Formulas-Gauss-Legendre Quadrature

Example 4.4 Integration Formulas - Gauss-Legendre Quadrature. Write a general 

MATLAB function for integrating a function using a general Gauss-Legendre quadrature. Apply this function for the solution of the following problem  

A cold liquid film, initially at temperature Tg, is falling down (in z-direction) a vertical solid wall (xz-plane). The solid wall is maintained at a temperature T ) higher than that of the falling film. Tt is desired to know the temperature profile of the fluid as a function of y and z, near the wall. The partial differential equation that describes the temperature of the liquid for this problem is 

Putting this velocity profile into the energy balance equation, we get 

Using Gauss-Legendre quadrature calculate the above temperature profile and plot it 

---

These algorithms are frequently stated in terms of integrals over [-1,1], termed Gauss-Legendre quadrature, and the general formula then is... 

In Fig. 2 we report the spherical average potential Vq(R) for the Ne-N2/ O2 and NO systems compared with the corresponding isotropic Ne-Ar case. The Vq(R) term, i.e., thefirst term in the Legendre expansion or the potential surface, is easily obtained from the full PES by a Gauss-Legendre quadrature, according to the formula ... 

The Gauss quadrature formula developed in this section is known as the Gauss-Legendre quadrature because of the use of the Legendre polynomials. Other orthogonal polynomials, such as Chebychev, Laguerre, or Hermite, may be used in a similar manner to develop a variety of Gauss quadrature formulas. 

Typically, the quadrature formulae that can be used are either of the non-nested t3q>e such as the Gauss-Legendre and Gauss-Hermite quadrature rules or nested t)q)e. A brief discussion on a couple of nested quadrature rules are presented next. 

---