By J. Albrecht (auth.), J. Li. Morris (eds.)
Read Online or Download Conference on the Numerical Solution of Differential Equations: Held in Dundee/Scotland, June 23–27, 1969 PDF
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Extra info for Conference on the Numerical Solution of Differential Equations: Held in Dundee/Scotland, June 23–27, 1969
Where sj = Wj, j = O, 1, ... , k-1. ~ If ~q it can be proved, under the above assumptions, using the methods of , , that (~-) - (a) 28 - ~r = O(hP) + O(h q+1 ) , (1 5) (b) e(t r) : Fr + ~r + 0(%), This last result means that the dominant part of the error in the computed solution obtained from (3) is given by e r. There is a temptation at this point to use the following line of argument. Consider the integral equation i p(s) - L k(s, t) p(t) dt = q(s) , s >~0 (I 6) Discretize this equation (2) in the same way as (3) was obtained from (I)° Then an approximation Pr to P(tr) is given by r Pr - h jZ=o Wrj k(tr, tj)Pj = q(tr) , r = k, k+l, ...
16) 51 - Chebyshev-series-approxiillation P(tl = Q(t) = R(t) ~ Un Pn Tn n=O ~ Un n=O = Qn Tn and put (t), (t), ~ u n Rn T n ( t ) . 17) t h e n by ( 2 . 181 and ( 4 . 1 5 ) , (n=O,l,2,... 191 Ill b' = n s--O Us_ n Vs_n'Sb s (n=O,1,2,... ,ill-l) , - 52 - m qn=+ ~ s=0 ( U n - s Us Qn-s + Un+s Us Qn+s + U s . 20) 1 rn = 7 m ; s=O (Un- s u s Rn- s + Un+ s u s Rn+ s + Us. n u s Rs- n + U . n u . s Rs) b s (n=0,1,2, . . ,m-l) . 7), from ' an-i - a 'n+l = 2na n ' I = 0. 18), (n--l,2,.. 24) d for n < m-2, 0 for n _< m-l, n bin+ 1 = O.
The derivation of ( ~ ) from (43) is a generallzation of the derivation of (37), (38) from (33), (34). , from (42), the sum of the elements in each row of C is unity. , one of ~he k. in (~4) is unity. The I corresponding integrsl equation has exactly the same form as the perturbation equation (6). This means that one component of th~ discretization error behaves as if it were produced by a perturbation of K or g, as we should expect. The interesting result that we have obtained is that the error depends on a system of equations (4~), each of which has the form of the perturbation equation (6), except for the presence of the k..