Principles of Mathematics in Operations Research

14.3 Difference Equations

14.3 Difference Equations

Let us start with first-order difference equations:

»<* + '> = »« + >«} Ay{k) = mk = h%^

The initial value problem of the above equation can be solved by the

following recurrence relation:

y(k)=y(k+l)-f(k), A: = -1,-2,...

Therefore, we find

( 2/o + £*lS/(«)>* = 1,2,3,...;

V(k) = < 2/o, k = 0;

U-£:=*/(«), fc =-i,-2,...

For second-order equations, we will consider first the homogeneous case:

y(fc + 2) + axy{k + 1) + a 0 y(fc) = 0; j/(0) = yQ, 2/(1) = 2/1 •

We seek constants

Ai, A 2 3 z(k + 1) = A 2 z(fc); z(0) = yi - Ait/ 0

which are the roots of

A^2 + aiA + ao = 0.

If Ai 7^ A 2 , then y(k) = ciAf + C2A2 where Ci, c 2 are the unique solutions

of

ci + c 2 = yo, C1A1+ c 2 A 2 = 2/1.

If Ai = A 2 = A, then y(k) = ci\h + c 2 Afc where ci,c 2 are the unique

solutions of

c\ — 2/o, ciA + c 2 A = yx.

When the roots are non-real, A = pel6 and A = pe~l6, then

y(k) — c\pk cos kO + c 2 /9fc sin k6,

where c\ and c 2 are the unique solutions of

ci = j/oi cicos0 +C2sin# = 2/1.

If we have systems of equations,

y{k + 1) = ^(fc), fc = 0,1,2,...; 2/(0) = j/ 0 ,

we, then, have as a recurrence relation