### Recent Posts

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##### Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« Last post by Victor Ivrii 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 $|k|\to \infty$
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##### Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« Last post by Zengyue Lin on March 17, 2019, 01:59:15 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
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##### Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« Last post by MikeMorris on March 17, 2019, 12:27:26 PM »
What condition is given at infinity? I don't see any condition given in the question for the behaviour of $u$ at infinity.
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##### Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« Last post by Zengyue Lin on March 16, 2019, 10:22:00 PM »
So for this question why we don’t use cos and sin for the general solution?
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##### Home Assignment 2 / Re: 2.4 problem4
« Last post by shuxian on March 14, 2019, 02:42:23 PM »
Here is my work.
For the source, the function should be f (y,t).Hoping it could help you.
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##### Home Assignment 6 / Re: Laplace Fourier Transform S5.3.P Q1
« Last post by Victor Ivrii 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)
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##### Home Assignment 6 / Laplace Fourier Transform S5.3.P Q1
« Last post by Sebastian Lech on March 12, 2019, 01:00:45 PM »
I was wondering if anyone could let me know how to move forward on problem one:

Consider Dirichlet problem:
$$$$u_{xx}+u_{yy}=0$$, -\infty<x<\infty, y>0$$
$$$$u|_{y=0}=f(x)$$$$
We need to make a Fourier Transform by x and leave the solution in the form of a Fourier Integral.
What I did first was make the Fourier transform:
$$$$\hat{u}_{yy}-\xi^2\hat{u}=0$$$$
$$$$\hat{u}|_{y=0}=\hat{f}(\xi)$$$$

Which has general solution:

$$$$\hat{u}(\xi, y)=A(\xi)e^{-|\xi|y}+B(\xi)e^{|\xi|y}$$$$
and using equation (4):
$$$$\hat{u}(\xi,0)=A(\xi)+B(\xi)=\hat{f}(\xi)$$$$
Which I am now stuck on, how do we solve for $A, B$ ?

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##### APM346--Misc / Scope of TT2?
« Last post by JUNJING FAN on March 11, 2019, 05:17:23 PM »
As titled, what chapters will be on TT2?
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##### APM346--Misc / Re: Midterm 2018 (spring night) Prob 3
« Last post by Aaron Pan on February 28, 2019, 02:50:44 AM »
I believe that is a typo. The question does not have anything to do with sine.
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##### Home Assignment 2 / 2.4 problem4
« Last post by Wanying Zhang on February 27, 2019, 10:30:59 PM »
The problem is given:
$$u_{tt} - u_{xx} = (x^2 -1)e^{-\frac{x^2}{2}}$$
$$u(x,0) = -e^{-\frac{x^2}{2}}, u_t(x,0) = 0$$

I have already got the general solution as followed, but I have trouble solving the integral,
$$\int_{0}^{t} \int_{x-t+s}^{x+t-s} (y^2-1)e^{-\frac{x^2}{2}}dyds$$
and I tried $\Delta$ method, but it seems to make the equation more complex. Professor, could you please give a hint of solving this problem?
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