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3.8: Problems

  • Page ID
    90931
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    Exercise \(\PageIndex{1}\)

    Write \(y(t)=3 \cos 2 t-4 \sin 2 t\) in the form \(y(t)=A \cos (2 \pi f t+\phi)\).

    Exercise \(\PageIndex{2}\)

    Derive the coefficients \(b_{n}\) in Equation (3.2.2).

    Exercise \(\PageIndex{3}\)

    Let \(f(x)\) be defined for \(x \in[-L, L]\). Parseval’s identity is given by \[\frac{1}{L} \int_{-L}^{L} f^{2}(x) d x=\frac{a_{0}^{2}}{2}+\sum_{n=1}^{\infty} a_{n}^{2}+b_{n}^{2} .\nonumber \] Assuming the the Fourier series of \(f(x)\) converges uniformly in \((-L, L)\), prove Parseval’s identity by multiplying the Fourier series representation by \(f(x)\) and integrating from \(x=-L\) to \(x=L\). [In Section 9.6.3 we will encounter Parseval’s equality for Fourier transforms which is a continuous version of this identity.]

    Exercise \(\PageIndex{4}\)

    Consider the square wave function \[f(x)=\left\{\begin{array}{cc} 1, & 0<x<\pi, \\ -1, & \pi<x<2 \pi . \end{array}\right.\nonumber \]

    1. Find the Fourier series representation of this function and plot the first 50 terms.
    2. Apply Parseval’s identity in Problem 3 to the result in part a.
    3. Use the result of part \(b\) to show \(\frac{\pi^{2}}{8}=\sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{2}}\).

    Exercise \(\PageIndex{5}\)

    For the following sets of functions: i) show that each is orthogonal on the given interval, and ii) determine the corresponding orthonormal set.

    1. \(\{\sin 2 n x\}, \quad n=1,2,3, \ldots, \quad 0 \leq x \leq \pi\).
    2. \(\{\cos n \pi x\}, \quad n=0,1,2, \ldots, \quad 0 \leq x \leq 2\).
    3. \(\left\{\sin \frac{n \pi x}{L}\right\}, \quad n=1,2,3, \ldots, \quad x \in[-L, L]\).

    Exercise \(\PageIndex{6}\)

    Consider \(f(x)=4 \sin ^{3} 2 x\).

    1. Derive the trigonometric identity giving \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta\) using DeMoivre’s Formula.
    2. Find the Fourier series of \(f(x)=4 \sin ^{3} 2 x\) on \([0,2 \pi]\) without computing any integrals.

    Exercise \(\PageIndex{7}\)

    Find the Fourier series of the following:

    1. \(f(x)=x, x \in[0,2 \pi]\).
    2. \(f(x)=\frac{x^{2}}{4},|x|<\pi\).
    3. \(f(x)=\left\{\begin{array}{cc}\frac{\pi}{2}, & 0<x<\pi, \\ -\frac{\pi}{2}, & \pi<x<2 \pi .\end{array}\right.\)

    Exercise \(\PageIndex{8}\)

    Find the Fourier Series of each function \(f(x)\) of period \(2 \pi\). For each series, plot the \(N\) th partial sum, \[S_{N}=\frac{a_{0}}{2}+\sum_{n=1}^{N}\left[a_{n} \cos n x+b_{n} \sin n x\right],\nonumber \] for \(N=5,10,50\) and describe the convergence (is it fast? what is it converging to, etc.) [Some simple Maple code for computing partial sums is shown in the notes.]

    1. \(f(x)=x,|x|<\pi\).
    2. \(f(x)=|x|,|x|<\pi\).
    3. \(f(x)=\left\{\begin{array}{cc}0, & -\pi<x<0, \\ 1, & 0<x<\pi .\end{array}\right.\)

    Exercise \(\PageIndex{9}\)

    Find the Fourier series of \(f(x)=x\) on the given interval. Plot the \(N\) th partial sums and describe what you see.

    1. \(0<x<2\).
    2. \(-2<x<2\).
    3. \(1<x<2\)

    Exercise \(\PageIndex{10}\)

    The result in problem \(7 \mathrm{~b}\) above gives a Fourier series representation of \(\frac{x^{2}}{4}\). By picking the right value for \(x\) and a little arrangement of the series, show that [See Example 3.3.2.]

    1. \[\frac{\pi^{2}}{6}=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots\nonumber \]
    2. \[\frac{\pi^{2}}{8}=1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\cdots .\nonumber \] Hint: Consider how the series in part a. can be used to do this.

