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Note: 2 lectures, §9.5 in [EP], §10.5 in [BD]

Let us recall that a partial diﬀerential equation or PDE is an equation containing the partial derivatives with respect to several independent variables. Solving PDEs will be our main application of Fourier series.

A PDE is said to be linear if the dependent variable and its derivatives appear at most to the ﬁrst power and in no functions. We will only talk about linear PDEs. Together with a PDE, we usually have speciﬁed some boundary conditions, where the value of the solution or its derivatives is speciﬁed along the boundary of a region, and/or some initial conditions where the value of the solution or its derivatives is speciﬁed for some initial time. Sometimes such conditions are mixed together and we will refer to them simply as side conditions.

We will study three speciﬁc partial diﬀerential equations, each one representing a more general class of equations. First, we will study the heat equation, which is an example of a parabolic PDE. Next, we will study the wave equation, which is an example of a hyperbolic PDE. Finally, we will study the Laplace equation, which is an example of an elliptic PDE. Each of our examples will illustrate behavior that is typical for the whole class.

Let us ﬁrst study the heat equation. Suppose that we have a wire (or a thin metal rod) of length that is insulated except at the endpoints. Let denote the position along the wire and let denote time. See Figure 4.13.

Let denote the temperature at point at time . The equation governing this setup is the so-called one-dimensional heat equation:

where is a constant (the thermal conductivity of the material). That is, the change in heat at a speciﬁc point is proportional to the second derivative of the heat along the wire. This makes sense; if at a ﬁxed the graph of the heat distribution has a maximum (the graph is concave down), then heat ﬂows away from the maximum. And vice-versa.

We will generally use a more convenient notation for partial derivatives. We will write instead of , and we will write instead of . With this notation the heat equation becomes

For the heat equation, we must also have some boundary conditions. We assume that the ends of the wire are either exposed and touching some body of constant heat, or the ends are insulated. For example, if the ends of the wire are kept at temperature 0, then we must have the conditions

If, on the other hand, the ends are also insulated we get the conditions

In other words, heat is not ﬂowing in nor out of the wire at the ends. We always have two conditions along the axis as there are two derivatives in the direction. These side conditions are called homogeneous (that is, or a derivative of is set to zero).

Furthermore, suppose that we know the initial temperature distribution at time . That is,

for some known function . This initial condition is not a homogeneous side condition.

The heat equation is linear as and its derivatives do not appear to any powers or in any functions. Thus the principle of superposition still applies for the heat equation (without side conditions). If and are solutions and , are constants, then is also a solution.

Superposition also preserves some of the side conditions. In particular, if and are solutions that satisfy and , and , are constants, then is still a solution that satisﬁes and . Similarly for the side conditions and . In general, superposition preserves all homogeneous side conditions.

The method of separation of variables is to try to ﬁnd solutions that are sums or products of functions of one variable. For example, for the heat equation, we try to ﬁnd solutions of the form

That the desired solution we are looking for is of this form is too much to hope for. What is perfectly reasonable to ask, however, is to ﬁnd enough “building-block” solutions of the form using this procedure so that the desired solution to the PDE is somehow constructed from these building blocks by the use of superposition.

Let us try to solve the heat equation

Let us guess . We plug into the heat equation to obtain

We rewrite as

This equation must hold for all and all . But the left hand side does not depend on and the right hand side does not depend on . Hence, each side must be a constant. Let us call this constant (the minus sign is for convenience later). We obtain the two equations

In other words

The boundary condition implies . We are looking for a nontrivial solution and so we can assume that is not identically zero. Hence . Similarly, implies . We are looking for nontrivial solutions of the eigenvalue problem , , . We have previously found that the only eigenvalues are , for integers , where eigenfunctions are . Hence, let us pick the solutionsThe corresponding must satisfy the equation

By the method of integrating factor, the solution of this problem is

It will be useful to note that . Our building-block solutions are

We note that . Let us write as the sine series

That is, we ﬁnd the Fourier series of the odd periodic extension of . We used the sine series as it corresponds to the eigenvalue problem for above. Finally, we use superposition to write the solution as

Why does this solution work? First note that it is a solution to the heat equation by superposition. It satisﬁes and , because or makes all the sines vanish. Finally, plugging in , we notice that and so

Example 4.6.1: Suppose that we have an insulated wire of length 1, such that the ends of the wire are embedded in ice (temperature 0). Let . Then suppose that initial heat distribution is . See Figure 4.14.

