1549301742-The_Theory_of_Difference_Schemes__Samarskii

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