Temperature/heat equation

Question

I solved this problem $$\left\{\begin{array}{ll} u_{t}=ku_{xx}, & x\in(0,1), t>0 \\ u(0,t)=2, u(1,t)=3, & t>0 \\ u(x,0)=x^{2}+x+2, & x\in(0,1) \end{array}\right.$$ and I got this $$u(x,t)=2+x+\sum_{n=1}^{\infty} c_{n}e^{-n^{2}\pi^{2}kt}\sin(n\pi x)$$

About this, I had no problem.

My questions are (actually, my teacher's questions):

1- After a long time, is there any point in the bar which the temperature is $10^ºC$? If yes or no, why?

2- Can I use the fact that both temperatures in the bar's extremes are less than 10? If not, how could I answer this?

Answer 1

There is a property of certain PDEs known as the Maximum Principle. (See these notes (pdf warning) for its application to the heat equation.)

The maximum principle states that any solution $u(x,t)$ of the heat equation in a region of $(x,t)$ must have its extrema (minimum and maximum values) on the boundary of that region.

In this case, the region is $(x,t)\in[0,1]×[0,\infty)$. The boundary of this region has three parts: On $(x=0,t>0)$ the solution is fixed at 2. On $(x=1,t>0)$ the solution is fixed at 3. On $(x,t=0)$ the maximum value of 4 occurs at $x=1$. Thus 4 is the maximum value of u(x,t) which occurs on the boundary. By the maximum principle 4 is therefore the maximum value for all $x$ and $t$.

So $u(x,t) = 10$ cannot occur for any $x$ and $t$.

Answer 2

The "Maximum Principle" is a very useful tool to answer to such a question (as John Barber rightly did). I will not come back with this principle to repeat what was already said and which is more general for all times from $t=0$ to $t\to\infty$.

In the present case, since the wording of the question concerns only $t\to\infty$ and where it is not question of the intermediate times, it is sufficient to observe that : $$u(x,t)=2+x+\sum_{n=1}^{\infty} c_{n}e^{-n^{2}\pi^{2}kt}\sin(n\pi x)$$ tends to : $$u(x,\infty)=2+x$$ because $e^{-n^{2}\pi^{2}k\:\infty}=0$

So, after a long time, the temperature profile tends to become linear, from $2$ (at $x=0$) to $3$ (at $x=1$). $$2\leq u(x,\infty)\leq 3$$

NOTE :

(For information, of no use to answer to the question)

In the above equation, the condition $u(x,0)=x^2+x+2$ in not taken into account.

In order to compute the values of the coefficients $c_n$ : $$u(x,0)=2+x+\sum_{n=1}^{\infty} c_{n} \sin(n\pi x)=x^2+x+2$$ $$x^2=\sum_{n=1}^{\infty} c_{n} \sin(n\pi x)$$ This is a Fourier series which coefficients are : $$c_n=2\int_0^1 x^2\sin(n\pi x)dx= 4\frac{(-1)^n-1}{\pi^3n^3}-2\frac{(-1)^n}{n\pi}$$ $$u(x,t)=2+x+\sum_{n=1}^{\infty} \left(4\frac{(-1)^n-1}{\pi^3n^3}-2\frac{(-1)^n}{n\pi} \right)e^{-n^{2}\pi^{2}kt}\sin(n\pi x)$$