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Comparison of Waves and Diffusions
Last day to turn in the project: Monday, Dec 5

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Purpose: The purpose of this assignment is not only to identify the main
differences between the two classical problems: heat equation and wave equation,
but also to learn about the history that underlies these equations, about the minds
working on these problems, and to be aware of their struggles before success.

Skills: This homework will help you in the analysis on why certain properties are
associated with one of the equations and not the other, by reflecting on these
differences, you will gain a deeper understanding which will allow you to propose
possible problems that could be of interest to explore in the future.

Task: For this assignment, you are to write the solutions of the problems described
below in a framework that includes the following sections:

• Introduction
o Here you are to introduce the reader about the topic you will discuss,

what is to be expected and make your work easy to follow.

o In this portion of the work, you are to identify the time line where
these two classical problems first were discussed, and when
breakthroughs were achieved.

• Problems to solve
o This is just to include the problems to be solved given below.

• Solutions
o On this section, for each problem, you will write down your solution.

• Questions to explore in the future
o In the process of solving the problems, or in general, thinking about

these classical equations, you may wonder about generalizations, or
why something works, or anything that comes to your mind, here is
the place to phrase those questions that you find may be of interest to

• Conclusion
o With the conclusion you wrap up the framework open to your reader

at the introduction.
• Bibliography

o List all the sources that you used in the creation of your paper, using
the style of your choice, APA, MLA, or AMS. Decide on one format, and
be consistent.

Here are the problems to solve:

1. State carefully the maximum principle for the heat equation
2. Show that there is no maximum principle for the wave equation
3. Consider 𝑢(𝑥, 𝑡) = 𝑓(𝑥 − 𝑎𝑡) where 𝑓 is a given function of one variable

(a) If 𝑢(𝑥, 𝑡) is a solution of the wave equation, show that the speed must be
𝑎 = ±𝑐 (unless f is a linear function)

(b) If 𝑢(𝑥, 𝑡) is a solution of the heat equation, find f and show that the speed
a is arbitrary

4. What is the behavior of the solution to the wave equation as t goes to infinity
5. What is the behavior of the solution to the heat equation as t goes to infinity
6. Explain what does it mean that the information is transported in the wave

7. Explain what does it mean that the information is lost gradually in the heat

*8. Let 𝑢(𝑥, 𝑡) solve the wave equation on the whole line with bounded second

derivatives. Show that

Solves the diffusion equation.

Criteria for Success: Make sure you turn in a complete assignment, one of the most
important things is that needs to be your work, your own words, and the length
could be as long or as little as you want, think that this work will be read carefully,
so you want to make sure you make your meaning clear. A suggestion is that after
you finish, read it, and test if you can follow easily, and clearly.

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