The sum1narized approxiinatiou rnethod 625The second reduction of the Cauchy proble1n. On the whole segment
tj < t < tj+l we must solve sequentially p Cauchy problems(55)with the initial dataj = l, 2, ... '
(56)By definition, an ele1nentj = 0, 1, 2,. .. '
gives a solution of problen1 (55) fort= tj+i · Fort= 0 we agree to consider
(57) v(l)(O) = u(O) = u 0.Knowing v(tj), it is possible to cletennine vl 11 (tj+il from the first equation
entering the above collection with v(l)(tj) = v(tj) incorporated. At the next
stage vc 11 (tj+i) is taken as the initial value of v( 2 )(t) fort = tj, allowing
to solve the second equation for ex = 2, etc. The outcome of solving all
the p problerns is v(p)(tj+l) = v(tj+l ), giving a solution of the system of
equations (55)-(57) fort= tj+i ·
\l\1hen the operators A 0 happen to be independent of t and f = 0,
problen1s (53)-(54) and (55)-(57) becon1e equivalent. \^1 Vith this in n1ind,
we are going to show that proble1n ( 51) is approxi1nated by problem (55 )-
(56) in a summarized sense. To that encl, the differences