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### Messages - Tianyi Zhang

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##### Chapter 4 / 4.2 Example 4ï¼ˆperiodicï¼‰
« on: November 02, 2016, 02:10:55 PM »
$$X^{''} + \lambda X = 0$$
with condition  $$X(0) = X(l), X^{'}(0) = X^{'}(l)$$
how to get the answer $$\lambda_{2n-1} = \lambda_{2n} = (\frac{n\pi}{2l})^{2}$$ and the corresponding eigenfunctions?

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##### TT1 / Re: TT1-P2
« on: October 20, 2016, 04:22:17 PM »
For the new question,
(a)We can just apply the formula here to get:
$$u = {1 \over 2}( {e^{ - {{{{(x + t)}^2}} \over 2}}} + {e^{ - {{{{(x - t)}^2}} \over 2}}}) + {{1 \over {2c}}\int_0^t {\int_{x - t + t'}^{x + t - t'} {({y^2} - 1)} } {e^{ - {{{y^2}} \over 2}}}dydt'}$$

I skip all the integral parts because Shentao had it.

Lastly we can get,
$$u = {e^{ - {{{{(x + t)}^2}} \over 2}}} + {e^{ - {{{{(x - t)}^2}} \over 2}}}- {e^{ - {{{x^2}} \over 2}}}$$

(b)
Since the first and second items will vanish as t goes to infinity
$$u = - {e^{ - {{{x^2}} \over 2}}}$$

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##### Q3 / Re: Q3
« on: October 13, 2016, 09:39:13 PM »
Do we have to get the constant right? I don't remember u has to be continuous in the question.

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##### Chapter 2 / Re: Question 1 in section 2.4
« on: October 07, 2016, 05:11:51 PM »
Thank you, now I get it.

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##### Chapter 2 / Question 1 in section 2.4
« on: October 06, 2016, 10:40:50 PM »
In question 1, I have two integration problems.

(1): With condition line (4), first by using the d'Alembert I got:
$$\frac{1}{2c}\int_{0}^{t} \int_{x-c(t-t')}^{x+c(t-t')}\sin(\alpha x')sin(\beta t')dx'dt'$$
To integrate with respect to $x'$ is easy, I got this:
$$-\frac{1}{2\alpha c}\int_{0}^{t}[\cos(\alpha x+\alpha ct-\alpha ct')\sin(\beta t')-\cos(\alpha x-\alpha ct+\alpha ct')\sin(\beta t')]dt'$$
I don't know how to integrate with respect to $t'$ since now I have a mixture of $\sin$ and $\cos$ and they are both about $t$'.

(2): With condition line (7), first by using the d'Alembert I got:
$$\frac{1}{2c}\int_{0}^{t}\int_{x-c(t-t')}^{x+c(t-t')} F'''(x')t' dx'dt'$$
Integrate with respect to x', I got:
$$\frac{1}{2c}\int_{0}^{t}[F''(x+ct-ct')-F''(x-ct+ct')]t'dt'$$
I don't know how to integrate with respect to $t'$ either.

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##### Chapter 2 / About d'Alembert formula
« on: October 05, 2016, 04:35:43 PM »
Will we be tested on how to derive d'Alembert formula? Or do we just need to use it to solve problems? I really have trouble understanding this process.

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##### Chapter 2 / Textbook Section2.3 Question2
« on: October 04, 2016, 05:35:24 PM »
When g(x) = 0 and h(x) is a piecewise function, I'm totally lost. How to deal with the integral of h(x), especially when we have absolute value in this question? I know I have to break this into several cases. However, in our general solution, we always integral from x-ct to x+ct. In this question, I found it's possible x + ct <0 while x -ct > 0, then how do we integral?

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##### Chapter 1 / Re: How to determine what is semilinear and what is quasilinear?
« on: September 26, 2016, 02:29:26 PM »
Thank you SO MUCH! Now it's much clearer to me.

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##### Chapter 1 / How to determine what is semilinear and what is quasilinear?
« on: September 25, 2016, 01:00:20 PM »
In the textbook, the definitions are not very clear. I have trouble doing my week1 homework now.

Our TA said that if an equation is linear in its highest order, it's semilinear.

If coefficients of the derivatives depend on u, it's quasilinear.

But I think these definitions are different from what the textbook said.

Can anyone help me?

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