Approximate Solutions to Equations

Since equation (39) represents an approximate solution to equation (35), the only differential equation that remains to be solved is equation (36), in which F (t, (p) = F (t, (p(z)) depends only on t. By utilizing equations (38) and (41) in equation (34), we can obtain the t dependence of explicitly. When this result is substituted into equation (36), we find that... 

This gives estimates for k and Y2 that can now be used to estimate p. Given these estimated values of k and Y2 we seek the least-squares approximate solution to equation. (7.18), subject to the linear constraint in equation 7.20. This solution turns out to be (see the mathematics section below) ... 

In the previous sections we several times sought the least-squares approximate solution to some linear equation subject to some linear constraint. In other words, we sought the best approximate solution to equation 7.4 subject to the constraint equation 7.3. Given kp> find p to solve... 

Levenspiel and Bischoff (3) used a derivation by Pasqnon and Dente (20) to obtain an expression that gives an approximate solution to equation Q 1.2.9) for small T) /uL and an arbitrary rate law ... 

Here i//(r) is set equal to zero at points where the concentration of counterions equals its average value n. An approximate solution to Equation (2.61) for the case of dilute suspensions has been obtained by Imai and Oosawa [64-66] and later by Ohshima [67]. They showed that there are two distinct cases separated by a certain critical value of the particle charge Q, that is, (case 1) the low-charge case and (case 2) the high-charge case. In the latter case, counterions are condensed near the particle surface (counterion condensation). For the dilute case (< < C 1), approximate values of i//(a) and i//(/f) together with the particle surface potential defined by — i//(F) are given below. 

Most previous studies concentrated on the solution of Equation 3 by providing an approximate functional form for the term in the square brackets. Normally, the assumption is made that at the discharge end hj 0 for a drum with no end constriction, otherwise is the depth of the constriction lip, i.e., hj = (R - R ), where R is the radius of the constriction opening. The assumption for the values of h is particularly correct for lightly loaded drums, and it becomes less accurate as the drum loading is increased. Various approximate solutions to Equation 3 are given below for different authors using a unified nomenclature. 

The approximate solution to Equation 9.41 is readily available, since the multipliers for the Gaussian elimination method have been calculated (and retained). Now, then y, the approximate solution of Ay = r, satisfies... 

Approximate solutions to Equation 5.111 can be obtained using the following arguments(KulikovskyandEikerling, 2013). Ase > 1 (Table 5.6), it can be assumed that ejo 1, therefore, unity may be neglected under the square root in Equation... 

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

Caleulations that employ the linear variational prineiple ean be viewed as those that obtain the exaet solution to an approximate problem. The problem is approximate beeause the basis neeessarily ehosen for praetieal ealeulations is not suffieiently flexible to deseribe the exaet states of the quantnm-meehanieal system. Nevertheless, within this finite basis, the problem is indeed solved exaetly the variational prineiple provides a reeipe to obtain the best possible solution in the space spanned by the basis functions. In this seetion, a somewhat different approaeh is taken for obtaining approximate solutions to the Selirodinger equation. 

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The term ah initio is Latin for from the beginning. This name is given to computations that are derived directly from theoretical principles with no inclusion of experimental data. This is an approximate quantum mechanical calculation. The approximations made are usually mathematical approximations, such as using a simpler functional form for a function or finding an approximate solution to a dilferential equation. 

In practice this relationship is only approximately correct because most plastics are not linearly viscoelastic, nor do they obey completely the power law expressed by equation (2.62). However this does not detract from the considerable value of this simple relationship in expressing the approximate solution to a complex problem. For the purposes of engineering design the expression provides results which are sufficiently accurate for most purposes. In addition. 

A theoretical model should be uniquely defined for any given configuration of nuclei and electrons. This means that specifying a molecular structure is all that is required to produce an approximate solution to the Schrodinger equation no other parameters are needed to specify the problem or its solution. 

Equation 6 was used to correlate the data of this paper however, a more accurate approximation solution of Equation 3 is given by ... 

In view of the above as well as the fact that all the various approximate solutions of Equation 3 give about the same answer when the reactant concentration is low, it did not seem worthwhile to seek better accuracy in the solution of Equation 3. 

Let now Q(e) be the total number of arithmetic operations necessary for obtaining a solution to equation (1) with a prescribed accuracy e > 0 regardless of the initial approximation in the iteration scheme (3). Its ingredients Bk and Tj. should be so chosen as to minimize the quantity (5(e). If the desirable accuracy can be attained in a minimal number of the iterations n = n e), then... 

Perturbation theory provides a procedure for finding approximate solutions to the Schrodinger equation for a system which differs only slightly from a system for which the solutions are known. The Hamiltonian operator H for the system of interest is given by... 

Goldman [42], Kakiuchi derives the following approximate solution to the Nernst-Planek equation for the ion transport across the interfaee and the concomitant current density ... 

There are several closed form approximate solutions to both the general and first-order forms of the dispersion equations (11.2.9 and 11.2.10). For example, Levenspiel and Bischoff... 

---