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1.8: Problems

Last updated

Sep 4, 2024
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- 1.6: Modern Nonlinear PDEs
- 2: Second Order Partial Differential Equations

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Exercise $\PageIndex{1}$

Write the following equations in conservation law form, $ut + φ_x = 0$ by finding the flux function $φ(u)$ .

$u_t+cu_x=0.$
$u_t+uu_x-\mu u_{xx}=0.$
$u_t+6uu_x+u_{xxx}=0.$
$u_t+u^2u_x+u_{xxx}=0.$

Exercise $\PageIndex{2}$

Consider the Klein-Gordon equation, $u_{tt} − au_{xx} = bu$ for $a$ and $b$ constants. Find traveling wave solutions $u(x, t) = f(x − ct)$ .

Exercise $\PageIndex{3}$

Find the general solution $u(x, y)$ to the following problems.

$u_x=0.$
$yu_x-xu_y=0.$
$2u_x+3u_y=1.$
$u_x+u_y=u.$

Exercise $\PageIndex{4}$

Solve the following problems.

$u_x+2u_y=0, \: u(x,0)=\sin x.$
$u_t+4u_x=0,\: u(x,0)=\frac{1}{1+x^2}.$
$yu_x-xu_y=0,\: u(x,0)=x.$
$u_t+xtu_x=0,\: u(x,0)=\sin x.$
$yu_x+xu_y=0,\: u(0,y)=e^{-y^2}.$
$xu_t-2xtu_x=2tu,\: u(x,0)=x^2.$
$(y − u)u_x + (u − x)u_y = x − y,\: u = 0\text{ on }xy = 1.$
$yu_x + xu_y = xy,\: x, y > 0,\text{ for }u(x, 0) = e^{−x^2},\: x > 0\text{ and }u(0, y) = e^{−y^2},\: y > 0.$

Exercise $\PageIndex{5}$

Consider the problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition $u(x, 0) = \frac{1}{1+x^2}$ .

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Plot solutions $u(x, t)$ at times before and after the breaking time.

Exercise $\PageIndex{6}$

Consider the problem $u_t + u^2u_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition $u(x, 0) = \frac{1}{1+x^2}$ .

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Plot solutions $u(x, t)$ at times before and after the breaking time.

Exercise $\PageIndex{7}$

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Find the shock wave solution.

Exercise $\PageIndex{8}$

Consider the problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}1, &x\leq 0, \\ 2, &x>0.\end{array}\right.\nonumber$

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Find the shock wave solution.

Exercise $\PageIndex{9}$

Consider the problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}0, &x\leq -1, \\ 2, &|x|<1, \\ 1, &x>1.\end{array}\right.\nonumber$

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Find the shock wave solution.

Exercise $\PageIndex{10}$

Solve the problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}1, &x\leq 0, \\ 1-\frac{x}{a}, &0<x<a, \\ 0, &x\geq a.\end{array}\right.\nonumber$

Exercise $\PageIndex{11}$

Solve the problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}0, &x\leq 0, \\ \frac{x}{a}, &0<x<a, \\ 1, &x\geq a.\end{array}\right.\nonumber$

Exercise $\PageIndex{12}$

Consider the problem $u_t + u^2u_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}2, &x\leq 0, \\ 1,&x>0.\end{array}\right.\nonumber$

Find and plot the characteristics.
Graphically locate where a gradient catastrophe might occur. Estimate from your plot the breaking time.
Analytically determine the breaking time.
Find the shock wave solution.

Exercise $\PageIndex{13}$

Consider the problem $u_t + u^2u_x = 0,\: |x| < ∞,\: t > 0$ satisfying the initial condition

$u(x,0)=\left\{\begin{array}{cc}1, &x\leq 0, \\ 2,&x>0.\end{array}\right.\nonumber$

Find and plot the characteristics.
Find and plot the fan characteristics.
Write out the rarefaction wave solution for all regions of the $xt$ -plane.

Exercise $\PageIndex{14}$

Solve the initial-value problem $u_t + uu_x = 0,\: |x| < ∞,\: t > 0$ satisfying

$u(x,0)=\left\{\begin{array}{cc}1, &x\leq 0, \\ 1-x,&0\leq x\leq 1, \\ 0, &x\geq 1.\end{array}\right.\nonumber$

Exercise $\PageIndex{15}$

Consider the stopped traffic problem in a situation where the maximum car density is $200$ cars per mile and the maximum speed is $50$ miles per hour. Assume that the cars are arriving at $30$ miles per hour. Find the solution of this problem and determine the rate at which the traffic is backing up. How does the answer change if the cars were arriving at $15$ miles per hour.

Exercise $\PageIndex{16}$

Solve the following nonlinear equations where $p = u_x$ and $q = u_y$ .

$p^2+q^2=1,\: u(x,x)=x.$
$pq=u,\: u(0,y)=y^2.$
$p+q=pq,\: u(x,0)=x.$
$pq=u^2.$
$p^2+qy=u.$

Exercise $\PageIndex{17}$

Find the solution of $xp + qy − p^2q − u = 0$ in parametric form for the initial conditions at $t = 0$ :

$x(t,s)=s,\quad y(t,s)=2,\quad u(t,s)=s+1\nonumber$

Exercise 1.8.1\PageIndex{1}

Exercise 1.8.2\PageIndex{2}

Exercise 1.8.3\PageIndex{3}

Exercise 1.8.4\PageIndex{4}

Exercise 1.8.5\PageIndex{5}

Exercise 1.8.6\PageIndex{6}

Exercise 1.8.7\PageIndex{7}

Exercise 1.8.8\PageIndex{8}

Exercise 1.8.9\PageIndex{9}

Exercise 1.8.10\PageIndex{10}

Exercise 1.8.11\PageIndex{11}

Exercise 1.8.12\PageIndex{12}

Exercise 1.8.13\PageIndex{13}

Exercise 1.8.14\PageIndex{14}

Exercise 1.8.15\PageIndex{15}

Exercise 1.8.16\PageIndex{16}

Exercise 1.8.17\PageIndex{17}

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