We want to ﬁnd the temperature function . Let us suppose we also want to ﬁnd when (at what ) does the maximum temperature in the wire drop to one half of the initial maximum of 12.5.

We are solving the following PDE problem:

We write for as a sine series. That is, whereThe solution , plotted in Figure 4.15 for , is given by the series:

Finally, let us answer the question about the maximum temperature. It is relatively easy to see that the maximum temperature will always be at , in the middle of the wire. The plot of conﬁrms this intuition.

If we plug in we get

For and higher (remember is only odd), the terms of the series are insigniﬁcant compared to the ﬁrst term. The ﬁrst term in the series is already a very good approximation of the function. Hence

The approximation gets better and better as gets larger as the other terms decay much faster. Let us plot the function , the temperature at the midpoint of the wire at time , in Figure 4.16. The ﬁgure also plots the approximation by the ﬁrst term.

After or so it would be hard to tell the diﬀerence between the ﬁrst term of the series for and the real solution . This behavior is a general feature of solving the heat equation. If you are interested in behavior for large enough , only the ﬁrst one or two terms may be necessary.

Let us get back to the question of when is the maximum temperature one half of the initial maximum temperature. That is, when is the temperature at the midpoint . We notice on the graph that if we use the approximation by the ﬁrst term we will be close enough. We solve

That is,

So the maximum temperature drops to half at about .

We mention an interesting behavior of the solution to the heat equation. The heat equation “smoothes” out the function as grows. For a ﬁxed , the solution is a Fourier series with coeﬃcients . If , then these coeﬃcients go to zero faster than any for any power . In other words, the Fourier series has inﬁnitely many derivatives everywhere. Thus even if the function has jumps and corners, then for a ﬁxed , the solution as a function of is as smooth as we want it to be.

Now suppose the ends of the wire are insulated. In this case, we are solving the equation

Yet again we try a solution of the form . By the same procedure as before we plug into the heat equation and arrive at the following two equations

At this point the story changes slightly. The boundary condition implies . Hence . Similarly, implies . We are looking for nontrivial solutions of the eigenvalue problem , , . We have previously found that the only eigenvalues are , for integers , where eigenfunctions are (we include the constant eigenfunction). Hence, let us pick solutionsThe corresponding must satisfy the equation

For , as before,

For , we have and hence . Our building-block solutions will be

and

We note that . Let us write using the cosine series

That is, we ﬁnd the Fourier series of the even periodic extension of .

We use superposition to write the solution as

Example 4.6.2: Let us try the same equation as before, but for insulated ends. We are solving the following PDE problem

For this problem, we must ﬁnd the cosine series of . For we have

The calculation is left to the reader. Hence, the solution to the PDE problem, plotted in Figure 4.17, is given by the series

Note in the graph that the temperature evens out across the wire. Eventually, all the terms except the constant die out, and you will be left with a uniform temperature of along the entire length of the wire.

Exercise 4.6.2: Imagine you have a wire of length 2, with and an initial temperature distribution of . Suppose that both the ends are embedded in ice (temperature 0). Find the solution as a series.

Exercise 4.6.6: Find a series solution of

Hint: Use the fact that is a solution satisfying , , . Then use superposition.Exercise 4.6.7: Find the steady state temperature solution as a function of alone, by letting in the solution from exercises 4.6.5 and 4.6.6. Verify that it satisﬁes the equation .

Exercise 4.6.9 (challenging): Suppose that one end of the wire is insulated (say at ) and the other end is kept at zero temperature. That is, ﬁnd a series solution of

Express any coeﬃcients in the series by integrals of .