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Mathematics LibreTexts

4.E: Fourier series and PDEs (Exercises)

These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence. 

4.1: Boundary value problems

Hint for the following exercises: Note that when \( \lambda>0\), then \( \cos( \sqrt{\lambda}(t-a))\) and \( \sin( \sqrt{\lambda}(t-a))\) are also solutions of the homogeneous equation.

Exercise 4.1.2: Compute all eigenvalues and eigenfunctions of \( x''+ \lambda x=0, x(a)=0, x(b)=0\) (assume \(a<b\)).

Exercise 4.1.3: Compute all eigenvalues and eigenfunctions of \( x''+ \lambda x=0, x'(a)=0, x'(b)=0\) (assume \(a<b\)).

Exercise 4.1.4: Compute all eigenvalues and eigenfunctions of \( x''+ \lambda x=0, x'(a)=0, x(b)=0\) (assume \(a<b\)).

Exercise 4.1.5: Compute all eigenvalues and eigenfunctions of \( x''+ \lambda x=0, x(a)=x(b), x'(a)=x'(b)\) (assume \(a<b\)).

Exercise 4.1.6: We have skipped the case of \( \lambda <0\) for the boundary value problem \( x''+ \lambda x=0, x(- \pi)=x( \pi), x'(- \pi)=x'(\pi)\). Finish the calculation and show that there are no negative eigenvalues.

Exercise 4.1.101: Consider a spinning string of length 2 and linear density 0.1 and tension 3. Find smallest angular velocity when the string pops out.

Exercise 4.1.102: Suppose \( x''+ \lambda x=0\) 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).

Exercise 4.1.103: Suppose \( x''+ x=0\) and  \( x(0)=0, x'(\pi)=1\). Find all the solution(s) if any exist.

Exercise 4.1.104: Consider \( x'+ \lambda x=0\) and  \( x(0)=0, x(1)=0\). Why does it not have any eigenvalues? Why does any first order equation with two endpoint conditions such as above have no eigenvalues?

Exercise 4.1.105 (challenging): Suppose \( x'''+ \lambda x=0\) and  \( x(0)=0, x'(0)=0, x(1)=0\). Suppose that \( \lambda >0\). Find an equation that all such eigenvalues must satisfy. Hint: Note that \( - \sqrt[3]{\lambda}\) is a root of \( r^3+\lambda =0\).

4.2: The Trigonometric Series

Exercise 4.2.3: Suppose \(f(t)\) is defined on \([- \pi,\pi]\) as \(\sin(5t)+ \cos(3t)\). Extend periodically and compute the Fourier series of \(f(t)\).

Exercise 4.2.4: Suppose \(f(t)\) is defined on \([- \pi,\pi]\) as \( |t|\). Extend periodically and compute the Fourier series of \(f(t)\).

Exercise 4.2.5: Suppose \(f(t)\) is defined on \([- \pi,\pi]\) as \( |t|^3\). Extend periodically and compute the Fourier series of \(f(t)\).

Exercise 4.2.6: Suppose \(f(t)\) is defined on \((- \pi,\pi]\) as

\[  f(t)= \left\{ \begin{array}{cc} -1&~~~~ {\it{~if~}} - \pi < t \leq 0, \\ 1& {\it{~if~}} 0 < t \leq \pi. \end{array} \right. \]

Extend periodically and compute the Fourier series of \(f(t)\).

Exercise 4.2.7: Suppose \(f(t)\) is defined on \((- \pi,\pi]\) as \(t^3\). Extend periodically and compute the Fourier series of \(f(t)\).

Exercise 4.2.8: Suppose \(f(t)\) is defined on \([- \pi,\pi]\) as \(t^2\). Extend periodically and compute the Fourier series of \(f(t)\).

There is another form of the Fourier series using complex exponentials that is sometimes easier to work with.

Exercise 4.2.9: Let

\[  f(t)= \frac{a_0}{2} + \sum^{\infty}_{n=1} a_n \cos(nt)+b_n \sin(nt).\]

Use Euler’s formula \( e^{i \theta}= \cos(\theta) + i \sin(\theta)\) to show that there exist complex numbers \( c_m\) such that

\[  f(t)=\sum^{\infty}_{m=- \infty} c_m e^{imt}.\]

Note that the sum now ranges over all the integers including negative ones. Do not worry about convergence in this calculation. Hint: It may be better to start from the complex exponential form and write the series as

\[  c_0+ \sum^{\infty}_{m=1} c_m e^{imt}+c_{-m}e^{-imt}.\]

Exercise 4.2.101: Suppose \(f(t)\) is defined on \([- \pi,\pi]\) as \( f(t)= \sin(t)\). Extend periodically and compute the Fourier series.

Exercise 4.2.102: Suppose \(f(t)\) is defined on \((- \pi,\pi]\) as \( f(t)= \sin(\pi t)\). Extend periodically and compute the Fourier series.

