Finite difference approach - Principles of Chemical Separations with Environmental Applications - page 302

We can next define a shifting operator ( E ) which moves the index forward:

= +

E 1

Ey n = (1 + ) y n = y n + y n + 1 − y n = y n + 1 .

Again, generalizing

E k y n =

y n + k .

There are three rules which can also be useful.

Index rule

r s y n = s r y n = r + s y n .

Commutative rule

( Cy n ) = Cy n + 1 − Cy n = C y n C = constant .

Distributive rule

( y n + z n ) = y n + z n .

Equilibrium-stage mass balances give rise to general linear difference equations. For a
difference equation of order k :

y n + k + P n + k − 1 y n + k − 1 +···+ P n + 1 y n + 1 + P n y n = Q ,

where each P , and Q , are constants.

The complete solution includes a homogeneous and particular component:

y n + y n . y n =

Using the shifting operator, we can rewrite the homogeneous difference equation as:

( E k P n + k − 1 E k − 1 P n + 1 E 1 P n E 0 ) y n = + +···+ + 0 .

This equation can be factored for the k roots:

( E − α 1 )( E − α 2 ) ... ( E − α k ) y n = 0 .

Each root can be evaluated:

( E − α i ) y n = 0 ⇒ y n + 1 − α i y n = 0 .

n

Assume a solution of the form: y n = C β .

Substituting:

n + 1 n

C β − α i C β = 0 ⇒ β = α i ;

i

∴ y n = C α .