Jacobi Polynomials in an Orthogonal Collocation Method
Partial differential equations in rectangular, cylindrical, and spherical coordinates with symmetric boundary conditions occur in many fields of science and engineering. It is often possible to solve such equations using an orthogonal collocation method with roots of Jacobi polynomials as the points of collocation.
Symmetric Jacobi polynomials used in the orthogonal collocation method can be obtained from
where applies to a rectangular, cylindrical, or spherical coordinate, respectively, is the hypergeometric function, is the order in the polynomial, is the Jacobi polynomial, and is the independent variable.
Reference: J. V. Villadsen and W. E. Stewart, "Solution of Boundary-Value Problems by Orthogonal Collocation," Chemical Engineering Science, 22, 1967 pp. 3981–3996.