In this section we need to take a look at the equation of a line in R 3 R 3 . As we saw in the previous section the equation y = m x + b y = m x + b does not describe a line in R 3 R 3 , instead it describes a plane. This doesn’t mean however that we can’t write down an equation for a line in 3-D space. We’re just going to need a new way of writing down the equation of a curve. So, before we get into the equations of lines we first need to briefly look at vector functions. We’re going to take a more in depth look at vector functions later. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example. So, consider the following vector function. → r ( t ) = ⟨ t , 1 ⟩ r → ( t ) = ⟨ t , 1 ⟩ A vector function is a function that takes one or more variables, one in this case, and returns a vector. Note as we
So, we’ve finally seen an example where the constant of separation didn’t have a minus sign and again note that we chose it so that the boundary value problem we need to solve will match one we’ve already seen how to solve so there won’t be much work to there.
All the examples worked in this section to this point are all problems that we’ll continue in later sections to get full solutions for. Let’s work one more however to illustrate a couple of other ideas. We will not however be doing any work with this in later sections however, it is only here to illustrate a couple of points.
Example 6 Use Separation of Variables on the following partial differential equation.
Note that this is a heat equation with the source term of
and is both linear and homogenous. Also note that for the first time we’ve mixed boundary condition types. At we’ve got a prescribed temperature and at we’ve got a Newton’s law of cooling type boundary condition. We should not come away from the first few examples with the idea that the boundary conditions at both boundaries always the same type. Having them the same type just makes the boundary value problem a little easier to solve in many cases.
So, we’ll start off with,
and plugging this into the partial differential equation gives,
Now, the next step is to divide by and notice that upon doing that the second term on the right will become a one and so can go on either side. Theoretically there is no reason that the one can’t be on either side, however from a practical standpoint we again want to keep things a simple as possible so we’ll move it to the side as this will guarantee that we’ll get a differential equation for the boundary value problem that we’ve seen before.
So, separating and introducing a separation constant gives,
The two ordinary differential equations that we get are then (with some rewriting),
Now let’s deal with the boundary conditions.
and we can see that we’ll only get non-trivial solution if,
So, here is what we get by applying separation of variables to this problem.
So, we’ll start off with,
and plugging this into the partial differential equation gives,
Now, the next step is to divide by and notice that upon doing that the second term on the right will become a one and so can go on either side. Theoretically there is no reason that the one can’t be on either side, however from a practical standpoint we again want to keep things a simple as possible so we’ll move it to the side as this will guarantee that we’ll get a differential equation for the boundary value problem that we’ve seen before.
So, separating and introducing a separation constant gives,
The two ordinary differential equations that we get are then (with some rewriting),
Now let’s deal with the boundary conditions.
and we can see that we’ll only get non-trivial solution if,
So, here is what we get by applying separation of variables to this problem.
On a quick side note we solved the boundary value problem in this example in Example 5 of the Eigenvalues and Eigenfunctions section and that example illustrates why separation of variables is not always so easy to use. As we’ll see in the next section to get a solution that will satisfy any sufficiently nice initial condition we really need to get our hands on all the eigenvalues for the boundary value problem. However, as the solution to this boundary value problem shows this is not always possible to do. There are ways (which we won’t be going into here) to use the information here to at least get approximations to the solution but we won’t ever be able to get a complete solution to this problem.
Okay, that’s it for this section. It is important to remember at this point that what we’ve done here is really only the first step in the separation of variables method for solving partial differential equations. In the upcoming sections we’ll be looking at what we need to do to finish out the solution process and in those sections we’ll finish the solution to the partial differential equations we started in Example 1 – Example 5 above.
Also, in the Laplace’s Equation section the last two examples show pretty much the whole separation of variable process from defining the product solution to getting an actual solution. The only step that’s missing from those two examples is the solving of a boundary value problem that will have been already solved at that point and so was not put into the solution given that they tend to be fairly lengthy to solve.
We’ll also see a worked example (without the boundary value problem work again) in the Vibrating String section.
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