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Quiz-6 / Qz6-TWO-C
« Last post by duoyizhang on March 25, 2021, 01:45:34 PM »
Question:Find Fourier transformation of the function $$e^{-\alpha x^2/2}sin(\beta x)$$
with $$\alpha>0,\beta>0$$
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Quiz-6 / Section 0201 quiz6-question 2B
« Last post by Houze Xu on March 18, 2021, 01:09:00 PM »
f(x) = e^−α|x|sin(βx)
Answer: f1(x) = e^−α|x|
f1^(k) = (2/pi)^1/2(α/α^2 + k^2)
f^(k) = F(f(x)) = -i/2(F(e^−α|x|e^iβx)-F(e^−α|x|e^-iβx))
F(e^−α|x|e^iβx) = (2/pi)^1/2(α/α^2 +(k^2- β^2)^2)
F(e^−α|x|e^-iβx))= (2/pi)^1/2(α/α^2 +(k^2+β^2)^2)
f^(k) = -i/(2pi)^1/2(α/α^2 +(k^2- β^2)^2 - α/α^2 +(k^2+β^2)^2 )
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Quiz-5 / Qz5-THREE-E
« Last post by duoyizhang on March 16, 2021, 01:27:02 AM »
The question is:Decompose f(x) = xcos(x)  into full Fourier series on interval [0, pi].
My confusion is how to decompose it on an interval like  [0,𝑙] rather than[-l,l]
Firstly,I compute f(x) into  full Fourier series on interval [-pi, pi] by the formula,Which is $$f1=f(x)=-\frac{sinx}{2}+\sum_{n=2}^\infty\frac{2n(-1)^{n}}{n^{2}-1}$$
Then what should we do to compute f(x) on[0,pi],I tried to use the property that f(x)is an odd function but it seems to be wrong.
Any help will be appreciated!
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Test-2 / Re: Test-2 problem-1 confusion regarding boundary conditions
« Last post by Victor Ivrii on March 07, 2021, 04:12:41 AM »
Solution is allowed to be discontinuous.
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Test-2 / Test-2 problem-1 confusion regarding boundary conditions
« Last post by aryakim on March 06, 2021, 10:07:36 AM »
I had a question about problem $1$ of my Test$2$. (My version was called alternative-F, night section).There was something confusing about this problem that I realized during the test.
Here is the problem:

\begin{equation}
\nonumber
\left\{ \begin{aligned}
& u_{tt}-u_{xx}=0, &&0< t < \pi, 0 < x < \pi, &(1.1) \\\
&u|_{t=0}= 2\cos (x),   && 0< x < \pi, &(1.2)\\
&u_t|_{t=0}= 0,  && 0< x < \pi,  &(1.3)\\
&u|_{x=0}= u|_{x= \pi}=0, && 0< t < \pi.  &(1.4)
\end{aligned}
\right.
\end{equation}

So, on the $t-x$ diagram, the lines $t = 0$ and $x = \pi$ intersect at ($t=0,x=\pi$), which will be a boundary point of the region where $0<t<x<\pi$.
If this point (i.e. ($t=0,x=\pi$)) on the diagram is approached by the line $t = 0$, equation $(1.2)$ is used to conclude that the value of $u(x,t)$ approaches $-2$.
On the other hand, if the point is approached from the line $x = \pi$,  $u(x,t)$ should become zero, which is in contradiction with the other boundary condition. It seems this would make it impossible to incorporate conditions $(1.2)$ and $(1.4$) at the same time. I was hoping someone could clear my confusion regarding this problem.


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Quiz-3 / QUIZ3 5301 TWO-C
« Last post by Jin Qin on February 19, 2021, 06:30:49 PM »
Hi, this is my answer for QUIZ3 TWO-C in section 5301. Hope this can help you out!
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Chapter 3 / Re: 3.1 Heaviside step function
« Last post by Victor Ivrii on February 09, 2021, 05:11:06 AM »
Everything is correct. You need to look carefully at limits in the integrals
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Chapter 3 / Re: 3.2 Theorem1
« Last post by Victor Ivrii on February 09, 2021, 05:10:26 AM »
$\int_0^\infty$. I fixed it
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Chapter 3 / 3.2 Theorem1
« Last post by SelinaW on February 09, 2021, 04:05:58 AM »
I am not sure what should this integral integrate over. Is it -∞ to +∞ or 0 to +∞?
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Chapter 3 / 3.1 Heaviside step function
« Last post by SelinaW on February 09, 2021, 04:00:59 AM »
I believe as t->0+ and x>0, the integral I marked in red should be sqrt(pi), then U(x, t) should be 1/2. I do not know how we get 1.
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