Parabolic PDE Example

We saw in the last example for the elliptic PDE that the orthogonal collocation was applied on two spatial domains (sometime called double collocation). Here, we wish to apply it to a parabolic PDE. The heat or mass balance equation used in Example 11.3 (Eq. 11.55) is used to demonstrate the technique. The difference between the treatment of parabolic and elliptic equations is significant. The collocation analysis of parabolic equations leads to coupled ODEs, in contrast to the algebraic result for the elliptic equations. 

The quantity of interest is the mean concentration or temperature, which is calculated from the integral 

We note that this problem is symmetrical at x = 0. Therefore, the application of the symmetry transformation 

Therefore, to use the Radau quadrature with the exterior point ( = 1) included, the N interior collocation points are chosen as roots of the Jacobi polynomial with a = 1 and jS = 0. Once N -I- 1 interpolation points are chosen, the first and second order derivative matrices are known. 

Evaluating Eq. 12.205a at the interior collocation point i gives 

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Chapter 12 Approximate and Numerical Solution Methods for PDEs 12.4.2 Parabolic PDE Example 1... 

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