Heat equation with peculiar boundary conditions

[Skirdge]

Homework Statement

Find the solution to the heat equation for the following conditions:

Homework Equations

The Attempt at a Solution

Not sure. I've only encountered the following scenarios:

temperatures of both ends are arbitrary values
both ends are insulated (so the first partials with respect to x are both zero)
the rod is a circle so the temperatures of both ends and their rate of change are equal.

I tried using separation of variables but wasn't sure how to proceed after solving the equation for x.

 

[mighty2000]

Any help would be appreciated.



Thank you for your post. The heat equation is a fundamental equation in physics that describes the flow of heat in a given system. In order to solve this equation, we need to know the boundary conditions and initial conditions of the system. In your specific case, you mentioned three scenarios, and each one requires a different approach to find the solution to the heat equation.

1. For the first scenario where the temperatures of both ends are arbitrary values, we can use separation of variables to solve the equation. The solution will involve an infinite series of terms, and the coefficients of these terms can be determined by applying the boundary conditions at the two ends of the rod.

2. In the second scenario where both ends are insulated, the solution will involve a single term and the coefficients can be determined by applying the initial conditions at the starting time. This case is known as the homogeneous solution.

3. For the third scenario where the rod is a circle, we can use the method of images to solve the equation. This involves creating a mirror image of the rod and using the boundary conditions to determine the coefficients of the terms in the solution.

I hope this helps you in solving the heat equation for your specific case. If you need further assistance, please don't hesitate to ask. As scientists, it is important for us to work together and share our knowledge to advance our understanding of the world around us. Keep up the good work!