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Messages - Victor Ivrii

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1
Web Bonus Problems / Web Bonus problem -- Reading Week
« on: February 16, 2018, 08:27:43 AM »
$\renewcommand{\Re}{\operatorname{Re}}$
Find Fourier transform of $e^{-\frac{1}{2}a x^2}$, where $a\in \mathbb{C}$ with $\Re a>0$.

To do this

(a) Check that $u=e^{-\frac{1}{2}a x^2}$ satisfies $u'(x) = -ax u(x)$; then using Fourier Transform properties conclude that $\hat{u}$ satisfies $i \omega \hat{u}(\omega)= -i a\hat{u}'(\omega)$ and solving this equation find $\hat{u}$ up to a constant factor $C$.

(b) Then calculate $I:=\hat{u}(0)=\int e^{-\frac{1}{2}a x^2}\,dx$ in the same way as we did it for real $a>0$ but justify the correct "sign" (since $I$ is recovered up to a factor $\pm 1$), comparing with the case of real $a>0$.

2
Term Test 1 / P5 Night
« on: February 15, 2018, 08:27:08 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Find the solution $u(x,t)$ to
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,\tag{1}\\[2pt]
&u|_{t=0}=\left\{\begin{aligned} &x &&|x|<1,\\ &0 &&|x|\ge 1,\end{aligned}\right.\tag{2}\\
&\max |u|<\infty.\tag{3}
\end{align}
Calculate the integral.

Hint: For $u_t=ku_{xx}$ use
\begin{equation}
G(x,y,t)=\frac{1}{\sqrt{4\pi kt}}\exp (- (x-y)^2/4kt).
\tag{4}
\end{equation}
To calculate integral make change of variables and use $\erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-z^2}\,dz$.

3
Term Test 1 / P4 Night
« on: February 15, 2018, 08:23:58 PM »
Consider the PDE  with boundary conditions:
\begin{align}
&u_{tt}+c^2u_{xxxx}  + a u=0,\qquad&&0<x<L, \tag{1}\\
&u|_{x=0} =u_{x}|_{x=0}=0,\tag{2}\\
&u_{xx}|_{x=L} =u_{xxx}|_{x=0}=0|_{x=L}=0, tag{3}
\end{align}
where  $c>0$ and $a$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{xx}^2 +au^2)\,dx
\tag{4}
\end{equation}
does not depend on $t$.

4
Term Test 1 / P3 Night
« on: February 15, 2018, 08:21:20 PM »
Find   solution to
\begin{align}
&u_{tt}-u_{xx}=0, &&   t>0,\; x>  2-2\sqrt{t+1} \tag{1}\\
&u|_{t=0}=0, && x>0,\tag{2}\\
&u_t|_{t=0}=0, && x>0,\tag{3}\\
&u|_{x=2-2\sqrt{t+1}}= t, &&t>0.\tag{4}
\end{align}

5
Term Test 1 / P2 Night
« on: February 15, 2018, 08:19:03 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Find solution $u(x,t)$ to
\begin{align}
&u_{tt}-9u_{xx}=18 e^{-x^2},\tag{1}\\
&u|_{t=0}=0, \quad u_t|_{t=0}=0.\tag{2}
\end{align}

Hint: Change order of integration over characteristic triangle.
Use $\erf(x)=\frac{2}{\sqrt{\pi}}\int _0^x e^{-z^2}\,dz$.

6
Term Test 1 / P1 Night
« on: February 15, 2018, 08:18:09 PM »
 Consider the first order equation:
\begin{equation}
u_t +  xt  u_x =  x te^{-t^2/2}.
\tag{1} 
\end{equation}

(a)  Find the characteristic curves and sketch them in the $(x,t)$ plane.

(b) Write the general solution.

(c) Solve  equation (1)  with the initial condition $u(x,0)= x$. Explain why the solution is fully  determined by the initial condition.

