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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 $$\alpha>0,\beta>0$$
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 )
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!
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.
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.