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
We’ve got one more section that we need to take care of before we actually start solving partial differential equations. This will be a fairly short section that will cover some of the basic terminology that we’ll need in the next section as we introduce the method of separation of variables.
Let’s start off with the idea of an operator. An operator is really just a function that takes a function as an argument instead of numbers as we’re used to dealing with in functions. You already know of a couple of operators even if you didn’t know that they were operators. Here are some examples of operators.
Or, if we plug in a function, say , into each of these we get,
These are all fairly simple examples of operators but the derivative and integral are operators. A more complicated operator would be the heat operator. We get the heat operator from a slight rewrite of the heat equation without sources. The heat operator is,
Now, what we really want to define here is not an operator but instead a linear operator. A linear operator is any operator that satisfies,
The heat operator is an example of a linear operator and this is easy enough to show using the basic properties of the partial derivative so let’s do that.
You might want to verify for yourself that the derivative and integral operators we gave above are also linear operators. In fact, in the process of showing that the heat operator is a linear operator we actually showed as well that the first order and second order partial derivative operators are also linear.
The next term we need to define is a linear equation. A linear equation is an equation in the form,
where is a linear operator and is a known function.
Here are some examples of linear partial differential equations.
The first two from this list are of course the heat equation and the wave equation. The third uses the Laplacian and is usually called Laplace’s Equation. We’ll actually be solving the 2-D version of Laplace’s Equation in a few sections. The fourth equation was just made up to give a more complicated example.
Notice as well with the heat equation and the fourth example above that the presence of the and do not prevent these from being linear equations. The main issue that allows these to be linear equations is the fact that the operator in each is linear.
Now just to be complete here are a couple of examples of nonlinear partial differential equations.
We’ll leave it to you to verify that the operators in each of these are not linear however the problem term in the first is the while in the second the product of the two derivatives is the problem term.
Now, if we go back to and suppose that then we arrive at,
We call this a linear homogeneous equation (recall that is a linear operator).
Notice that will always be a solution to a linear homogeneous equation (go back to what it means to be linear and use with any two solutions and this is easy to verify). We call the trivial solution. In fact, this is also a really nice way of determining if an equation is homogeneous. If is a linear operator and we plug in into the equation and we get then we will know that the operator is homogeneous.
We can also extend the ideas of linearity and homogeneous to boundary conditions. If we go back to the various boundary conditions we discussed for the heat equation for example we can also classify them as linear and/or homogeneous.
The prescribed temperature boundary conditions,
are linear and will only be homogenous if and .
The prescribed heat flux boundary conditions,
are linear and will again only be homogeneous if and .
Next, the boundary conditions from Newton’s law of cooling,
are again linear and will only be homogenous if and .
The final set of boundary conditions that we looked at were the periodic boundary conditions,
and these are both linear and homogeneous.
The final topic in this section is not really terminology but is a restatement of a fact that we’ve seen several times in these notes already.
Principle of Superposition
If and are solutions to a linear homogeneous equation then so is for any values of and .
Now, as stated earlier we’ve seen this several times this semester but we didn’t really do much with it. However, this is going to be a key idea when we actually get around to solving partial differential equations. Without this fact we would not be able to solve all but the most basic of partial differential equations.
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