Consider the heat equation with homogenous boundry condition
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Okay, it is finally time to completely solve a partial differential equation. In the previous section we applied separation of variables to several partial differential equations and reduced the problem down to needing to solve two ordinary differential equations. In this section we will now solve those ordinary differential equations and use the results to get a solution to the partial differential equation. We will be concentrating on the heat equation in this section and will do the wave equation and Laplace’s equation in later sections.
The first problem that we’re going to look at will be the temperature distribution in a bar with zero temperature boundaries. We are going to do the work in a couple of steps so we can take our time and see how everything works.
The first thing that we need to do is find a solution that will satisfy the partial differential equation and the boundary conditions. At this point we will not worry about the initial condition. The solution we’ll get first will not satisfy the vast majority of initial conditions but as we’ll see it can be used to find a solution that will satisfy a sufficiently nice initial condition.
Explanation:
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