Quadrature, solution

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

In the case of 3b, Gaussian quadrature can be used, choosing the weighting function to remove the singularities from the desired integral. A variable step size differential equation integration routine [38, Chapter 15] produces the only practicable solution to 3c. 

The dependence of the in-phase and quadrature lock-in detected signals on the modulation frequency is considerably more complicated than for the case of monomolecular recombination. The steady state solution to this equation is straightforward, dN/dt = 0 Nss — fG/R, but there is not a general solution N(l) to the inhomogeneous differential equation. Furthermore, the generation rate will vary throughout the sample due to the Gaussian distribution of the pump intensity and absorption by the sample... 

The accuracy of this method increases nhen increasing M in equation 33, i. e. the dimension of the system of ordinary differential equations 36. Usually, due to the monomiodal sh ie of the PSD considered in this work, M = 3 provides a satisfactory approxinatlon of the solution for the same reason, a low mmher of quadrature points (<5) is required in the evaluation of the integral terms in equations 20, 21 and 36. 

Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia (1985). There are methods that are analogous to the usual methods for... 

The variables of the differential equation are separable, but to find C as a function of t requires a numerical quadrature and application of "root solver" many times. A direct solution by ODE program is simpler. 

Fig. 23. The in-phase (/LF) an< quadrature (Qlf ) component of the demodulation response pertaining to the tris-oxalato Fe(III)/ tris-oxalato Fe(II) electrode reaction at the dropping mercury electrode in aqueous IMK2C2O4 + 0.05 M H2C2O4 solution. High frequency 100 kHz low frequency 170 Hz, = 0.21 A cm-2 Fe(III) concentration 1 mM. The thin solid lines represent the demodulation response in the absence of the redox couple [72].

It is very practical to work with the depth as an independent variable, for, if we need it, we can find the time of rise from the solution of the equations by quadrature. The pressure is given by the gas law and is, in fact, equal to either side of the differential equation for . The depth was calculated from the point at which the extrapolated hydrostatic pressure is zero, so that if P is the pressure at the surface, the equations must be integrated until... 

It is surprising how often some of the special forms of easily soluble differential equations turn up. For single equations in which the derivative is a function only of the state, dxldt = f(x), x(0) = X, the solution by quadratures is immediate... 

Equations (102)-(104) are the Riccati differential equations that have no solutions in quadratures for arbitrary jq and k2. They can, however, be easily solved numerically to obtain the desired second moment of the distribution (and hence the weight average degree of polymerization) as a function of time, according to Eq. (101). 

In this section, the nonlinear problem of forming a least squares solution for the sine-wave amplitudes, phases, and frequencies is transformed into a linear problem. This is accomplished by assuming the sine-wave frequencies are known apriori, and by solving for the real and imaginary components of the quadrature representation of the sine waves, rather than solving for the sine-wave amplitudes and phases. The... 

When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, nucleation,47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50... 

B. D. Shizgal and H. Chen,/. Chem. Phys., 107,8051 (1997). The Quadrature Discretization Method (QDM) in the Solution of the Fokker-Planck Equation with Nonclassical Basis Functions. 

A good quadrature is one where the number of evaluations of the function is kept as small as possible in order to achieve an accurate solution. There are two different types of integrations methods, closed and open formula, as schematically depicted in Fig. 7.15. Those that use the value of the function at the lower and upper limits, f(a) and /( ), called closed formula and those that do not include these function values, called open formulas. The lattter are used when the function presents a singularity in one of the limits. 

Had we used a higher precision for the Gauss points, the quadrature would have rendered a solution even closer to the exact integral. 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

The system of equations (2.27) is seen to be rather complicated. Its solution, if obtainable at all in quadratures, must probably be even more complicated. However, in experiments certain conditions which enable the initial equations to be simplified are usually fulfilled. Consider limiting cases of particular interest from both theoretical and practical viewpoints.134,136,139,140 The process of growth of the ApBq and ArBs layers will be analysed in its development with time from the start of the interaction of initial substances A and B up to the establishment of equilibrium at which, according to the Gibbs phase rule (see Refs 126-128), no more than two phases should remain in any two-component system at constant temperature and pressure. 

APPLIED ANALYSIS, Cornelius Lanczos. Classic work on analysis and design ol linite processes for approximating solution of analytical problems. Algebraic equations, matrices, harmonic analysis, quadrature methods, much more. 559pp. 5H x 8H. 65656-X Pa. 11.95... 

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