Basics
We can use differenence equations whenever we have a system that can be modelled as quantity that changes "discretely" from one time period to the next and the changes can be expressed in terms of rates and amounts. By rate we mean an addition or subtraction that is based on current value while an amount is an absolute number to be added or subtracted from the quantity.
Suppose, for example, that our quantity is money in the bank. It can change from month to month based on interest earned (a change based on a rate) and funds deposited or withdrawn (amounts).
Generic Procedure
- Describe the situation
- Identify the quantity
- Note of the periodicity — what are our time units?
- Identify the increases and the decreases. Take note of which ones are amounts and which ones are rates.
- Write an equation that relates the quantity next time to the value of the quantity this time
- Solve this equation in terms of an initial or known value of the variable.
Examples
It can be useful to think in terms of a bathtub analogy — difference equations describe in-flows and out-flows and we are interested in the net result. The flows can be either rates or amounts. Rate here means a proportion of the variable of interest — leakage from the tub might be 10% of volume per time unit. Amount means an absolute number per time period — the faucet might supply 3 gallons per minute.
If we start with that distinction between amounts and rates, there are four basic types of difference equations, two simple and two mixed:Type | Description | Example | Base | Rates | Amounts | Equation |
---|---|---|---|---|---|---|
Simple | amount in and amount out | Enrollment (E), recruits (R), graduates (G) | E_{i} | +R-G | E_{i+1}=E_{i}+R-G | |
Simple | rate in and rate out | Population (P) with a birth rate (B) and a death rate (D) | P_{i} | B, D | P_{i+1}=P_{i}(1+B-D) | |
Mixed | amount in and rate out | Reservoir volume (V), rainfall (R), leakage (L) | V_{i} | L | R | V_{i+1}=V_{i}(1-L)+R |
Mixed | rate in and amount out | Endowment balance (B), interest (I), expenditure (E) | B_{i} | I | E | B_{i+1}=(1+I)B-E |