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    One dimensional heat equation

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    I hope you and your family are well.

    I am currently taking a course in PDE's and would like a few things explained if possible.

    • Consider a bar insulated on both sides with the ends held at some constant temperature (other than 0) my analysis gives, that as times goes to infinity, the temperature goes to 0 which is definitely not accurate and I don't know where my mistake is.
    • How do I solve the same problem when the ends are at different temperatures?
    • How do I solve the same problem if the ends are insulated?

    Please show the analysis for each situation....I am using separation of variables and Fourier Series as the method of solution.

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    Solution Preview

    Hi C. Great to see you again.
    Here is the solution . I start by solving the most basic case (ends at zero temperature) and show how to generalize from there.
    Here is a hint: Convert the system to a system that you already solved - by adding a dummy function to the heat function. Choose the dummy function in such a way that it will not change the heat equation, but will change the boundary conditions to the one you already know the solution to.
    An at last I solve for the Neumann boundary conditions.

    The one dimensional homogenous heat equation on a finite domain.
    The heat equation is given by:
    Where the function is the temperature across the finite domain
    To solve this equation one must have three conditions.
    At time the temperature across the domain has an initial spatial distribution
    This is called the initial condition.
    The other two conditions come from the situation at the ends of the bar. These are called boundary conditions.
    The most basic boundary conditions is when the ends are kept at constant temperature. This is called "Dirichlet boundary condition":
    A special case is when both ends are kept at the same temperature.
    Another type of boundary condition is that of the heat flux through the ends. The heat flux is the change in the temperature across the boundary. This is called the "Neuman boundary condition":
    In most cases with Neumann conditions the ends are adiabatically insulated - that is there is no flux through the ends:
    And now we can start solving the homogenous heat equation.
    Let's start with the most simple case - both ends are kept at
    The system becomes:
    We assume we can write the function as a product of two single-variable ...

    Solution Summary

    The solution (15 pages) shows how to solve the heat homogeneous equation with ends at zero temperature, then when the ends are at constant different temperatures and then when the ends are insulated.