Quadrature Gaussian

The problem with classical integration techniques is that they require a large number of integration points, which for two- and three-dimensional domains, where area and volume integrals are calculated, requires large amounts of storage and computation time. 

Numerical integration of a known function. To illustrate the Gauss quadrature integration technique and compare it to classical integration techniques, we will evaluate the following integral 

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Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

Gaussian Quadrature Gaussian quadrature provides a highly accurate formula based on irregularly spaced points, but the integral needs to be transformed onto the intei val 0 to I. 

Using the Gaussian quadrature formulas gives the following values for various ... 

Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)... 

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. 

Some other typical Gaussian quadrature formulas are (a,b) W(x) Gauss-... 

Weights and zeros for the above formulas (and for other Gaussian formulas) may he found in references such as Stroud Gaussian Quadrature Formulas, Prentice-Hall, 1966). 

Using the Gaussian quadrature formulas gives the following values for various values ofm. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results. 

The integrals in (28) and (54) are approximated by q Gaussian quadrature points, so that, for each quadrature energy point in (28), there is a set of 9 + 1 first-order differential equations to be solved, since... 

Two features of the trapezoidal method are that we use a uniform spacing between the positions where we evaluate fix) and that every evaluation of fix) (except the end points) is given equal weight. Neither of these conditions is necessary or even desirable. An elegant class of integration methods called Gaussian quadrature defines methods that have the form... 

For an isotropic medium, by substituting the Gaussian quadrature formula for the integral in Eq. (4.78), the integral-differential equation may be reduced into a system of ordinary linear differential equations. Specifically, the integral can be treated as [Chandrasekhar, 1960]... 

The notion of an effectiveness factor introduced by Thiele, Amundson s exploitation of the phase plane (34), Gavalas use of the index theorem (41), the Steiner symmetrization principle used by Amundson and Luss (42) and the latter s exploitation of the formula for Gaussian quadrature (43)—perhaps the prettiest connection ever made in the chemical engineering literature—are theoretical counterparts, large and small, of the careful craft of the experimentalist. So perhaps also the very important insight that Danckwerts contributed in his formulation of the residence time distribution is a happy foil to his heroic ambition to trace a blast furnace (44). 

These residuals can be calculated by fitting a polynomial to the converged solution and then evaluating the integrals by Simpson s rule or Gaussian quadrature. The number of quadrature points to be used should be of the order of 3N in the region 0 to X. 

The inner-integral of Equation (2) was numerically integrated using a four-point Gaussian quadrature. The mean bubble length was calculated from the first moment of the frequency distribution function given in Equation (2). 

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