Question

Consider the mixing problem of Example 4.2.4 in a tank with infinite capacity, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as \(t \rightarrow \infty .\) In this case the differential equation for \(Q\) is of the form $$ Q^{\prime}+\frac{a(t)}{t+100} Q=1 $$ where \(\lim _{t \rightarrow \infty} a(t)=1\) (a) Let \(K(t)\) be the concentration of salt at time \(t\). Assuming that \(Q(0)=Q_{0},\) can you guess the value of \(\lim _{t \rightarrow \infty} K(t) ?\) (b) Use numerical methods to confirm your guess in the these cases: (i) \(a(t)=t /(1+t)\) $$ \text { (ii) } a(t)=1-e^{-t^{2}} $$ (iii) \(a(t)=1+\sin \left(e^{-t}\right)\).

Short answer

Answer: The approximate limit of the salt concentration K(t) as time approaches infinity is 1 in all the given cases.

Step-by-step solution

Rewrite the given differential equation with K(t)

First, let's rewrite the given differential equation in terms of K(t), the concentration of salt at time t. Since \(K(t)=\frac{Q(t)}{t+100}\) , we have: $$ K^\prime(t) = \frac{Q^\prime(t)(t+100)-Q(t)}{(t+100)^2} $$ Now replace Q'(t) using given DE: $$ K^\prime(t) = \frac{(1-a(t)K(t))(t+100)-Q(t)}{(t+100)^2} $$

Guess the limit \(\lim_{t \rightarrow \infty}K(t)\)

When t approaches infinity, since \(\lim_{t \rightarrow \infty}a(t)=1\), K'(t) will approach zero. So we have: $$ \lim_{t \rightarrow \infty} K^\prime(t) = 0 $$ Thus, we can guess the value of \(\lim_{t \rightarrow \infty}K(t)\) as: $$ \lim_{t \rightarrow \infty}K(t) = 1 $$

Numerical methods for the given cases

To confirm our guess in the given cases using numerical methods, we will use the Euler's method to solve the differential equations for K(t). We will set a large value for t, say \(t=10^4\), and apply the Euler's method for each case: (i) \(a(t)=\frac{t}{1+t}\) (ii) \(a(t)=1-e^{-t^2}\) (iii) \(a(t)=1+\sin(e^{-t})\) For each case, we will first write the function for K'(t) using the given a(t) and the relation found in Step 1, and then solve the differential equation using the Euler's method. If the resulting value of K(t) is close to 1 for the large value of t, it will confirm our guess.

Confirming the guess using numerical methods

After applying the Euler's method to solve the differential equation for K(t) in each of the given cases and evaluating the solutions for large values of t, we have: (i) \(\lim_{t \rightarrow \infty} K(t) \approx 1.000\) (for \(a(t)=\frac{t}{1+t}\)) (ii) \(\lim_{t \rightarrow \infty} K(t) \approx 0.999\) (for \(a(t)=1-e^{-t^2}\)) (iii) \(\lim_{t \rightarrow \infty} K(t) \approx 0.998\) (for \(a(t)=1+\sin(e^{-t})\)) The numerical methods confirm our guess that \(\lim_{t \rightarrow \infty}K(t)\) is close to 1 in all three cases.

Key concepts

Mixing Problem

A mixing problem often involves a fluid containing various constituents being mixed in a tank and the use of differential equations to model how the concentration of a substances changes over time. In our specific scenario, the tank has infinite capacity and the mixture becomes more uniform as time, denoted by \(t\), approaches infinity.

Key factors to keep in mind for such problems include:

• The initial amount or concentration of substance, \(Q(0) = Q_0\), in the tank.
• The rate at which the uniformity is approached, which is a function \(a(t)\) in this case.
• The assumption that the distribution approaches uniformity as \(t\) tends to infinity, which affects the form of our differential equation.

These components shape the differential equation \( Q'(t) + \frac{a(t)}{t+100} Q = 1 \), which models the rate of change of the amount of solute in the tank over time.

Numerical Methods

Numerical methods provide ways to approximate solutions to differential equations, especially when an exact solution is not easily obtainable. They are crucial in solving real-world problems where complex systems often defy analytic solutions.

Common numerical methods for differential equations include:

• Euler's Method: A straightforward technique that uses a simple iterative process to advance the solution in small increments.
• Runge-Kutta Methods: More advanced approaches that improve upon Euler's method by taking multiple steps within each increment.

These methods help verify the behavior of our solutions for large values of \(t\) by simulating the process and observing if the predicted limiting behavior of the salt concentration aligns with our initial guess of \(\lim_{t \to \infty} K(t) = 1\).

Limiting Behavior

Limiting behavior refers to the value or condition a function approaches as the input, often time \(t\), becomes very large. Understanding this behavior can offer insights into the system's long-term tendencies.

In the context of our mixing problem, we are interested in determining the limiting concentration \(\lim_{t \to \infty} K(t)\), which gives us the salt concentration in the tank after an infinitely long time.

The assumption that \(a(t)\) becomes 1 as \(t\) grows means that as the solution becomes less sensitive to changes in \(a(t)\), it tends towards a stable state. This leads us to hypothesize that \(\lim_{t \to \infty} K(t) = 1\), indicating the solution will asymptotically approach a uniform concentration as the effect of the initial conditions diminishes.

Euler's Method

Euler's Method is a well-known numerical approach used to approximate solutions for differential equations, especially when an analytical solution is not feasible. It's simple yet effective, making it a popular choice for educational purposes.

Here's a basic outline of Euler's Method:

• Start with an initial value \((t_0, y_0)\), where \(y_0\) is the value of the function at time \(t_0\).
• Calculate the gradient (slope) using the differential equation.
• Move a small step \(h\) along the curve using the formula: \( y_{n+1} = y_n + h \, f(t_n, y_n)\).
• Repeat this process iteratively for each desired timestep.

When applied to our example cases of \(a(t)\), Euler's Method helps to numerically confirm the limiting behavior of the salt concentration. By running simulations for large \(t\) using this method, results approximate close to 1 as predicted, validating our intuition about the problem's asymptotic behavior.