Chapter 5: Fourier series and PDEsContent check — Differential Equations (Lebl), Chapter 5: Fourier series and PDEs

Consider a spinning string of length 2 and linear density 0.1 and tension 3. Find the smallest angular velocity when the string pops out.

Suppose $x^{\prime \prime }+\lambda x=0$ and $x(0)=1$⁠, $x(1)=1$⁠. Find all $\lambda$ for which there is more than one solution. Also find the corresponding solutions (only for the eigenvalues).

Suppose $x^{\prime \prime }+x=0$ and $x(0)=0$⁠, $x^{\prime }(\pi )=1$⁠. Find all the solution(s) if any exist.

Consider $x^{\prime }+\lambda x=0$ and $x(0)=0$⁠, $x(1)=0$⁠. Why does it not have any eigenvalues? Why does every first-order equation with two endpoint conditions such as the one above have no eigenvalues?

(challenging) Suppose $x^{\prime \prime \prime }+\lambda x=0$ and $x(0)=0$⁠, $x^{\prime }(0)=0$⁠, $x(1)=0$⁠. Suppose that $\lambda >0$⁠. Find an equation that all such eigenvalues must satisfy. Hint: Note that $-\sqrt[3]{\lambda }$ is a root of $r^3+\lambda =0$⁠.

Suppose $f(t)$ is defined on $[-\pi ,\pi ]$ as $f(t)=\sin (t)$⁠. Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi ,\pi ]$ as $f(t)=\sin (\pi t)$⁠. Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi ,\pi ]$ as $f(t)={\sin }^2(t)$⁠. Extend periodically and compute the Fourier series.

Suppose $f(t)$ is defined on $(-\pi ,\pi ]$ as $f(t)=t^4$⁠. Extend periodically and compute the Fourier series.

Let

$$f(t)=t^2\text{for }-2<t\le 2$$

extended periodically.

a) Compute the Fourier series for $f(t)$⁠.

b) Write out the series explicitly up to the $3^{\text{rd}}$ harmonic.

Let

$$f(t)=t\text{for }-\lambda <t\le \lambda \text{ (for some }\lambda >0\text{)}$$

extended periodically.

a) Compute the Fourier series for $f(t)$⁠.

b) Write out the series explicitly up to the $3^{\text{rd}}$ harmonic.

Let

$$f(t)=\frac{1}{2}+\sum _{n=1}^{\infty }\frac{1}{n(n^2+1)}\sin (n\pi t).$$

Compute $f^{\prime }(t)$⁠.

Let

$$f(t)=\frac{1}{2}+\sum _{n=1}^{\infty }\frac{1}{n^3}\cos (nt).$$

a) Find the antiderivative.

b) Is the antiderivative periodic?

Let

$$f(t)={}^t/{}_2\text{for }-\pi <t<\pi$$

extended periodically.

a) Compute the Fourier series for $f(t)$⁠.

b) Plug in $t={}^{\pi }/{}_2$ to find a series representation for ${}^{\pi }/{}_4$⁠.

c) Using the first 4 terms of the result from part b) approximate ${}^{\pi }/{}_4$⁠.

Let

$$f(t)=\{\begin{array}{cc}0\hfill & \text{if }-2<t\le 0,\hfill \\ 2\hfill & \text{if }\phantom{-}0<t\le 2,\hfill \end{array}$$

extended periodically. Suppose $F(t)$ is the function given by the Fourier series of $f$⁠. Without computing the Fourier series evaluate

a) $F(0)$

b) $F(-1)$

c) $F(1)$

d) $F(-2)$

e) $F(4)$

f) $F(-9)$

Let $f(t)={}^t/{}_3$ on $0\le t<3$⁠.

a) Find the Fourier series of the even periodic extension.

b) Find the Fourier series of the odd periodic extension.

Let $f(t)=\cos (2t)$ on $0\le t<\pi$⁠.

a) Find the Fourier series of the even periodic extension.

b) Find the Fourier series of the odd periodic extension.

