1549301742-The_Theory_of_Difference_Schemes__Samarskii

Stability of a difference sche1ne 91

so that

With these entries, we look for a solution of problem (6) in the form

n

u(t) = 2= ocs(t) es.

s=l

Substitution of this expression into equation (6) yields

which implies that

providing the same grounds for the series

n

(7) u(t) = L Ock(O) exp {->.kt} ek.

k=l

Under the initial condition

n

u(O) = Uo = L ock(O) ek'

k=l

we arrive at the relations

n

II Uo 112 = L oc~(O).

k=l

From (7) it follows that

n n

II u(t) 112 = ~ ~ ock(O) ocs(O) exp {->.kt} exp {->.st} (ek 'es)

k=l s=l

n n

k=l k=l