Recent Posts

1

Chapter 10 / Re: S 10.1 Lemma

« Last post by Victor Ivrii on April 02, 2023, 05:15:13 AM »

This condition is not necessary. However more conditions to $\phi$ lead to a lesser class of admissible $\phi$ in the assumption  and thus to weaker assumption of Lemma 1, and thus to stronger and more general statement.

2

Chapter 10 / S 10.1 Lemma

« Last post by Issra on April 01, 2023, 07:31:40 AM »

Posted here since there is no section for Chapter 10.

In Lemma 1 in S10.1, why is the condition $\phi(t_0) = \phi(t_1) = 0 $ necessary?

3

Test 2 / Re: 2022 midterm 2 solution: Possible typos?

« Last post by Victor Ivrii on March 25, 2023, 06:48:46 AM »

Thanks! Fixed

4

Chapter 6 / Re: Question while calculating the solution to Laplace using separation of vars

« Last post by Victor Ivrii on March 24, 2023, 08:28:50 PM »

Section 4.1ODE is not enough. You need to take into account also boundary conditions. See Section 4.1

5

Test 2 / 2022 midterm 2 solution: Possible typos?

« Last post by ravanasa on March 22, 2023, 10:02:57 PM »

I've been looking through the midterm solutions from 2022 to study for the midterm coming up. I was wondering if it's possible there are typos in the solutions, or if you can explain the following if I am misunderstanding something:

1. In problem 1 of Main Sitting 2022, there is the equation T' + 4λT = 0. I believe this should be T'' + 4λT = 0, with the first T as a second derivative.

2. In problem 1 of Morning Sitting 2022, there is equation 4T'' + λT = 0. I believe this should be T'' + 9λT = 0.
In the same question, T_n = A_n cos(nt) + B_n sin(nt). I believe this should be T_n = A_n cos(3nt) + B_n sin(3nt).
In the same question, there are initial boundary conditions dependent on the value of t. I believe in the solution when plugging these in, they are swapped.

3. In problem 3 of Main Sitting 2022, there u(r,θ) =B₀ /2 + (a sum). Then u(1,θ) = A₀ /2 + (a sum). I'm not sure how B₀ turned into A₀ in this.

Thank you!

6

Chapter 6 / Question while calculating the solution to Laplace using separation of vars

« Last post by Zhonghan CHEN on March 19, 2023, 12:46:09 AM »

Hi Prof:

I have a question for solving the Laplacian operation using the separation of variables.
Slide: Week 9 Lecture 1 Page 4.

The queston happens in solving ${\theta}'' + \lambda \theta = 0$.

Q: I do not understand how do you derive the $\lambda_{n} = n^2$.

Also, I do not understand why we can get a expression for $\lambda$ ? To my understading, $\lambda$ is just a constant in the ODE.

Thanks

7

Chapter 4 / Re: When is it a good idea to assume solutions are separable?

« Last post by Victor Ivrii on February 27, 2023, 09:28:36 AM »

If domain is "simple" in the appropriate coordinate system it is a good idea to try to find such solutions. For linear equations we even can construct general solutions as linear combinations of such solutions. Details in the class later.

8

Chapter 4 / When is it a good idea to assume solutions are separable?

« Last post by Issra on February 23, 2023, 07:56:27 PM »

When is it a good idea to assume solutions are separable? Are there any implied assumptions about the nature of the solutions when we assume that they are separable?

9

Chapter 3 / Re: Solving Heat equation

« Last post by Victor Ivrii on February 13, 2023, 08:26:20 PM »

I will post solutions to show that all the integrals can be computed using elementary functions and $\operatorname{erf}(.)$

10

Chapter 3 / Solving Heat equation

« Last post by Yuchen Wu on February 12, 2023, 11:14:19 PM »

Dear Prof Ivrii,
I did some home assignment for heat equation problems to prepare the quiz, and I found some integrals seems hard to compute. So I am wondering if we need to compute the integral completely correct to get the full mark when do the quiz, or just simplify the heat formula is fine? Thank you.