- #1
kairama15
- 31
- 0
I have recently been curious about heat diffusion. If there is space in one dimension with any kind of temperature dispersed throughout, then the heat equation states that the derivative of the temperature with respect to time at any point equals some constant (k) multiplied by the second derivative of temperature with respect to space at that point... or:
dT/dt=k*d^2T/d^2x
This is expressed in the Wikipedia page regarding the heat equation. This is intuitive, and I understand this differential equation.
I was reading up on ‘heat kernels’, which seem to be a tiny point of heat inside an object (let’s say a long metal rod with a point somewhere in it that is very hot). Through a lot of math it can be shown that the function that describes the temperature as a function of position throughout the rod and time is:
(1/sqrt(4*pi*k*t)) * e^(-x^2/(4*k*t)) (found on wikipedia)
This function makes sense. At time t, the function expresses the temperature as being concentrated at a point in the rod and as time goes by the temperature moves out of the point along the length of the rod.
However, I am curious if anyone has ever attempted to formulate an equation for temperature as a function of position and time f(x,t) for an infinitely sized rod whose one end is attached to an object that remains at constant temperature? So, instead of a heat kernel being placed somewhere in the middle of the rod, an object of always constant temperature is placed at one end of the rod, slowly heating the rod from one end. I have a suspicion the function will likely be something like:
T=e^(-k*x/t)
but I don't know. Does anyone know if a solution was ever developed for this?
dT/dt=k*d^2T/d^2x
This is expressed in the Wikipedia page regarding the heat equation. This is intuitive, and I understand this differential equation.
I was reading up on ‘heat kernels’, which seem to be a tiny point of heat inside an object (let’s say a long metal rod with a point somewhere in it that is very hot). Through a lot of math it can be shown that the function that describes the temperature as a function of position throughout the rod and time is:
(1/sqrt(4*pi*k*t)) * e^(-x^2/(4*k*t)) (found on wikipedia)
This function makes sense. At time t, the function expresses the temperature as being concentrated at a point in the rod and as time goes by the temperature moves out of the point along the length of the rod.
However, I am curious if anyone has ever attempted to formulate an equation for temperature as a function of position and time f(x,t) for an infinitely sized rod whose one end is attached to an object that remains at constant temperature? So, instead of a heat kernel being placed somewhere in the middle of the rod, an object of always constant temperature is placed at one end of the rod, slowly heating the rod from one end. I have a suspicion the function will likely be something like:
T=e^(-k*x/t)
but I don't know. Does anyone know if a solution was ever developed for this?