Let $f(t)$ be defined on $0\le t<1$⁠. Consider the average of the two extensions $g(t)=\frac{F_{\text{odd}}(t)+F_{\text{even}}(t)}{2}$⁠.

a) What is $g(t)$ if $0\le t<1$ (Justify!)

b) What is $g(t)$ if $-1<t<0$ (Justify!)

Let $f(t)={\sum }_{n=1}^{\infty }\frac{1}{n^2}\sin (nt)$⁠. Solve $x^{\prime \prime }-x=f(t)$ for the Dirichlet conditions $x(0)=0$ and $x(\pi )=0$⁠.

(challenging) Let $f(t)=t+{\sum }_{n=1}^{\infty }\frac{1}{2^n}\sin (nt)$⁠. Solve $x^{\prime \prime }+\pi x=f(t)$ for the Dirichlet conditions $x(0)=0$ and $x(\pi )=1$⁠. Hint: Note that $\frac{t}{\pi }$ satisfies the given Dirichlet conditions.

Let $F(t)=$ $\sin (2\pi t)+$ $0.1\cos (10\pi t)$⁠. Find the steady periodic solution to $x^{\prime \prime }+\sqrt{2}x=F(t)$⁠. Express your solution as a Fourier series.

Let $F(t)={\sum }_{n=1}^{\infty }{e}^{-n}\cos (2nt)$⁠. Find the steady periodic solution to $x^{\prime \prime }+3x=F(t)$⁠. Express your solution as a Fourier series.

Let $F(t)=|t|$ for $-1\le t\le 1$ extended periodically. Find the steady periodic solution to $x^{\prime \prime }+\sqrt{3}x=F(t)$⁠. Express your solution as a series.

Let $F(t)=|t|$ for $-1\le t\le 1$ extended periodically. Find the steady periodic solution to $x^{\prime \prime }+{\pi }^{2}x=F(t)$⁠. Express your solution as a series.

Find a series solution of

$$\begin{array}{cc}{u}_t=3u_{xx}\hfill & \text{for }0<x<\pi \text{ and }t>0,\hfill \\ u(0,t)=u(\pi ,t)=0\hfill & \text{for }t>0,\hfill \\ u(x,0)=5\sin (x)+2\sin (5x)\hfill & \text{for }0<x<\pi .\hfill \end{array}$$

Find a series solution of

$$\begin{array}{cc}{u}_t=0.1u_{xx}\hfill & \text{for }0<x<\pi \text{ and }t>0,\hfill \\ u_x(0,t)=u_x(\pi ,t)=0\hfill & \text{for }t>0,\hfill \\ u(x,0)=1+2\cos (x)\hfill & \text{for }0<x<\pi .\hfill \end{array}$$

Use separation of variables to find a nontrivial solution to $u_{xt}=u_{xx}$⁠.

Use a variation on separation of variables to find a nontrivial solution to $u_x+u_t=u$⁠. Hint: Try $u(x,t)=X(x)+T(t)$⁠.

Suppose that the temperature on the wire is fixed at $0$ at the ends, $L=1$⁠, $k=1$⁠, and $u(x,0)=100\sin (2\pi x)$⁠.

a) What is the temperature at $x={}^1/{}_2$ at any time?

b) What is the maximum and the minimum temperature on the wire at $t=0$⁠?

c) At what time is the maximum temperature on the wire exactly one half of the initial maximum at $t=0$⁠?