7
Term Test 1 / P5
« on: February 15, 2018, 07:08:45 PM »
$\newcommand{erf}{\operatorname{erf}}$
Find the solution $u(x,t)$ to
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,
\tag{1}\\[2pt]
&u|_{t=0}=e^{-|x|}
\tag{2}\\
&\max |u|<\infty.
\tag{3}
\end{align}
Calculate the integral.


Hint: For $u_t=ku_{xx}$ use
\begin{equation}
G(x,y,t)=\frac{1}{\sqrt{4\pi kt}}\exp (- (x-y)^2/4kt).
\tag{4}
\end{equation}
To calculate integral make change of variables and use $\erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-z^2}\,dz$.

8
Term Test 1 / P4
« on: February 15, 2018, 07:06:24 PM »
Consider the PDE  with boundary conditions:
\begin{align}
&u_{tt}-c^2u_{xx}  + a u =0,\qquad&&0<x<L,\tag{1}\\
&(u_x -\alpha u_{tt})|_{x=0}=0,\tag{2}\\
&(u_x +\beta u_{tt})|_{x=L}=0\tag{3}
\end{align}
where  $c>0$, $\alpha>0$, $\beta>0$ and $a$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{x}^2 +au^2)\,dx +\frac{\alpha c^2}{2}u_t(0,t)^2+
\frac{\beta c^2}{2}u_t(L,t)^2
\tag{4}
\end{equation}
does not depend on $t$.

9
Term Test 1 / P3
« on: February 15, 2018, 07:05:01 PM »
Find  continuous solution to
\begin{align}
&u_{tt}-4u_{xx}=0, &&   t>0, x>-t,\tag{1}\\
&u|_{t=0}=0, && x>0,\tag{2}\\
&u_t|_{t=0}=0, && x>0,\tag{3}\\
&u_x|_{x=-t}= \sin(t), &&t>0.\tag{4}
\end{align}

10
Term Test 1 / P2
« on: February 15, 2018, 07:03:41 PM »
Find solution $u(x,t)$ to
\begin{align*}
&u_{tt}-4u_{xx}= \frac{8}{x^2+1},\tag{1}\\
&u|_{t=0}=0, \quad u_t|_{t=0}=0.\tag{2}
\end{align*}

Hint: Change order of integration over characteristic triangle

11
Term Test 1 / P1
« on: February 15, 2018, 06:56:34 PM »
Consider the first order equation:
\begin{equation}
u_t +  3(t^2-1)  u_x =  6t^2.
\tag{1} 
\end{equation}

(a)  Find the characteristic curves and sketch them in the $(x,t)$ plane.

(b)  Write the general solution.

(c) Solve  equation (1)  with the initial condition $u(x,0)= x$. Explain why the solution is fully  determined by the initial condition.


12
Term Test 1 / P4-Morning
« on: February 15, 2018, 05:12:34 PM »
Find the general solution for equation
\begin{equation*}
y''(t)+8y'(t)+25y(t)=9 e^{-4t}+ 104\sin(3t).
\end{equation*}

13
Term Test 1 / P3-Morning
« on: February 15, 2018, 05:12:13 PM »
(a) Find the general solution for equation
\begin{equation*}
y''(t)-5y'(t)+6y(t)=4e^{t}+e^{2t} .
\end{equation*}

(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

14
Term Test 1 / P2-Morning
« on: February 15, 2018, 05:10:58 PM »
(a)  Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE 
\begin{equation*}
-x^2\bigl(\ln(x)-1\bigr)y''+xy'-y=0
\end{equation*}

(b) Check that $y_1(x)=x$ is a solution and find another linearly independent solution.
 
(c) Write the general solution,  and find solution such that ${y(1)=1, y'(1)=0}$.


15
Term Test 1 / P1-Morning
« on: February 15, 2018, 05:09:54 PM »
Find integrating factor and then a general solution of ODE
\begin{equation*}
(4x y^2+3\ln(x)+1)+2x^2yy'=0 \ .
\end{equation*}
 
Also, find a solution satisfying $y(1)=1$.

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