1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations

the sum of which gives

Ui+l - 2 Ui + Ui-l

h2

5

Neglecting 0( h^2 ) we derive rough expressions for u" within the notations:

,6.^2 Ui-1

h2

Substituting the first expression into the expansion Ui+l

l 2 h^2 u" z + O(h^3 ) instead of u" z J we deduce that

(1) U· I z

Ui + h u; +

Replacing in (1) u; by fi we cancel O(h^2 ) and multiply the resulting equa-

tion by 2h. As a final result we get the second-order difference equation

b. \1 Yi - 2 b. Yi = - 2 h fi.

Its modification gives the approximation of the first-order differential equa-

tion

dy = f.

dx

Difference equations possess remarkable properties which will be given spe-

cial investigation in the near future.

3. The first-order difference equations and inequalities. Of our concern is
   the first-order difference equation

(2) b b. Yi + a Yi = fi ,

which is a formal analog of the first-order differential equation

du

b - + au= f.

dt

Alternative forms of equation (2) are

b (Yi+l - Yi) + a Yi = fi or byi+1=cy;+fi, c=b-a.