Ordinary differential equations, boundary value orthogonal collocation

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

C0LL0CATI0N Solves a boundary value set of ordinary differential % equations by the orthogonal collocation method. 

The concept of orthogonal collocation for ordinary differential equations can be easily extended to solve parabolic partial differential equations. The difference is that the application of orthogonal collocation method on the two-point boundary-value differential equation discussed earlier results in a set of algebraic equations, whereas application of orthogonal collocation method on parabolic partial differential equations results in a set of ordinary differential equations. 

Since we have not used the symmetry boundary condition that dTfdy = 0 at y = 1/2, we will solve the problem by performing an orthogonal collocation in the spatial domain using the shifted Legendre polynomials as the basis function. This will reduce the problem to a set of initial-value ordinary differential equations that can be solved using IMSL ordinary differential equation routines. As discussed by Cooper et al. (1986), seven internal collocation points accurately describe the solution to the partial differential equations. Therefore, we use equation (8.12.14) to approximate the second spatial derivative. This reduces the original partial differential equation of (8.12.15) to... 

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

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