 

When several mathematical models fulfill the requirements, then, in principle, a simpler model is accepted. For us, it is the polynomial model that in the case of two factors may have these forms ... 

Apart from optimization, a problem is often set for mathematical modeling or interpolation. The optimum does not interest us in that case but the model that adequately describes the obtained results in the experimental field. A subdomain is not chosen in that case, but the polynomial order is moved up until an adequate model is obtained. When a linear or incomplete square model (with no members with a square factors) is adequate it means that the research objective corresponds to the optimization objective. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

The approach of using a mathematical model to map responses predictively and then to use these models to optimize is limited to cases in which the relatively simple, normally quadratic model describes the phenomenon in the optimum region with sufficient accuracy. When this is not the case, one possibility is to reduce the size of the domain. Another is to use a more complex model or a non-polynomial model better suited to the phenomenon in question. The D-optimal designs and exchange algorithms are useful here as in all cases of change of experimental zone or mathematical model. In any case, response surface methodology in optimization is only applicable to continuous functions. 

The non existence of oscillation in the proposed discretization allied with the robustness of the computer code indicate that the association of these techniques seems to be adequate to solve the mathematical model of gasoil cracking. Other discretization methods, such as global polynomial approximation [17], could bring about oscillatory profiles through the bed length. 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

Modeling the Experimental Data. The data collected from each experiment in a given experimental design can be mathematically modeled so that the response, such as migration time, resolution, and so on, can be correlated with the experimental conditions that produced it. This way, by using the model, the desired output can be maximized and the corresponding experimental conditions defined in a predictive manner. Frequently, data are fitted to quadratic polynomial functions similar to Equation 5.7,... 

Mathematical modeling was applied in interpreting textural properties (surface area, pore volume, average pore diameter) as the functions of temperature and time as two independent variables. Based on an analysis of shapes of the surfaces, Sp(x, t), Vp(x, t) and R(x,t), different polynomials models have been tested and the simplest (linear in terms of time but quadratic in terms of heating temperature) was accepted y(x,t) = bi + b2X + bst + bft ... 

A study of the principal ABS plastics on the market was made in order to determine the relations between some main physical-mechanical properties and composition specifications. The statistical methodology of multiple regression was used in the investigation. The mathematical models obtained for most of the tested properties were well explained by second order and linear polynomials. The expansion of mathematical models enabled us to calculate the best estimates expected for the physical properties whenever the ABS compositions were known. The plotting of these polynomials obviously represents a qualitative picture of the physical properties-composition relations in the wide experimental range of the variables. 

Whether it is to understand or interpret a phenomenon, as in the factor study (chapter 3), or to predict results under different conditions by response surface modelling (as we will see in the next chapter), we require a mathematical model that is close enough in its behaviour to that of the real system. The models we u.se are polynomials of coded (normalised) variables, representing factors that are all transformed to the same scale and with constant coefficients. It is the unknown value of each coefficient that must be estimated with the best possible precision by experiments whose position in the experimental factor space is chosen according to the form of the mathematical model postulated. For this to be possible, the number of distinct experiments (that is not counting replications) must be at least equal to the number of coefficients in the model. 

---