Backward difference, first derivative

For the second derivative, we note that this means derivative of the first derivative or rate of change of slope, at. We have two slopes about P - the forward and backward difference first derivatives - and it... 

When the user, whether working on stand-alone software or through a spreadsheet, supplies only the values of the problem functions at a proposed point, the NLP code computes the first partial derivatives by finite differences. Each function is evaluated at a base point and then at a perturbed point. The difference between the function values is then divided by the perturbation distance to obtain an approximation of the first derivative at the base point. If the perturbation is in the positive direction from the base point, we call the resulting approximation a forward difference approximation. For highly nonlinear functions, accuracy in the values of derivatives may be improved by using central differences here, the base point is perturbed both forward and backward, and the derivative approximation is formed from the difference of the function values at those points. The price for this increased accuracy is that central differences require twice as many function evaluations of forward differences. If the functions are inexpensive to evaluate, the additional effort may be modest, but for large problems with complex functions, the use of central differences may dramatically increase solution times. Most NLP codes possess options that enable the user to specify the use of central differences. Some codes attempt to assess derivative accuracy as the solution progresses and switch to central differences automatically if the switch seems warranted. 

Dekker et al. [170] studied the extraction process of a-amylase in a TOMAC/isooctane reverse micellar system in terms of the distribution coefficients, mass transfer coefficient, inactivation rate constants, phase ratio, and residence time during the forward and backward extractions. They derived different equations for the concentration of active enzyme in all phases as a function of time. It was also shown that the inactivation took place predominantly in the first aqueous phase due to complex formation between enzyme and surfactant. In order to minimize the extent of enzyme inactivation, the steady state enzyme concentration should be kept as low as possible in the first aqueous phase. This can be achieved by a high mass transfer rate and a high distribution coefficient of the enzyme between reverse micellar and aqueous phases. The effect of mass transfer coefficient during forward extraction on the recovery of a-amylase was simulated for two values of the distribution coefficient. These model predictions were verified experimentally by changing the distribution coefficient (by adding... 

Using 4 grid points to represent d/dx Instead of using the first order (backward or forward) or the second order (central) finite difference approximation for the first derivative, let us calculate the derivative using four grid points (see Fig. 8.2)... 

The boundary condition at x = L or n = 5 has a first derivative that comes from the balance between the conduction and the convection at the end of the fin. We can use a backward finite difference to approximate this derivative, i.e.,... 

It is apparent that the central difference approximations converge 0(Ax2). The forward and backward approximations to the first derivative converge O(Ax). This is because they are really approximating the derivatives at the points x = Ax rather than at x = 0. 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. When a first-order backward difference approximation (c > 0) for the convective term is used, then the FDE of the PDE Eq. (10.2) is... 

This implicit method uses a first-order backward difference approximation for the time derivative and a second-order central difference approximation for the spatial derivatives. The FDE is... 

We now have three two-point approximations for a first derivative, all in fact being the same expression, (y2 — Vi)/h, but depending on where this formula is intended to apply, being, respectively a forward difference of 0(h) if applied at xt, a backward difference of 0(h) if applied at x2 and a central difference of 0(h2) if applied at (.iq +. r2)/2. In subsequent chapters, all these will be used to approximate, among others (2.3)-(2.8). 

If centered differences are also used for the spatial derivatives in Equation (7) and backward differences are used to represent the time derivative describing the viscous Kelvin-Voigt effect (2 ) then the first term on the RHS of Equation (7), in two space dimensions (x,y) become,... 

Enter the three point backward difference expression (accurate to the order h ) for the first derivative at x = L ... 

Note that boundary condition at x = 0 is used at node point i = 0 and governing equation is used for the node points i = 1..N. Central difference is used for the first derivative for node points i = 1..N-1 and backward difference is used for the first derivative for the node point i = N. Equation 10.11 is solved in Maple... 

When backward finite difference accurate to the order h is used for the first derivative in the governing equation, the solution does not oscillate, and the following plots are obtained for N = 10 node points. 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

Eq. (10.71) cannot be solved directly and requires discretization of the other variable. In difference methods, the area between the boundaries is divided into grid points, where the values of derivatives are approximated. For instance, for the first derivative, the five-point backward difference formula can be used,... 

Solution of these two equations yields second-order-correct backward difference expressions for first and second derivatives of f x) atx = X2. The desired result for the first derivative is... 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

To resolve the problem of stability, we approach the problem in the same way we did in the backward Euler method. We evaluate Eq. 12.130 at the unknown time level fy+j and use the following backward difference formula for the time derivative term (which is first order correct)... 

Consider Eq. (27), which expresses the time derivative as a backward difference, that is, takes the simple two-point expression to pertain to the next point in time at t + St. This has an error that is dominated by the first-order term in 5f. It is... 

The second-order derivative in Eq. (1.13a) is approximated by the second-order difference derivative on the set Df = D r D,, which is a set of interior nodes in the grid in the following way. On the grid we put the first-order derivative (dldx)u(x) into correspondence with the first-order difference derivatives S z(x) and d z(x). Here 8 z(x) and S z(x) are, respectively, right (or forward) and left (or backward) difference derivatives determined by the relations... 

From these equations corresponding formulas for forward difference and backward difference expressions of the first derivative can be easily drawn out ... 

The squeeze velocity was represented implicitly by a first order backward difference. Thus, a discretized form of the equation was derived as follows ... 

Mathematically, the system consists of parabolic PDEs, which were solved numerically by discretization of the spatial derivatives with finite differences and by solving the ODEs thus created with respect to time (Appendix 2). Typically, 3-5-point difference formulae were used in the spatial discretization. The first derivatives of the concentrations originating from a plug flow (Equations 9.1 through 9.3) were approximated with BD formulae, whereas the first and second derivatives originating from axial dispersion in the bulk phases and diffusion inside the catalyst particles were approximated by central difference formulae. Some simple backward (Equation 9.14) and central difference (Equation 9.15) formulae are shown here as examples ... 

Alternative representations for the first derivative are the first-order accurate backward and forward difference formulas... 

Applying central differences to all spatial derivatives and backward differences to the first-order time derivative, we have... 

Equations (3.32), (3.36), and (3.37) express the backward difference operators in terms of infinite series of differential operators. In order to complete the set of relationships, equations that express the differential operators in terms of backward difference operators will also be derived. To do so, first rearrange Eq. (3.31) to solve for e ... 

The derivative matrix returned by the function deriv.m has the same number of elements as the vector of input data itself. However, it is important to note that, depending on the method of finite difference used, some elements at one or both ends of the derivative vector are evaluated by a different method of differentiation. For example, in first-order differentiation with the forward finite difference metliod with truncation error 0(h), the last element of the returned derivative vector is calculated by backward differences. Another example is the calculation of the second-order derivative of a vector by the central finite difference method with truncation error 0(h ), where the function evaluates tlie first two elements of the vector of derivatives by forward differences and the last two elements of tlie vector of derivatives by backward differences. The reader should pay special attention to the fact that when the function calculates the derivative by the central finite difference method with the truncation error of the order 0(h ), the starting and ending rows of derivative values are calculated by forward and backward finite differences, with truncation error of the order O(h ). 

---