The Method of Successive Approximations

Identify the functional H for the breakage process in Section 3.2.2 described by the Eq. (3.2.8). Determine the integral equation that must be satisfied by the population density. 

The existence of solution to Eq. (4.1.15) is generally established by a procedure that guarantees the convergence of the method of successive approximations (also called Picard s iteration). This method consists in substituting into the right-hand side of (4.1.15), the nth approximation for the population density denoted by in order to calculate the (n + l)st approximant. Thus we have 

The actual calculation has already been covered in the preceding section. If 

5 See Ramkrishna and Amundson (1985), pp. 95, 139, for a demonstration of existence of solution for a one-dimensional linear integro-differential equation. For a nonlinear integro-differential equation, the reader is referred to Naylor and Sell (1971). 

Consider the cell population of Section 2.11.2, which multiplies by binary division for the more general case of an arbitrary cell division rate T(t, t) with an initial age distribution of (t). The population density / (t, t) satisfies the equation. 

We then assume that x on the right-hand side of the equation is zero and calculate a first value, jcj. 

We then substitute this value into the original equation and derive a second value, JC2. That is. 

Note that after three iterations, is 5.99 X 10 , which is within about 0.8% of the final value of 5.94 X 10 M. 

The method of successive approximations is particularly useful when cubic or higher-power equations need to be solved. 

As shown in Chapter 5 of Applications of Microsoft Excel in Analytical Chemistry iterative solutions can be readily performed using a spreadsheet. 

The graphical method can give us only an approximate idea of the roots of a polynomial. To obtain a more accurate numerical result, the roots of 1/ = F(a ) can easily be obtained by trial and error, finding the values of x that make the function y equal to zero. 

As an example, let s calculate the solubility of barium carbonate in pure water. Barium carbonate, BaC03, is a sparingly soluble salt with Kgp = 5.1 x 10  

If the hydrolysis of the carbonate ion to form bicarbonate is taken into account (equation 10-2, = 2.1 x lO ), solving the system of mass- and charge-balance 

You can locate the positive root in a systematic fashion by changing the value of a in increments and observing the value of the function y. When y exhibits a sign change, you ll know that it has passed through zero. Then decrease the size of the increment and repeat the process. In this way you can determine the value of X to any desired degree of accuracy. 

You can start with any value of X. Eventually you ll get the right answer. But if you use an educated guess, you ll get there sooner. Since the K p of BaC03 is 

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In most of the problems you will work, the approximation a — x a is valid, and you can solve for [H+] quite simply, as in Example 13.7, where x = 0.012a. Sometimes, though, you will find that the calculated [H+] is greater than 5% of the original concentration of weak acid. In that case, you can solve for x by using either the quadratic formula or the method of successive approximations. 

The method of successive approximations. In the first approximation, you ignored the x in the denominator of the equation. This time, you use the approximate... 

The second answer is physically ridiculous the concentration of H+ cannot be a negative quantity. The first answer is the same one obtained by the method of successive approximations. 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

This can be solved with the quadratic equation, and the result is = 0.0118 moles. We can attempt the method of successive approximations. First, assume that 0.0240. We obtain ... 

The method of successive approximations is a good way to deal with difficult equations that do not have simple solutions. For example, Equation 10-11 is not a good approximation when the concentration of the intermediate species of a diprotic acid is not close to F, the formal concentration. This situation arises when Kt and K2 are nearly equal and F is small. Consider a solution of 1.00 X 10 1M HM, the intermediate form of malic acid. 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

In calculations of ionic equilibria, the required activity coefficients depend on ionic strength and, thus, on the equilibrium. As a result, such calculations are usually carried out by the method of successive approximations. Fortunately, convergence to a unique solution is usually achieved very rapidly in these problems. 

This technique is called successive approximation as the digitized value is better approximated with each step of conversion. Compared with the word-at-a-time method the method of successive approximation needs only ld(n) reference voltages but it is slower than the above mentioned technique by a factor of ld(n) (n = number of discrete steps, Id = logarithm with basis 2). 

To make a complete calculation of the phase composition of the test sample, this method can take the experimentally measured value of p or the mass absorption coefficient can be calculated with the help of the method of successive approximations [30,31,39], However, the previously explained method is complex therefore, it is easier to avoid the microabsorption effect by milling the test sample to get particles of about 1 pm [39],... 

The method of successive approximations has been conveniently described by Wigley ( ) where either a "brute force" method or a "continued fraction" method can be used. 

To further illustrate the method of successive approximations, we will solve Example 7.9 by using this procedure. In solving for [H+] for 0.010 M H2SO4, we obtain the following expression ... 

The method of successive approximations is especially useful for solving polynomials containing % to a power of 3 or higher. The procedure is the same as for quadratic equations Substitute a guessed value for % into the equation for every x term but one, and then solve for x. Continue this process until the guessed and calculated values agree. 

Since the value of [Brd was small, the equation was easily solved by the method of successive approximations. In a preliminary calculation [Brj] was neglected and an approximate value of [Brsl was obtained. The cube of the value thus obtained was then substituted in equation 39 and the resulting quadratic equation again solved for [Brzl. After repeating this process several times, the numerical value of [Brsl approached a constant which was the correct root of the equation. 

Lemma 58 If C is a contraction defined on a complete metric space X then C has a unique fixed point which can be determined by the method of successive approximations as follows. 

The method of successive approximations is often faster to apply than the quadratic formula. Keep in mind that the accuracy of a result is limited both by the accuracy of the input data (values of and initial concentrations) and by the fact that solutions are not ideal. It is pointless to calculate equilibrium concentrations to any degree of accuracy higher than 1% to 3%. 

Ten percent error is a somewhat arbitrary cutoff, but since we do not consider activity coefficients incur calculation.s, which often create eiTors of at iea,st 10%, our choice is reasonable. Many general chemistry and analytical chemistry texts suggest that 5% error is appropriate, but such decisions should be based on the goal of the calculation. If you require an exact answer, the method of successive approximations presented in Feature 9-4 may be used, a spreadsheet solution may be appropriate for complex examples. 

In general, it is good practice to make the simplifying assumption and obtain a trial value for [H O ] that can be compared with c a in Equation 9-17. If the trial value alters [HA] by an amount smaller than the allowable error in the calculation, the solution may be considered satisfactory. Otherwise, the quadratic equation must be solved to obtain a better value for [H3O+J. Alternatively, the method of successive approximations (see Feature 9-4) may be used. 

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