Summary of Collocation Methods

Equation 8.171 is a cubic equation in terms of y hence, depending on the values of f , 8, and y, there may exist three solutions (multiple steady states) for this problem. A knowledge of this will then help the comprehensive computation using more collocation points. 

This example serves as a means to allow workers to quickly study the topology of a system before more time is spent on the detailed computation of the governing equations. 

We have presented a family of approximate methods, called weighted residuals, which are quite effective in dealing with boundary value problems. The name suggests that we need to generate residuals obtained when the approximate solution is substituted into the governing equation. Then, we try to minimize the residuals or force it to be asymptotically close to zero at certain points. A number of methods have appeared, depending on how we minimize this residual. 

We now summarize the steps taken in the application of the orthogonal collocation procedure. 

Step 1 For the given problem, normalize the range of the independent variable to (0,1). Any domain (a, b) can be transformed to a (0,1) by the transformation 

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