Solve

$$\begin{array}{cc}{y}_{tt}=y_{xx}\hfill & \text{for }0<x<\pi ,t>0,\hfill \\ y(0,t)=y(\pi ,t)=0\hfill & \text{for }t>0,\hfill \\ y(x,0)=\sin (x)\hfill & \text{for }0<x<\pi ,\hfill \\ y_t(x,0)=\sin (x)\hfill & \text{for }0<x<\pi .\hfill \end{array}$$

Solve

$$\begin{array}{cc}{y}_{tt}=25y_{xx}\hfill & \text{for }0<x<2\text{ and }t>0,\hfill \\ y(0,t)=y(2,t)=0\hfill & \text{for }t>0,\hfill \\ y(x,0)=0\hfill & \text{for }0<x<2,\hfill \\ y_t(x,0)=\sin (\pi x)+0.1\sin (2\pi x)\hfill & \text{for }0<x<2.\hfill \end{array}$$

Solve

$$\begin{array}{cc}{y}_{tt}=2y_{xx}\hfill & \text{for }0<x<\pi \text{ and }t>0,\hfill \\ y(0,t)=y(\pi ,t)=0\hfill & \text{for }t>0,\hfill \\ y(x,0)=x\hfill & \text{for }0<x<\pi ,\hfill \\ y_t(x,0)=0\hfill & \text{for }0<x<\pi .\hfill \end{array}$$

What happens when $a=0$⁠? Find a solution to $y_{tt}=0$⁠, $y(0,t)=y(\pi ,t)=0$⁠, $y(x,0)=\sin (2x)$⁠, $y_t(x,0)=\sin (x)$⁠.

Using the d’Alembert solution solve $y_{tt}=9y_{xx}$⁠, $0<x<1$⁠, $t>0$⁠, $y(0,t)=y(1,t)=0$⁠, $y(x,0)=\sin (2\pi x)$⁠, and $y_t(x,0)=\sin (3\pi x)$⁠.

Take $y_{tt}=4y_{xx}$⁠, $0<x<1$⁠, $t>0$⁠, $y(0,t)=y(1,t)=0$⁠, $y(x,0)=x-x^2$⁠, and $y_t(x,0)=0$⁠. Using the d’Alembert solution find the solution at

a) $t=0.1$⁠,

b) $t={}^1/{}_2$⁠,

c) $t=1$⁠.

You may have to split your answer up by cases.

Take $y_{tt}=100y_{xx}$⁠, $0<x<4$⁠, $t>0$⁠, $y(0,t)=y(4,t)=0$⁠, $y(x,0)=F(x)$⁠, and $y_t(x,0)=0$⁠. Suppose that $F(0)=0$⁠, $F(1)=2$⁠, $F(2)=3$⁠, $F(3)=1$⁠. Using the d’Alembert solution find

a) $y(\mathrm{1,1})$⁠,

b) $y(\mathrm{4,3})$⁠,

c) $y(\mathrm{3,9})$⁠.

In the region described by $0<x<1$ and $0<y<1$⁠, solve the problem

$$\mathrm{\Delta }u=0,u(x,0)=\sum _{n=1}^{\infty }\frac{1}{n^2}\sin (n\pi x),u(x,1)=0,u(0,y)=0,u(1,y)=0.$$

In the region described by $0<x<1$ and $0<y<2$⁠, solve the problem

$$\mathrm{\Delta }u=0,u(x,0)=0.1\sin (\pi x),u(x,2)=0,u(0,y)=0,u(1,y)=0.$$

Using series solve $\mathrm{\Delta }u=0$⁠, $u(1,\theta )=$ $1+$ $\sum _{n=1}^{\infty }\frac{1}{n^2}\sin (n\theta )$⁠.

Using the series solution find the solution to $\mathrm{\Delta }u=0$⁠, $u(1,\theta )=1-\cos (\theta )$⁠. Express the solution in Cartesian coordinates (that is, using $x$ and $y$⁠).

a) Try and guess a solution to $\mathrm{\Delta }u=-1$⁠, $u(1,\theta )=0$⁠. Hint: Try a solution that only depends on $r$⁠. Also, don’t worry about the boundary condition at first.

b) Now solve $\mathrm{\Delta }u=-1$⁠, $u(1,\theta )=\sin (2\theta )$ using superposition.

(challenging) Derive the Poisson kernel solution if the region is a circle of radius $\rho$ rather than 1. That is, solve $\mathrm{\Delta }u=0$⁠, $u(\rho ,\theta )=g(\theta )$⁠.