Difference equations 3
Lu = u' known from differential calculus. The same procedure works for
the second, the third and other differences
!:!.^2 Yi = !:!. ( !:J.yi) = !:!. (Yi+l - Yi)
= (Yi+2 - Yi+i) - (Yi+l - Yi)= Yi+2 - 2 Yi+l +Yi,
We see that one more right point is captured every time when the operator
!:!. is applied. Consequently, applying !:!. for in times we justify that !:!. m Yi
contains the values Yi, Yi+l, ... , Yi+m at the points i, i + 1, ... , i +in.
A very simple rule could be useful: the left difference operation at any
point i coincides with the right difference operation at the point i - ] so
that
!:!. V' Yi = t:i.f Yi-1 = Yi+1 - 2 Yi +Yi-l ·
Let a linear equation with the entering differences and coefficients
0: 0 (i), 0: 1 (i), ... , o:m(i) be composed:
Substituting the expressions for differences t:J.kyi, k = 1, 2, ... , m, one can
modify it to an mth order linear difference equation related to an
unknown y;:
This definition 1s a formal analog of an mth order ordinary differential
equation
As one possible example we consider the simplest ordinary differential equa-
ti on
- du = f ( ) x
dx