Fitting Polynomials Using Taylor Expansions

A kind of cross-over between tabulation and pol5tnomial storage methods is the application of a collection of Taylor expansions. The exact values are tabulated at some fixed points x of the input vector, but the values in between the tabulated points are determined not by linear interpolation but according to the following Taylor expansion  

Here T,(x) is the stored exact value, Ax is the deviation of the queried point from the stored point, and d YJdxj and d YJdx dxj are the first-order and second-order local sensitivity coefficients, respectively. There are several efficient numerical methods for the calculation of the first-order local sensitivity coefficients (see Sect. 5.2). The second-order local sensitivity coefficients can be calculated from the first-order coefficients using a finite-difference approximation. The Taylor series approximations have the general disadvantage that the accuracy significantly decreases further from the central point. 

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Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

The above formulation produces a set of differential and algebraic equations. These equations are solved by utilizing a Gear predictor-corrector algorithm [16-19]. The predictor operations estimate the system response at the ne rt time step (n+1) based on the response at previous points. A polynomial of order k is fitted to the previous values of the system states of each of the generalized coordinates of the system, and the polynomial thus calculated is used to evaluate a truncated Taylor series expansion to obtain the system response at time n+1... 

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