139 where u j is the coeﬃcient of φ j in the representation of u h and f j = Z 10 f ( t ) φ j ( t )d t = Z ( j +1) h ( j − 1) h f ( t ) φ j ( t )d t, which is h times a weighted average of f over the j th interval. The only diFerence between the ±nite diFerence system and this system is that we have replaced point samples of f by average values. Note that if a ( t ) is not constant, then the systems look even more diFerent. CHALLENGE 23.5. See the program posted on the website. CHALLENGE 23.6. See the program posted on the website. CHALLENGE 23.7. Here are the results, in dull tables with interesting entries: PROBLEM 1 Using coeﬃcient functions a (1) and c (1) with true solution u (1) In±nity norm of the error at the grid points for various methods and numbers of interior grid points M M = 9 99 999 1st order ±nite diFerence
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