Useful Taylor Series Expansions

Equation (C.6) is the binomial series. When p is a positive integer all the terms after the (p + 1) equal zero, and the series becomes hnite  

---

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

The plot in Fig. 3.2 of the acid dissociation constant for acetic acid was calculated using equation 3.2-21 and the values of standard thermodynamic properties tabulated by Edsall and Wyman (1958). When equation 3.2-21 is not satisfactory, empirical functions representing ArC[ as a function of temperature can be used. Clark and Glew (1966) used Taylor series expansions of the enthalpy and the heat capacity to show the form that extensions of equation 3.2-21 should take up to terms in d3ArCp/dT3. 

But over wider ranges of temperature, ArS ° and AtH ° are functions of temperature. Clarke and Glew (1966) have used Taylor series expansions of the enthalpy... 

Using Taylor series expansion, find the forward second-order accurate finite difference expansion for the first derivative of the... 

The error analysis is conveniently done by using Taylor series expansions. For the concentrations at the four points of the grid which are the neighbors of the point... 

Error estimate for Euler method) In this question you ll use Taylor series expansions to estimate the error in taking one step by the Euler method. The exact... 

Using Taylor-series expansions about the collision contact point. 

This assumption is usually verified by expanding the numerical and exact solutions in powers of h, using Taylor series expansions. 

To second-order approximation, the Taylor series expansion is fix) = 1 - hx -I- ibx)-/2, showm in Figure 4.2(c). Taylor series expansions for various functions are given in Appendix C, Useful Taylor Series Expansions. 

EXAMPLE 4.5 Two first-order approximations. Find a first-order approximation for 1/(1 + x). From Appendix C, Useful Taylor Series Expansions, use Equation (C.6) with p = -1 to get... 

For small values of r (typically less than 15 mV at room temperature), the exponential term can be approximated using Taylor series expansion as e 1 + X, for small x. For small n, then Equation 5.77 can be written as... 

A common approach in the various finite difference methods used to integrate the equations of motions for classical molecular dynamics simulations is that it is assumed that the positions, velocities, and accelerations (as well as all other dynamic properties) can be approximated using Taylor series expansions ... 

Interval Analysis Using Taylor Series Expansion... 

An approximate objective function/ using Taylor series expansion up to second order around the nominal model can be expressed as... 

In the method using Taylor series expansion as explained in previous sections, the accuracy of the robustness evaluation depends directly on the reliability of the numerical sensitivity analysis. For this reason, when the evaluation of numerical sensitivities has some difficulties resulting from the elastic-plastic structural property of isolators, another URP method should be introduced where the variation of the objective function is... 

---