Gauss Quadrature Rule of Integration Chemical Engineering Majors

Gauss Quadrature Rule of Integration Chemical Engineering Majors Authors: Autar Kaw, Charlie Barker http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates 12/16/2021 http: //numericalmethods. eng. usf. edu 1

Gauss Quadrature Rule of Integration http: //numericalmethods. eng. usf. edu

What is Integration? Integration The process of measuring the area under a curve. f(x) y Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration 3 a b lmethods. eng. usf. edu x http: //numerica

Two-Point Gaussian Quadrature Rule 4 lmethods. eng. usf. edu http: //numerica

Basis of the Gaussian Quadrature Rule Previously, the Trapezoidal Rule was developed by the method of undetermined coefficients. The result of that development is summarized below. 5 lmethods. eng. usf. edu http: //numerica

Basis of the Gaussian Quadrature Rule The two-point Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation where the arguments of the function are not predetermined as a and b but as unknowns x 1 and x 2. In the two-point Gauss Quadrature Rule, the integral is approximated as 6 lmethods. eng. usf. edu http: //numerica

Basis of the Gaussian Quadrature Rule The four unknowns x 1, x 2, c 1 and c 2 are found by assuming that the formula gives exact results for integrating a general third order polynomial, Hence 7 lmethods. eng. usf. edu http: //numerica

Basis of the Gaussian Quadrature Rule It follows that Equating Equations the two previous two expressions yield 8 lmethods. eng. usf. edu http: //numerica

Basis of the Gaussian Quadrature Rule Since the constants a 0, a 1, a 2, a 3 are arbitrary 9 lmethods. eng. usf. edu http: //numerica

Basis of Gauss Quadrature The previous four simultaneous nonlinear Equations have only one acceptable solution, 10 lmethods. eng. usf. edu http: //numerica

Basis of Gauss Quadrature Hence Two-Point Gaussian Quadrature Rule 11 lmethods. eng. usf. edu http: //numerica

Higher Point Gaussian Quadrature Formulas 12 lmethods. eng. usf. edu http: //numerica

Higher Point Gaussian Quadrature Formulas is called the three-point Gauss Quadrature Rule. The coefficients c 1, c 2, and c 3, and the functional arguments x 1, x 2, and x 3 are calculated by assuming the formula gives exact expressions for integrating a fifth order polynomial General n-point rules would approximate the integral 13 lmethods. eng. usf. edu http: //numerica

Arguments and Weighing Factors for n-point Gauss Quadrature Formulas In handbooks, coefficients and arguments given for n-point Gauss Quadrature Rule are given for integrals as shown in Table 1. 14 Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas. Points Weighting Factors 2 c 1 = 1. 00000 c 2 = 1. 00000 Function Arguments x 1 = -0. 577350269 x 2 = 0. 577350269 3 c 1 = 0. 55556 c 2 = 0. 88889 c 3 = 0. 55556 x 1 = -0. 774596669 x 2 = 0. 00000 x 3 = 0. 774596669 4 c 1 c 2 c 3 c 4 x 1 = -0. 861136312 x 2 = -0. 339981044 x 3 = 0. 339981044 x 4 = 0. 861136312 = = 0. 347854845 0. 652145155 0. 347854845 lmethods. eng. usf. edu http: //numerica

Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Table 1 (cont. ) : Weighting factors c and function arguments x used in Gauss Quadrature Formulas. Points 15 Weighting Factors Function Arguments 5 c 1 c 2 c 3 c 4 c 5 = = = 0. 236926885 0. 478628670 0. 568888889 0. 478628670 0. 236926885 x 1 = -0. 906179846 x 2 = -0. 538469310 x 3 = 0. 00000 x 4 = 0. 538469310 x 5 = 0. 906179846 6 c 1 c 2 c 3 c 4 c 5 c 6 = = = 0. 171324492 0. 360761573 0. 467913935 0. 360761573 0. 171324492 x 1 = -0. 932469514 x 2 = -0. 661209386 x 3 = -0. 2386191860 x 4 = 0. 2386191860 x 5 = 0. 661209386 x 6 = 0. 932469514 lmethods. eng. usf. edu http: //numerica

Arguments and Weighing Factors for n-point Gauss Quadrature Formulas So if the table is given for ? integrals, how does one solve The answer lies in that any integral with limits of can be converted into an integral with limits 16 If then Let Such that: lmethods. eng. usf. edu http: //numerica

Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Then Hence Substituting our values of x, and dx into the integral gives us 17 lmethods. eng. usf. edu http: //numerica

Example 1 For an integral derive the one-point Gaussian Quadrature Rule. Solution The one-point Gaussian Quadrature Rule is 18 lmethods. eng. usf. edu http: //numerica

Solution The two unknowns x 1, and c 1 are found by assuming that the formula gives exact results for integrating a general first order polynomial, 19 lmethods. eng. usf. edu ht

Solution It follows that Equating Equations, the two previous two expressions yield 20 lmethods. eng. usf. edu ht

Basis of the Gaussian Quadrature Rule Since the constants a 0, and a 1 are arbitrary giving 21 lmethods. eng. usf. edu ht

Solution Hence One-Point Gaussian Quadrature Rule 22 lmethods. eng. usf. edu ht

Example 2 In an attempt to understand the mechanism of the depolarization process in a fuel cell, an electro-kinetic model for mixed oxygen-methanol current on platinum was developed in the laboratory at FAMU. A very simplified model of the reaction developed suggests a functional relation in an integral form. To find the time required for 50% of the oxygen to be consumed, the time, T (s) is given by a) Use two-point Gauss Quadrature Rule to find the time required for 50% of the oxygen to be consumed. b) Find the true error, for part (a). c) Also, find the absolute relative true error, for part (a). 23 lmethods. eng. usf. edu http: //numerica

Solution a) 24 First, change the limits of integration from to [-1, 1] by previous relations as follows lmethods. eng. usf. edu http: //numerica

Solution (cont) Next, get weighting factors and function argument values from Table 1 for the two point rule, 25 lmethods. eng. usf. edu http: //numerica

Solution (cont. ) Now we can use the Gauss Quadrature formula 26 lmethods. eng. usf. edu http: //numerica

Solution (cont) since 27 lmethods. eng. usf. edu http: //numerica

Solution (cont) b) c) 28 The true error, , is The absolute relative true error, , is (Exact value = 190140 s) lmethods. eng. usf. edu http: //numerica

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/gauss_qua drature. html

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