Method of finite differences

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. 

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. 

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. 

At this stage, we need to discuss the actual task of calculating the Jacobian matrix J. It is always possible to approximate J numerically by the method of finite differences. In the limit as Akz approaches zero, the derivative of R with respect to k, is given by Equation 7.16. For sufficiently small Ak the approximation can be very good. 

One also calculates by the method of finite differences the vectors I nfEpili) and r Epiyi) of derivatives of Rmep and Pmep with respect to y. For each value y, the Hamiltonian is then rewritten in terms of the new set of conjugate variables... 

TTie solution of the 10 partial differential equations allows to determine temperature, composition and velocity- as a function of both time and space. The system of equations is solved by the method of finite difference. 

Various numerical methods are used to solve Laplace s equation for ECM including the method of finite differences, the finite element method, the boundary element method [9, 43, 44], and so forth. 

The quantity 77 is called the absolute, or chemical, hardness. From the method of finite differences we obtain the operational definition of 77 ... 

In order to minimize the deviations from nonlinearity in the computation of ACu by this method of finite differences, the time interval At must be short, such that each successive concentration increment always amounts to no more than a few percent of the concentration in the upper layer Ctt. The relative increment ACu/Cu (in percent) as a function of the dissolved solids concentration in the upper mass Cu is shown in Figure 7 for time increments At = 100 yr and At — 50 yr. For the concentration of the upper water mass in the range from 80 to 327 grams/liter, the concentration increments computed for 50-year time intervals are always under 10% of the concentration in the upper mass. For 100-year intervals, the corresponding increments are twice as large. The slight increase... 

Most commonly situations of this kind are attacked by variants of the method of finite differences. A numerical model of the electrochemical system is set up within a computer, and the model is allowed to evolve by a set of algebraic laws derived from the differential equations. In effect, one carries out a simulation of the experiment, and one can extract from it numeric representations of current functions, concentration profiles, potential transients, and so on. 

It is appropriate to notice that there are several very well-developed numerical techniques allowing us to solve this problem, such as the method of integral equations, the method of finite differences, and others. 

The method of finite differences enters the family of the first strategy with the overlapping of two points between adjacent elements and the use of the first alternative (elements with different size) with one single internal support point (three points for each element). 

If the first derivative is approximated with the most accurate formula that uses all three points, in practice only the two extreme points are used and the central point is skipped. In case of very large f ( ), the information to properly estimate the value of y at the central point is lost For this reason, all the programs based on the method of finite differences approximate the first derivative using a formula... 

The disadvantage to the method of finite differences is that it is unavoidably of... 

The main difficulty when working with thin conducting polymer membranes is the lack of quantitative theory of ion diffusion within the membrane. Various theoretical schemes and approximations have been suggested, but the most difficult problem seems to be in the analytical solution or even approximation for the boundary problem of the combined Nernst-Planck and Poisson equations. The latter equation comes from the fact that electroneutrality cannot be assumed to prevail inside the thin membrane. Doblhofer et al." have made an attempt to solve the problem numerically, but even then certain initial approximations were made. Also the brute force method of finite differences does not allow to see clearly the influence of different parameters. 

Currently, numerical methods are most used to solve heat transmission problems. The method of Finite Differences is being substituted by the Finite Element Method. Most Finite Element based mechanical calculation codes include the Thermal Analysis. The temperature distribution obtained from the thermal calculation is used as a load input to the mechanical stress and deformation problem. For that, the temperatures at the nodes are transformed into initial strain by means of the equation... 

Still, if the Parr-Pearson or chemical hardness is abstracted from Eq. (4.250) as the derivative of the Parr s electronegativity (3.1) (Parr Pearson, 1983), as in Eq. (3.3) presented, an operational definition of it from the method of finite differences ean be achieved ... 

We have demonstrated the method of finite differences for solving linear parabolic partial differential equations. But the utility of the numerical method is best appreciated when we deal with nonlinear equations. In this section, we will consider a nonlinear parabolic partial differential equation, and show how to deal with the nonlinear terms. 

It was demonstrated in the text that the number of equations to be solved using the method of finite difference depends on the type of boundary conditions. For example, if the functional values are specified at two end points of a boundary value problem, the number of equations to be solved is N — 1. On the other hand, if the functional value is specified at only one point and the other end involves a first derivative, then the number of equations is N. This homework problem will show that the variation in the number of equations can be avoided if the locations of the discrete point are shifted as... 

Eq. (65) is a nonlinear ordinary D.E. It was solved by quasi-linearisation which yields a linear boundary value problem. This was so ed by the method of finite differences (6). In Fig. is plotted as a function... 

PARSIM optionally provides the method of Finite Differences (FD) for space discretization. An advantage of this method is the lower bandwidth of the Jacobian matrix. Nevertheless, much more node points are needed to achieve the same accuracy compared to the OCFE method as demonstrated below. The method of global Orthogonal Collocation (OC) is provided additionally by PARSIM but should be used only for systems without steep gradients. 

Due to the steep gradients of the drying front this process is an excellent test case for an equation solver. The calculations were performed applying the method of Orthogonal Collocation on Finite Elements and the method of Finite Differences. The touchstone of the methods was the radial profile of... 

The complete derivations for this equation and different methods of finite-difference equations are to be foimd in the literature (e.g., in refs 27 and 28). 

The method of finite differences, which is involved in the building of Pascal s triangle, can sometimes be used to conjecture a formula/(n). Table 4 shows the first and second differences obtained fi-om sequence (a). 

Note that the second differences are constant (i.e., 15). If and when we reach a row that contains a constant value, we can write an explicit expression for fin). In Pythagorean terms one would say that if the method of finite differences is applicable, each difference can be considered as a gnomon for a previous series. We can consider numbers in Table 4 as Pascal s triangle modification C . 

Partial differential equations can also be solved numerically using the most commonly used method, the method of finite differences or the... 

The idea of the method of finite differences of solving boundary conditions for a two-dimensional Laplace or Poisson equation is as follows ... 

First are its mathematical foundations. In 1953 an intuitive commonplace amongst mathematicians was definitively proven by a Russian researcher. L.l. Kamynin showed that the method of finite differences, invented long before by the English mathematician and Royal Society alumnus Brook Taylor (1685-1731), was reliable. Small finite quantities, amenable to computer manipulation, replace the infinitesimal quotients of partial differential equations, mathematical models that describe physical processes, which would otherwise be insoluble. This intellectual tool took discrete points along a continuum of some kind, a mathematical function or a curving structural element, then found the conditions at those points and in between. 

The computer simulation in this paper was carried out by the method of finite differences using Eqs. 1 to 6. 

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