Question

We can approximate the continuous-time tank model of the previous problem by a discrete model as follows.

Assume that we only observe the tank contents each minute (time is now discrete). During each minute, 20 liters (or 10% of each tank’s contents) are transferred to the other tank.

Let x1(t) and x2(t) be the amounts of salt in each tank at time t. We then have:

x1(t + 1) = 9 /10 x1(t) + 1 /10 x2(t)

x2(t + 1) = 1 /10 x1(t) + 9 /10 x2(t)

Formulate the problem in the form x(t + 1) = Ax(t) where A is a 2 × 2 matrix, then solve for the amount of salt in each tank as a function of time using the eigenvalues and eigenvectors of A.

Sketch the graphs of the amount of salt in each tank as functions of time.

How does your solution compare to the continuous time model?

Answer 1