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Messages - Victor Ivrii

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16
Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« on: March 19, 2019, 02:14:40 PM »
I think we need to make assumption that the Fourier transformation u will be 0 as y goes to infinity,that’s what my TA did in tutorial
Or, at least, does not grow exponentially as $|k|\to \infty$

17
Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« on: March 12, 2019, 06:25:23 PM »
There was an explanation why one of the solutions in half-plane should be rejected (does not satisfy condition at infinity)

18
APM346--Misc / Re: Spring 2018 Midterm Problem 3 (Both Main and Late)
« on: February 26, 2019, 04:15:46 PM »
Misprints. Corrected

19
APM346--Misc / Re: Scope of Term Test 1
« on: February 15, 2019, 02:00:37 AM »
Ch 1--3

20
Home Assignment 3 / Re: S2.3 Problem 8
« on: February 05, 2019, 09:14:53 PM »
[quoute]Hello professor, could u give me a hint as to why we have to impose the restriction for absolute value of x to be smaller or equal to 1?[/quote]
Draw characteristic lines and plot $t=x^2/2$

21
Home Assignment 3 / Re: HW2.3 Problem 6
« on: February 02, 2019, 08:17:10 PM »
In 6(4) How to ensure the general solution is continuous as r = 0 since the denominator includes r and in the numerator there are 2 arbitrary fuctions? Thanks!
Find conditions to these functions which are necessary and sufficient for the continuity

22
Home Assignment 3 / Re: Problem2(17) even or odd?
« on: February 01, 2019, 04:56:18 PM »
If you change $t\mapsto -t$, equation does not change, and $u|_{t=0}$ does not change, but $u_t|_{t=0}$ acquires sign "$-$". However, since $u|_{t=0}=0$, if you replace in addition $u\mapsto -u$, then nothing changes. Thus $u(x,t)=-u(x,-t)$.

If on the other hand, $h(x)=0$ then $u$ would be even with respect to $t$

23
Home Assignment 3 / Re: S2.3 problem2(17)(18)
« on: January 31, 2019, 12:16:47 PM »
No, you draw lines according to conditions: $x=1+ct$, $x=-1+ct$, $x=1-ct$ and $x=-1-ct$.

And normally one need to consider all 9 domains. However, our problems have two symmetries and it is sufficient to consider only 4 domains intersecting with the 1st quadrant and extend solution to the remaining 5, using the fact that solution is either odd or even with respect to $t$ (you need to understand, what is the case), and also solution is either odd or even with respect to $x$ (you need to understand, what is the case).

24
Home Assignment 3 / Re: S2.3 problem2(17)(18)
« on: January 31, 2019, 07:19:41 AM »
Since in this problem nothing is said about $t>0$ the complete solution should cover all cases. However, since in the problems either $g(x)=0$, or $h(x)=0$ then $u(x,t)$ is odd or even with respect to $t$, respectively. At least in some problems you can observe that solution must be even or odd with respect to $x$ as well.

In such problem, as (17), we have several regions. But we need to work out only some of them and extend to the rest by above arguments.

25
Home Assignment 3 / Re: S2.3 problem2(17)(18)
« on: January 31, 2019, 03:47:08 AM »
duplicate removed.

Yes, you need to consider all cases and it is recommended to draw a plane and different regions there

26
Home Assignment 3 / Re: HW2.3 Problem 6
« on: January 31, 2019, 03:41:57 AM »
"and thereby solve the spherical wave   equation." So, find $u$ from expression for $v$

27
I have difficulties obtaining the general solution of the equation $u_{tt} - c^2u_{xx} = 0$. From the online textbook Section2.3, it mentions $v = u_t +cu_x$, and it gets the result $v_t - cv_x = 0$ by chain rule.
Not by a chain rule. Just from equation
Quote
But when I expand $v_t - cv_x = 0$, I get an extra term $x'(t)$ when applies chain rule to $v_t$ because I think $u_{t}$ in $v$ can be two separated into to parts which are $u_t(t)$ and $u_t(x(t))$. Then applying chain rule, it becomes $v_t = u_{tt} + x'(t) +c(u_{xt} + x'(t))$ where the term $x'(t)$ can not be cancelled. I wonder where is the problem of my thought.
Several days ago it was rewritten to make it more clear. Read online version

28
Home Assignment 3 / Re: S2.3 Problem 8
« on: January 27, 2019, 07:09:03 PM »
Find the general solution and then plug to initial conditions

29
Home Assignment 2 / Re: problem 5 (23)
« on: January 27, 2019, 07:07:51 PM »
There is NO root. You need to parametrize before integration

30
APM346--Misc / Re: analytic extension
« on: January 26, 2019, 11:31:38 AM »
If series has an infinite radius of convergence then it converges on the whole plane. If the radius of convergence is finite ...

However, even if the radius of convergence is infinite, it does not answer to many questions. F.e. from the decomposition of $e^z$ one cannot derive that $e^x$ fast rapidly as $\mathbb{R}\ni x\to +\infty$ and rapidly decays as $\mathbb{R}\ni x\to -\infty$.

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