In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

In this part we discuss applications of Fourier series to solving a certain type of partial differential equation (pde). In more detail, we discuss the heat equation. The aim is to show how Fourier series naturally come up in the solution of this equation.

The heat equation describes the heat distribution in space and time. To illustrate the method it is sufficient to consider only the case of one spatial variable here. Let denote the spatial variable, let denote time and let be a function of and . Then the heat equation is given by

where is a constant. The task is to find a function which satisfies the heat equation, some initial condition

where is, say, continuous, and (in our case) homogeneous boundary conditions

We solve the heat equation using separation of variables, that is, we assume that the solution to the problem is of the form

for a function and (note that does NOT depend on and that does NOT depend on ). In this case we have

where and Substituting this ansatz into the heat equation we obtain

We can now separate the functions and to obtain

Now observe that

is only a function of , whereas

is only a function of . Hence, the only way those two expressions can be equal is if both

are constant. Hence we have

Thus we obtain now two ordinary differential equations (ode)

where we set . The first ode has a solution of the form

(check this by substituting this equation into the ode). Now our solution should satisfy the boundary conditions

If or if , then for all and hence we only obtain the trivial solution for all and . Hence we assume now that , which implies that

The first equation implies that , hence we get from the second equation that

If , then for all and we only get the trivial solution. If , then . This holds if for some integer (for we only get the trivial solution again). Indeed, we get infinitely many solutions, where . Let now

Then we obtain infinitely many solutions to the ode (1) of the form

We now solve (2). We have ( is a constant given by the equation, but may take on different values depending on the solution). Hence for Hence for each natural number we obtain a solution to the ode (2) (where ) of the form

(check that this solves the ode (2)).

Thus we obtain solutions to the heat equation which satisfy the homogeneous boundary conditions of the form

We call these functions the **eigenfunctions** corresponding to the **eigenvalues**

We still need to find a solution which satisfies the initial condition. To do so, notice that any linear combination of the eigenfunctions is again a solution to the heat equation which satisfies the homogeneous boundary conditions. Hence, in general, we have a solution of the form

where are real numbers which we can choose such that satisfies the initial condition. More precisely, we need to choose such that

The last equation means that the are the Fourier coefficients of the Fourier sine series of . Hence

which completes our solution to the heat equation.

* Example* Solve the heat equation in one dimension with homogeneous boundary conditions, assuming that the initial temperature is given by

Is the process for solving heat equations something we should just remember and churn out and apply? From the lectures (and the one problem 13 on the heat equation in the Tute), it seems as if theres only ever one type of heat equation question where all that changes are the constants involved. That is, a solid bar with u=o at both ends and with a two piece linear function defined for f(x) = u(x,o)

Do we have to know about more complicated heat equation questions with more complex boundary conditions and functions? For example a more complex f(x) or a boundary condition where u(L,t) could be a function of t?

Hi Thien, you do not need to know about more complex boundary conditions, we only consider homogeneous boundary conditions . There might be other initial functions though and you should know how to find solutions in this case.

Here, the main purpose is to illustrate how Fourier (sine) series naturally come up in the solution of some pdes. More complex scenarios will be studied in Math2120 and Math2130.

Dear Josef,

Thank you for this useful notes. I have one question

my supervisor told me that the solution of the heat equation

(du/dt=d^2 u/dx^2) on the interval [0,2pi] with periodic

boundary condition u(0,t)=u(2pi,t)

and intial conditions

u(x,0)=1 when x in [pi/2, 3pi/2] and

u(x,0)=0 otherwise

the solution is (u(x,t) goes to 1/2) when ( t goes to infinity).

If you don’t mind, could you tell me is that right and why?

Thank you,

Mustafa