Linear Difference Equations

Difference equations arise in chemical engineering from staged operations, such as distillation or extraction, as well as from differential equations modeling adsorption and chemical reactors. The value of a variable in the n-th stage is noted by a subscript n. For example, if y_n,i denotes the composition of the i-th species on the n-th stage of a distillation column, x_n,i is the corresponding liquid composition, R is the reflux ratio (ratio of liquid returned to the column to product removed from the condenser), and K_n,i is the equilibrium constant, then the mass balances about the top of the column give

while the equilibrium equation gives

If these are combined we get

which is a linear difference equation. This particular problem is quite complicated and the interested reader is referred to Amundson [1966, Ch. 6]. However, the form of the difference equation is clear. Several examples are given here for solving difference equations. More complete information is available in Perry [1997].

An equation in the form

can be solved by

Usually difference equations are solved analytically only for linear problems. When the coefficients are constant and the equation is linear and homogeneous a trial solution of the form

is tried; f is raised to the power n. For example, the difference equation

coupled with the trial solution would lead to the equation

This gives

where A and B are constants which must be specified by boundary conditions of some kind.

When the equation is non-homogeneous, the solution is represented by the sum of a particular solution and a general solution to the homogeneous equation.

The general solution is the one found for the homogeneous equation and the particular solution is any solution to the non-homogeneous difference equation. This can be found by methods which are analogous to those used to solve differential equations: the method of undertermined coefficients and the method of variation of parameters.

The last method applies to equations with variable coefficients, too. For a problem like

the general solution is

This can then be used in the method of variation of parameters to solve the equation