    Exercise \(\PageIndex{11}\)

    Sketch (by hand) the graphs of each of the following functions over four periods. Then sketch the extensions each of the functions as both an even and odd periodic function. Determine the corresponding Fourier sine and cosine series and verify the convergence to the desired function using Maple

    1. \(f(x)=x^{2}, 0<x<1\).
    2. \(f(x)=x(2-x), 0<x<2\).
    3. \(f(x)= \begin{cases}0, & 0<x<1, \\ 1, & 1<x<2 .\end{cases}\)
    4. \(f(x)=\left\{\begin{array}{cc}\pi, & 0<x<\pi, \\ 2 \pi-x, & \pi<x<2 \pi .\end{array}\right.\)

    Exercise \(\PageIndex{12}\)

    Consider the function \(f(x)=x,-\pi<x<\pi\).

    1. Show that \(x=2 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin n x}{n}\).
    2. Integrate the series in part a and show that \[x^{2}=\frac{\pi^{2}}{3}-4 \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\cos n x}{n^{2}} .\nonumber \]
    3. Find the Fourier cosine series of \(f(x)=x^{2}\) on \([0, \pi]\) and compare it to the result in part b.

    Exercise \(\PageIndex{13}\)

    Consider the function \(f(x)=x, 0<x<2\).

    1. Find the Fourier sine series representation of this function and plot the first 50 terms.
    2. Find the Fourier cosine series representation of this function and plot the first 50 terms.
    3. Apply Parseval’s identity in Problem 3 to the result in part \(b\).
    4. Use the result of part c to find the sum \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\).

    Exercise \(\PageIndex{14}\)

    Differentiate the Fourier sine series term by term in Problem 18. Show that the result is not the derivative of \(f(x)=x\).

    Exercise \(\PageIndex{15}\)

    Find the general solution to the heat equation, \(u_{t}-u_{x x}=0\), on \([0, \pi]\) satisfying the boundary conditions \(u_{x}(0, t)=0\) and \(u(\pi, t)=0\). Determine the solution satisfying the initial condition, \[u(x, 0)=\left\{\begin{array}{cc} x, & 0 \leq x \leq \frac{\pi}{2}, \\ \pi-x, & \frac{\pi}{2} \leq x \leq \pi, \end{array}\right.\nonumber \]

    Exercise \(\PageIndex{16}\)

    Find the general solution to the wave equation \(u_{t t}=2 u_{x x}\), on \([0,2 \pi]\) satisfying the boundary conditions \(u(0, t)=0\) and \(u_{x}(2 \pi, t)=0\). Determine the solution satisfying the initial conditions, \(u(x, 0)=x(4 \pi-x)\), and \(u_{t}(x, 0)=0\).

    Exercise \(\PageIndex{17}\)

    Recall the plucked string initial profile example in the last chapter given by \[f(x)=\left\{\begin{array}{cc} x, & 0 \leq x \leq \frac{\ell}{2}, \\ \ell-x, & \frac{\ell}{2} \leq x \leq \ell, \end{array}\right.\nonumber \] satisfying fixed boundary conditions at \(x=0\) and \(x=\ell\). Find and plot the solutions at \(t=0, .2, \ldots, 1.0\), of \(u_{t t}=u_{x x}\), for \(u(x, 0)=f(x), u_{t}(x, 0)=0\), with \(x \in[0,1]\).

    Exercise \(\PageIndex{18}\)

    Find and plot the solutions at \(t=0, .2, \ldots, 1.0\), of the problem \[\begin{aligned} u_{t t} &=u_{x x}, \quad 0 \leq x \leq 1, t>0 \\ u(x, 0) &= \begin{cases}0, & 0 \leq x<\frac{1}{4}, \\ 1, & \frac{1}{4} \leq x \leq \frac{3}{4}, \\ 0, & \frac{3}{4}<x \leq 1,\end{cases} \\ u_{t}(x, 0) &=0, \\ u(0, t) &=0, \quad t>0, \\ u(1, t) &=0, \quad t>0 . \end{aligned}\]

    Exercise \(\PageIndex{19}\)

    Find the solution to Laplace’s equation, \(u_{x x}+u_{y y}=0\), on the unit square, \([0,1] \times[0,1]\) satisfying the boundary conditions \(u(0, y)=0, u(1, y)=\) \(y(1-y), u(x, 0)=0\), and \(u(x, 1)=0\).


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