Of finite differences

Fig. 11.28 Focusing can improve the accuracy of finite difference Poisson-Boltzmann calculations.

Applications of Finite Difference Poisson-Boltzmann Calculations... 

The well-known theorem of finite differences was used here. In particular, G (A, 1), 2 G (0, A), and therefore > 2- At the same time... 

A numerical solution of these equations maybe obtained in terms of finite difference equivalents, taking m radial increments and n axial ones. With the following equivalents for the derivatives, the solution maybe carried out by direct iteration ... 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

This approximation is called a forward difference since it involves the forward point, z + Az, as well as the central point, z. (See Appendix 8.2 for a discussion of finite difference approximations.) Equation (8.16) is the simplest finite difference approximation for a first derivative. 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

Gelfand, A. (1967) Calculus of Finite Differences. Nauka Moscow (in Russian),... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

If for example we discretize the region over which the PDE is to be solved into M grid blocks, use of finite differences (or any other discretization scheme) to approximate the spatial derivatives in Equation 10.1 yields the following system of ODEs ... 

We now consider how to compute the variance of AT, according to equation 44-68a. Ordinarily we would first discuss converting the summations of finite differences to... 

The absorbance spectrum in Figure 54-1 is made from synthetic data, but mimics the behavior of real data in that both are represented by data points collected at discrete and (usually) uniform intervals. Therefore the calculation of a derivative from actual data is really the computation of finite differences, usually between adjacent data points. We will now remove the quotation marks from around the term, and simply call all the finite-difference approximations a derivative. As we shall see, however, often data points that are more widely spread are used. If the data points are sufficiently close together, then the approximation to the true derivative can be quite good. Nevertheless, a true derivative can never be measured when real data is involved. 

The difference between this result and the thermodynamic perturbation result is that in the former (eq. (11.28)) the average is taken of (finite) differences in energy while eq. (11.31) averages over a differentiated energy function often the required derivative of the energy with respect to the coupling parameter can be obtained analytically and the averaging involved here is no more complicated than with eq. (11.28). 

In this book PDEs appear primarily in Section 8.1 and problem section P8.01. Some simpler methods of solution are mentioned there Separation of variables, application of finite differences and method of lines. Analytical solutions can be made of some idealized cases, usually in terms of infinite series, but the main emphasis in this area is on numerical procedures. Beyond the brief statements in Chapter 8, this material is outside the range of this book. Further examples are treated by WALAS (Modeling with Differential Equations in Chemical Engineering, 1991). 

Thus, the dimensionless current-potential curves depend on the dimensionless parameters 1, A, A , oq, and a2. Simulating the dimensionless cyclic voltammograms then consists of finite difference resolutions of equations (6.57) and (6.58), taking into account all initial and boundary conditions. Examples of such responses are given in Section 2.5.2 (Figure 2.35). 

As initial distribution corresponds to the linear mode (2.11) of the given waveguide, the deviation of T z) with respeet to unity may he eonsidered as a measure of the error in this method. The results presented in Fig.2 allow one to analyze the accuracy of the method depending on the type of finite-difference scheme (Crank-Nicholson" or Douglas" schemes have been applied) and on the method of simulation of conditions at the interface between the core and the cladding for both (2D-FT) and 2D problems. 

Figure 5.1 Vibrational frequencies of gas phase CO computed using DFT as a function of finite difference displacement, 8b, in angstroms.

Ong KG, Varghese OK, Mor GK, Grimes CA (2007) Application of finite-difference time domain to dye-sensitized solar cells The effect of nanotube-array negative electrode dimensions on light absorption. Solar Energy Materials Solar Cells 91 250-257... 

The solution of Eq. (2) can also be obtained by a numerical analysis similar to the calculus of finite differences. However, an analytical or semianalytical method based on Eq. (2) is not suitable for discussing the time-dependent distribution function because the calculation is lengthy. 

At each temperature, the method of finite differences was used and j evaluated for a series of x. The plot of ( x/ t)/l — x = k vs. x closely approximated a straight line (Figure 8). It is therefore considered that the rate constant, k, changes linearly with x, the fraction extracted. 

In these equations Vcr/v- rr, the residence time in tank r. Equation 1.46 may be compared with equation 1.37 for a tubular reactor. The difference between them is that, whereas 1.37 is an integral equation, 1.46 is a simple algebraic equation. If the reactor system consists of only one or two tanks the equations are fairly simple to solve. If a large number of tanks is employed, the equations whose general form is given by 1.46 constitute a set of finite-difference equations and must be solved accordingly. If there is more than one reactant involved, in general a set of material balance equations must be written for each reactant. 

One category of finite-difference method uses a rectangular grid. In this approach one covers the specified layout with a grid, or mesh, as shown in Figure i 15.2a. When curvilinear boundaries are involved, it is possible to sample the... 

The system of Eqs (4.10) - (4.13) was solved numerically by the method of finite differences, starting with Eq. (4.11) at the nodes of the network with P = 0.8. The process was assumed to be over when the minimum value of the "rheological" decree of conversion throughout the volume of the article had reached a preset level of conversion, q the calculations were ended at this time. 

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]. 

Taylor-series expansions allow the development of finite differences on a more formal basis. In addition, they provide tools to analyze the order of the approximation and the error of the final solution. In order to introduce the methodology, let s use a simple example by trying to obtain a finite difference expression for dp/dx at a discrete point i, similar to those in eqns. (8.1) to (8.3). Initially, we are going to find an expression for this derivative using the values of

backward difference equation). Thus, we are looking for an expression such as... 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

In performing calculations we are confronted with the situation that although we have no heat losses to the wall the adiabatic reactor has to be described by two dimensional differential equations The numerical solutions were obtained on a Cyber 175 with the method of finite differences. 

The variation in gas remaining in the solid is neglected since it is small compared to the amount of volatiles outflow. The equations are solved in dimensionless form by codes (21) using the method of finite differences. 

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