MAT 3724 Applied Analysis I

1. MAT 3724Applied Analysis I 2.1 Part ICauchy Problem for the Heat Equation http://myhome.spu.edu/lauw

2. Chapter 2 • PDE on Unbounded Region • In one dimension, it is the real line • Easier to solve than bounded region

3. Preview • Initial Value Problem with the Heat equation • Introduce Dimensional Analysis

4. Cauchy Problem for the Heat Equation • Note the change of notations (u instead of q)

5. Set Up Lateral side insulated Initially, the temp. distribution is given by ____

6. Example 1

7. Example 1:Thought Experiment Scenario 1Scenario 2

8. Example 1:Thought Experiment What would happen to w(x,t) as the time moves on?

9. Example 1: Simplification

10. Example 1: Simplification

11. Interpretations of HW 05 Problem 2 • The PDE model and inequality below appear in the last HW

12. Interpretations of HW 05 Problem 2 • Even though the set ups are not exactly the same, some calculations with this example may help us to understand what the inequality means.

13. Interpretations of HW 05 Problem 2

14. Interpretations of HW 05 Problem 2

15. Example 1 Solution Method: • Dimensional Analysis (units) • Guessing

16. Inspirations

17. Inspirations If we can find___________________, then we can recover_____________________.

18. Example 1 If we can find_________________________, then we can recover_____________________.

19. Notations Relaxation for Improper Integrals • Provided that you understand the correct concepts, you are allow to use less rigorous notations. Here is an illustration.