Tables Jacobi polynomial roots

This is one of the variants of the finite element methods. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their... 

The table below provides the roots of the Jacobi polynomials used as node points in orthogonal collocation, for some values of N. Values for X = 0 (i = 0) and X = 1 (i = N + 1) (the values are 0 and 1, resp.) are not included. The roots were computed using the subroutine JCOBI, modified from the original of Villadsen and Michelsen [562], discussed in Appendix C, using for a given N the call... 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

There are two common values for the weighting w x ). These are u = 1 and w = —x. Roots for these Jacobi polynomials are presented in Table 8.8 for all three geometries, both values of w and various orders of N. The roots presented in Table 8.8 specify specific collocation points to be used in the method of weighted residuals. The collocation method satisfies the describing differential equation at the collocation points. In order to implement this, we need to evaluate the function and its derivatives at the collocation points. The trial function of equation (8.12.21) can also be written as... 

Table 8.8 Roots of Jacobi Polynomials (Source Finlayson, 1972)...

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