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By Richard H. Bartels (auth.), David F. Griffiths (eds.)

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Additional info for Numerical Analysis: Proceedings of the 10th Biennial Conference held at Dundee, Scotland, June 28 – July 1, 1983

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Q - l ) . 2): g ( t ) = gl + tB " g2 ' where gl and replace g ( t ) ~ 1 g2 by the nonsmooth are constants, and choose B = 1 - ~. I t is e a s i l y v e r i f i e d t h a t a smooth s o l u t i o n e x i s t s only of the r e l a t i o n ~'gl + (I - ~)'g2 = 0 holds; t h i s s o l u t i o n is then Let us, at t h i s p o i n t , b r i e f l y assuming y(t) = gl " look at the Fredholm i n t e g r a l equations ( 1 . 2 ) , g ~ cm(1), K ~ cm(SF ).

I""\ ~'==o o o • '~~ V0 V P o - : J o V4 V3 V ..... 01 . p==O ~2=o --2 2 g5 \ ~ #t = 1 0 ~ . o Vo V3 gi,:~--o'~ ; ........ 5"... "%,°° , #2=0 •... o 0 V0 0 ......... :. V3 #2=0 0 . . . . V3 Vo o Vo . #2 = 0 ".... &=~ a==o 0 Vo g't ........ oo1~... V~ V4 29 The functions found in this way are nonnegative, have compact support, and can be verified symbolically to have the partition of unity property. A general proof of these results has not been found. The corresponding process has been carried out for the cubic case, in particular, and the final drawings of this presentation were all computed using Beta-splines defined by computational differencing.

I""\ ~'==o o o • '~~ V0 V P o - : J o V4 V3 V ..... 01 . p==O ~2=o --2 2 g5 \ ~ #t = 1 0 ~ . o Vo V3 gi,:~--o'~ ; ........ 5"... "%,°° , #2=0 •... o 0 V0 0 ......... :. V3 #2=0 0 . . . . V3 Vo o Vo . #2 = 0 ".... &=~ a==o 0 Vo g't ........ oo1~... V~ V4 29 The functions found in this way are nonnegative, have compact support, and can be verified symbolically to have the partition of unity property. A general proof of these results has not been found. The corresponding process has been carried out for the cubic case, in particular, and the final drawings of this presentation were all computed using Beta-splines defined by computational differencing.

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