Difference quotient

If S remains constant during the pumping process, then one can use the difference quotient instead of the differential quotient ... 

The basic principle in this technique is to replace derivatives by finite differences, i.e., dy/dx is replaced by Ay/Ax. The differential equation is then rewritten using these difference quotients in place of the derivatives and the boundary conditions of the problem introduced. The equations can then be solved analytically. Space and time... 

After dividing this equation by Al, rearranging to obtain difference quotients for n and Ni, and then taking limAi- o, we get... 

By rearranging, forming difference quotients, and taking the limit of Al we transform (5.8) to... 

Rearranging equation (6.50) and taking the difference quotient limit as Ay —> 0 gives us... 

For a decaying concentration-time function SER(t) is obtained from (5.1) by transition to the difference quotient according to Equation 5.2. 

The computer program PROG52 can be used to solve any number of nonlinear equations. The partial derivatives of the functions are estimated by the difference quotients when a variable is perturbed by an amount equal to a small value (A) used in the program to perturb the X-values. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

Neglecting higher-order derivatives and with equidistant discretization, we obtain the following expression for the central difference quotient of first order by the addition of the Eq. (71) and Eq. (73)... 

Using difference quotients, the partial differential equations become ordinary differential equations. The boundary conditions (e.g., inlet and outlet gas temperature of the fluidized bed) can easily be implemented using difference quotient in the entire differential equation system. So, initial/boundary value problems are transferred into initial value problems. Now, the ordinary differential equations of order 1... 

The statements about the time behavior of H(Z) or of an arbitrary F( Z) usually therefore refer to a discrete sequence of points selected from the step function, and one has to consider difference quotients whose At still contains a very large number of steps of the step function.189... 

This erroneous statement comes about by confusing the discrete //-curve with its smoothened interpolating curve. For the latter the left and right derivatives would be equal at every point, which leads immediately to an equation similar to Eq. (51). However, such a conclusion cannot be transferred to difference quotients, which correspond to Eq. (51) itself. 

The truncation error ex is estimated as the contribution of the second-derivative term in Eq. (6.B-2) to the difference quotient in Eq. (6.B-3). Show that this gives... 

With Hr defined as above, the rounding error in the difference quotient is dominated by the subtraction in the numerator of Eq. (6.B-3). GREGPLUS uses the estimate 6mS 6 ) for the rounding error of each numerator term, thereby obtaining the root-mean-square estimate... 

Give an expression for the minimum number of evaluations of S required to compute all the coefficients in Eq. (6.C-1) for a p-parameter model. For this calculation, assume that each partial derivative is approximated at 0 by a corresponding finite-difference quotient. 

The conversion-time functions were differentiated graphically the values of the difference quotient, AU/At, were plotted logarithmically vs. time. The conversion, U, is directly proportional to the concentration of the polymer, and aU is proportional to the decrease of the monomer concentration, —A[M]. Reactions of zero order (with respect to monomer concentration) thus plotted produce a parallel to the time abscissa ... 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

The derivatives which appear in (2.236) are replaced by difference quotients, whereby a discretisation error has to be taken into account... 

The second derivative in the -direction at position xf at time tk is replaced by the central second difference quotient... 

The writing of O (Ax2) indicates that the discretisation error is proportional to Ax2 and therefore by reducing the mesh size the error approaches zero with the square of the mesh size. The first derivative with respect to time is replaced by the relatively inaccurate forward difference quotient... 

The equations (2.238) and (2.239) for the replacement of the derivatives with difference quotients can be derived using a Taylor series expansion of the temperature field around the point (Xi,tk), cf. [2.53] and [2.57]. It is also possible to derive the finite difference formula(2.240) from... 

Its discretisation with the difference quotients according to (2.238) and (2.239) leads to the explicit finite difference formula... 

The local derivative which appears in (2.247) is replaced by the rather exact central difference quotient. It holds at the left hand boundary that... 

The derivative is formed in the outward normal direction, and a and can be dependent on time. For the discretisation of (2.253) it is most convenient if the boundary coincides with a grid line, Fig. 2.45, as the boundary temperature which appears in (2.253) can immediately be used in the difference formula. The replacement of the derivative d d/dn by the central difference quotient requires grid points outside the body, namely the temperatures i9k or ()k,, which, in conjunction with the boundary condition, can be eliminated from the difference equations. 

In the consideration of the heat transfer condition (2.253) a choice of grid different to that shown in Fig. 2.45 is frequently made. The grid is laid out according to Fig. 2.44 with the derivative dd/dx at xr = xq + Ax/2 being replaced by the central difference quotient from (2.248). The boundary temperature t (xR,h) is found using the approximation... 

A particularly accurate implicit difference method, which is always stable, has been presented by J. Crank and P. Nicolson [2.65]. In this method the temperatures at the time levels tk and tk+l are used. However the differential equation (2.236) is discretised for the time lying between these two levels tk + At/2. This makes it possible to approximate the derivative (dt>/dt)k+1 2 by means of the accurate central difference quotient... 

The second derivative (<92 d/<9x2)f+1 2 at time tk + At/2 is replaced by the arithmetic mean of the second central difference quotients at times tk and tk+l. This produces... 

Replacing the first derivatives with central difference quotients gives... 

In order to avoid complicated iterations it is recommended that an explicit difference method is applied to this case. We replace the time derivative with the first difference quotient according to (2.239) and obtain from (2.274) with (2.295) the difference equation... 

The two second derivatives in the x- or y-directions are approximated by the central difference quotient, so that... 

The restriction on the step size (2.304) due to the stability condition for the explicit difference method can be avoided by using an implicit method. This means that (2.298) is discretised at time tk+1 and the backward difference quotient is used to replace the time derivative. With... 

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