Question

Heat equation: Arbitrary temperatures at ends. If the ends x = 0 and x = L of the bar in the text are kept at constant temperatures U1 and U2, respectively. The initial temp distribution is given by u(x, 0) = f (x).

(a) What is the temperature u1(x) in the bar after a long time (theoretically, as t → ∞)? First guess, then calculate.

(b) What is the temperature at any t. Use the heat equation given by ut = c^2uxx

Answer 1

a) U(x) = U1 + x(U2-U1)/L

The edges are placed at temperatures U1 and U2 respectively. So, after a long time t, the temperature of the rod will change such that it varies linearly from U1 to U2 as x goes from 0 to L.

So,

b)

Using the heat equation,

Let

Substituting this in the above equation,

By the method of separation of variables,

With X(0) = U1 and X(L) = U2

Using the initial conditions,

and

Here, is an integer multiple of ​​​​​​​

So,

Using U(t=0) = f(x),

C = f(x)

Where is an integer multiple of