Integration is by successive approximation, using essentially Eulers method. For the first trial. [Pg.2080]

Iterative methods of successive approximation are in common usage for rather complicated cases of arbitrary domains, variable coefficients, etc. Throughout the entire section, the Dirichlet problem for Poisson s equation is adopted as a model one in the rectangle G = 0 < x < l, a = 1,2 with the boundary P [Pg.644]

FIGURE 4 Successive approximations to the true tangent are obtained as the two points defining the straight line come closer together and finally coincide. [Pg.913]

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 [Pg.365]

The potentiometer principle is used in the successive approximation instrument, where the unknown voltage is compared with a succession of known voltages until balance is obtained. The total time for measurement and display is about 5 ms. [Pg.240]

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3 [Pg.2200]

A similar approximation scheme, which gives successive approximations, in first order, to afl can be developed the thermal conductivity for the central force law is [Pg.39]

As the concentrations of dissolved water and EDC are small, the best approach to this problem is by successive approximation rather than by setting up and solving equations for the unknown concentrations. [Pg.149]

Models are usually formed through a process of successive approximations, i.e., refinements are introduced successively, based upon the worth and reliability of additional information and upon the suitability of the model for prediction. This refinement process may be described as follows 3 [Pg.251]

In practice, an initial guess, po x), is made and successive approximations to p x) are obtained by iterating equation 4.83. [Pg.208]

Olivieri, A. G. Solution of Acid-Base Equilibria by Successive Approximations, /. Chem. Educ. 1990, 67, 229-231. [Pg.178]

Most of the well-developed methods available for solving such a system falls within the categories of direct or exact-fitted methods and iterative or successive-approximate methods which are gaining increasing popularity. [Pg.2]

Substances, 255q density ofj 14 15 properties of, 13-19 solubility of, 15-17 Substitution reaction, 603 Successive approximations A technique [Pg.697]

In Fig. 4.4, the starting points (initial estimates) are denoted by circles in the A, B plane the evolution of the successive approximations is sketched by arrows that converge on the optimum, albeit in a roundabout way. The procedure is robust no local minima are apparent the minimum, marked by the square, is rapidly approached from many different starting points within the given A, B plane (black symbols) note that a B-value > 0, an A-value < 0, and combinations in the lower left comer (gray symbols) pose problems [Pg.183]

By repeating the calculation for various values of x, one can obtain y and Sx as functions of x and find the minimum of the latter by successive approximations. The value of x at this minimum (xo) gives the estimate of the isokinetic temperature Xo The corresponding values yo and So are obtained from eqs. (52) and (53) So has [Pg.448]

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. [Pg.366]

The family of hierarchical elements are specifically designed to minimize the computational cost of repeated computations in the p-version of the finite element method (Zienkiewicz and Taylor, 1994). Successive approximations based on hierarchical elements utilize the derivations of a lower step to generate the solution for a higher-order approximation. This can significantly reduce the [Pg.40]

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. [Pg.783]

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. [Pg.365]

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 [Pg.345]

Because of the convenient mathematical characteristics of the x -value (it is additive), it is also used to monitor the fit of a model to experimental data in this application the fitted model Y - ABS(/(x,. ..)) replaces the expected probability increment ACP (see Eq. 1.7) and the measured value y, replaces the observed frequency. Comparisons are only carried out between successive iterations of the optimization routine (e.g. a simplex-program), so that critical X -values need not be used. For example, a mixed logarithmic/exponential function Y=Al LOG(A2 + EXP(X - A3)) is to be fitted to the data tabulated below do the proposed sets of coefficients improve the fit The conclusion is that the new coefficients are indeed better. The y-column shows the values actually measured, while the T-columns give the model estimates for the coefficients A1,A2, and A3. The x -columns are calculated as (y- Y) h- Y. The fact that the sums over these terms, 4.783,2.616, and 0.307 decrease for successive approximations means that the coefficient set 6.499... yields a better approximation than either the initial or the first proposed set. If the x sum, e.g., 0.307, [Pg.79]

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