Partial derivatives understanding.

[Eugene Sizov]

Hello, I've just learned about partial derivatives and their apply in heat physics, but still I don't understand one thing.
Suppose we have a heated rod, each point of which lies on x-axis, and its temperature changes over time. For the heat distribution along the rod, T(x) is given by a linear function for t=0, so its second derivative is zero. The rule says that the derivative of T(x,t) with respect to t equals the second derivative of T(x,t) with respect to x. So my question is: can I get the equation T(x,t) knowing only T(x) for t=0? Intuititively, it seems possible to predict the temperature at any x and t knowing the initial conditions. But as we derive the linear function twice, we get zero, so by the rule it seems like T doesn't change over time. And in fact, it changes and at all points approaches to the mean of T(x,0).

 

[HallsofIvy]

What you have discovered is that if the heat distribution is linear then it will NOT change with time! If, for example, the temperature at x= 0 is A and is kept at that temperature, the temperature at x= 1 is B and is kept at that temperature, then temperature as a function of x is T(x)= (B- A)x+ A, a linear function that is independent of time- the temperature remains constant at every point as time goes on. If the temperature changes at either end, then the temperature will change along the length and will NOT be linear.

 

[Eugene Sizov]

What you have discovered is that if the heat distribution is linear then it will NOT change with time! If, for example, the temperature at x= 0 is A and is kept at that temperature, the temperature at x= 1 is B and is kept at that temperature, then temperature as a function of x is T(x)= (B- A)x+ A, a linear function that is independent of time- the temperature remains constant at every point as time goes on. If the temperature changes at either end, then the temperature will change along the length and will NOT be linear.

Thank you for the explaination. Perhaps I had an illusion that temperature always tends to change over time like it does in nature due to not ideal conditions. Now it seems clear that a linear T(x) won't change because its each point is a mean of two nearest ones.