Math 4557 Comparison of Waves and Diffusions
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Math 4557
Comparison of Waves and Diffusions
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.
• History
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 explore.
• 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 equation
7. Explain what does it mean that the information is lost gradually in the heat equation
*8. Let u(x, t) solve the wave equation on the whole line with bounded second
derivatives. Show that
∞
v(x, t) = √4几kt(c) ∫ e −
u(x, s)ds
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.
2022-11-19