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Messages - Victor Ivrii

Pages: [1] 2 3 ... 153
1
Chapter 1 / Re: chapter 1 Problem 4 (1)
« on: Today at 01:29:44 AM »
Display formulae are surrounded by double dollars and no empty lines. Multiline formulae use special environments (google LaTeX gather align

2
Chapter 1 / Re: home assignment1 Q3(1),(2)
« on: January 16, 2022, 05:47:56 PM »
OK. Remarks:

1. Do not use $*$ as a multiplication sign!
2. Do not use LaTeX for italic text (use markdown of the forum--button I)
3. Escape ln, cos, .... : \ln (x) to produce $\ln (x)$ and so on

3
Chapter 1 / Re: Classification of PDEs
« on: January 14, 2022, 01:47:15 PM »
Yes, all linear are also semilinear and all semilinear are also quasilinear. For full mark you need to provide the most precise classification. So, if equation is linear you say "linear", if it is semilinear but not  linear you say "semilinear but not  linear" and so on,... "quasilinear but not  semilinear" and "non-linear and not quasilinear".

4
Chapter 1 / Re: Classification of PDEs
« on: January 14, 2022, 02:45:57 AM »
In particular, the definition of a linear PDE, from the textbook, is: $au_{x}+bu_{y}+cu-f=0$, where $f= f(x,y)$. However, if we simply move the the $cu$ to the right-hand side, we get: $au_{x}+bu_{y}=f-cu$. Now, define $g(x,y,u) = f(x,y)-cu$, then $au_{x}+bu_{y}=g(x,y,u)$, and the right-hand side now depends on lower-order derivatives, so by definition, it's quasi-linear. Could someone help identify the issue with this argument?
First, it will be not just quasilinear, but also  semilinear. Second, it will also be linear since you can move $c(x,y)u$ to the left

Good job, you mastered some $\LaTeX$ basics. :)

5
Chapter 1 / Re: Second Order canonical Form
« on: January 13, 2022, 07:24:23 PM »
We replace differentiation by $x$, y$ by multiplication on $\xi,\eta$. So $\partial_x^2 \mapsto \xi^2$ (just square); as a result senior terms like $Au_{xx}+2Bu_{xy}+ Cu_{yy}$ are replaced by quadratic form $A\xi^2+2B\xi\eta+C\eta^2$.

In the Linear Algebra you studied quadratic forms, right? And you know that
  • if  $AC-B^2 >0$ the canonical form is $\pm (\xi^2+\eta^2)$ (as $\pm A>0$)
  • if  $AC-B^2 <0$ the canonical form is $ (\xi^2-\eta^2)$,
  • if  $AC-B^2 =0$, but at least one of coefficients is not $0$ the canonical form is $\pm \xi^2$.
 

6
Test-2 / Re: Test-2 problem-1 confusion regarding boundary conditions
« on: March 07, 2021, 04:12:41 AM »
Solution is allowed to be discontinuous.

7
Chapter 3 / Re: 3.1 Heaviside step function
« on: February 09, 2021, 05:11:06 AM »
Everything is correct. You need to look carefully at limits in the integrals

8
Chapter 3 / Re: 3.2 Theorem1
« on: February 09, 2021, 05:10:26 AM »
$\int_0^\infty$. I fixed it

9
Chapter 2 / Re: Example 6 from Week 3 Lec 2
« on: February 02, 2021, 04:31:46 PM »
Indeed

10
Quiz 1 / Re: Quiz 1 - Variant 2E - Problem 2
« on: January 29, 2021, 11:45:32 AM »
Try to avoid high-riser notations. Several years ago I was a referee for a paper which used notations like this $\widehat{\dot{\widetilde{\mathcal{D}}}}$ and sometimes this little pesky dot was missing  :D

11
Quiz 1 / Re: Quiz 1 - Variant 2E - Problem 1
« on: January 29, 2021, 11:41:45 AM »
Indeed, this equation has $u=0$ as a solution but the notion of "homogeneou"s does not apply to nonlinear.

12
Quiz 1 / Re: LEC9101 Quiz#1 B Question#2
« on: January 28, 2021, 08:01:17 PM »
not $sin(x)$ but $\sin(x)$ (it is \sin, \cos, \ln, \sum, \int and so on for "math operators")

13
Chapter 1 / Re: Math Notation Question
« on: January 24, 2021, 07:56:05 AM »
it depends: if $u=u(x)$ you can use either. If $u=u(x,y)$ then only partial derivative would be correct.

14
Chapter 3 / Re: Rouché's Theorem
« on: December 12, 2020, 06:25:05 PM »
You need to indicate that there are no zeroes on $\gamma$

15
Chapter 7 / Re: Drawing phase portrait with complex eigenvalues question
« on: December 01, 2020, 08:46:13 PM »
"How ellipses wouls look like" means the directions and relative size of their semi-axis. See frame 4 of MAT244_W8L3 handout

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