Exercise 4.2.103: Suppose \(f(t)\) is defined on \((- \pi,\pi]\) as \( f(t)= \sin^2(t)\). Extend periodically and compute the Fourier series.

Exercise 4.2.104: Suppose \(f(t)\) is defined on \((- \pi,\pi]\) as \( f(t)= t^4\). Extend periodically and compute the Fourier series.

4.3: More on the Fourier series

Exercise 4.3.3. Let

\[  f(t)= \left\{ \begin{array}{ccc} 0 & \it{if}& - 1 < t \leq 0, \\ t & \it{if} & 0 < t \leq 1, \end{array} \right. \]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\)  harmonic.

Exercise 4.3.4. Let

\[  f(t)= \left\{ \begin{array}{ccc} -t & \it{if}& - 1 < t \leq 0, \\ t^2 & \it{if} & 0 < t \leq 1, \end{array} \right. \]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic.

Exercise 4.3.5. Let

\[  f(t)= \left\{ \begin{array}{ccc} \dfrac{-t}{10} & \it{if}& - 10 < t \leq 0, \\ \dfrac{t}{10} & \it{if} & 0 < t \leq 10, \end{array} \right. \]

extended periodically (period is 20). a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic.

Exercise 4.3.6. Let \(f(t)= \sum_{n=1}^{\infty} \dfrac{1}{n^3} \cos(nt)\). Is \( f(t)\) continuous and differentiable everywhere? Find the derivative (if it exists everywhere) or justify why \(f(t)\) is not differentiable everywhere.

Exercise 4.3.7. Let \(f(t)= \sum_{n=1}^{\infty} \dfrac{(-1)^n}{n} \sin(nt)\). Is \( f(t)\) differentiable everywhere? Find the derivative (if it exists everywhere) or justify why \( f(t)\) is not differentiable everywhere.

Exercise 4.3.8. Let

\[  f(t)= \left\{ \begin{array}{ccc} 0 & ~~~\it{if}~ - 2 < t \leq 0, \\ t & \it{if} ~ 0 < t \leq 1, \\  -t+2 & \it{if} ~ 1 < t \leq 2, \end{array} \right. \]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic.

Exercise 4.3.9. Let

\[ f(t)=e^t ~~~~~ \it{for}~ -1<t \leq 1\]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic. c) What does the series converge to at \(t=1\).

Exercise 4.3.10. Let

\[ f(t)=t^2 ~~~~~ \it{for}~ -1<t \leq 1\]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) By plugging in \( t=0\), evaluate \( \sum_{n=1}^{\infty} \dfrac{(-1)^n}{n^2}= 1 - \dfrac{1}{4}+\dfrac{1}{9}- \cdots .\) c) Now evaluate \( \sum_{n=1}^{\infty} \dfrac{1}{n^2}= 1 + \dfrac{1}{4}+\dfrac{1}{9}+ \cdots .\).

Exercise 4.3.101. Let

\[ f(t)=t^2 ~~~~~ \it{for}~ -2<t \leq 2\]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic.

Exercise 4.3.102. Let

\[ f(t)=t ~~~~~ \it{for}~ \lambda<t \leq \lambda ~(\it{for~some} ~ \lambda)  \]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Write out the series explicitly up to the \( 3^{\it{rd}}\) harmonic.

Exercise 4.3.103. Let

\[ f(t)= \dfrac{1}{2}+ \sum_{n=1}^{\infty} \dfrac{1}{n(n^2+1)} \sin(n \pi t).\]

Compute \( f'(t)\).

Exercise 4.3.104. Let

\[ f(t)= \dfrac{1}{2}+ \sum_{n=1}^{\infty} \dfrac{1}{n^3)} \cos(nt).\]

a) Find the antiderivative. b) Is the antiderivative periodic?

Exercise 4.3.105. Let

\[ f(t)=\dfrac{t}{2} ~~~~~ \it{for}~ - \pi<t \leq \pi\]

extended periodically. a) Compute the Fourier series for \( f(t)\). b) Plug in \( t= \dfrac{\pi}{2}\) to find a series representation for \(\dfrac{\pi}{4}\). c) Using the first 4 terms of the result from part b) approximate \(\dfrac{\pi}{4}\).

4.4: Sine and Cosine Series

Exercise 4.4.4: Take \( f(t)=(t-1)^2\) defined on \( 0 \leq t \leq 1\). a) Sketch the plot of the even periodic extension of \(f\). b) Sketch the plot of the odd periodic extension of \(f\).

Exercise 4.4.5: Find the Fourier series of both the odd and even periodic extension of the function \( f(t)=(t-1)^2\) for \( 0 \leq t \leq 1\). Can you tell which extension is continuous from the Fourier series coefficients?

Exercise 4.4.6: Find the Fourier series of both the odd and even periodic extension of the function \( f(t)=t\) for \( 0 \leq t \leq \pi\).

Exercise 4.4.7: Find the Fourier series of the even periodic extension of the function \( f(t) = \sin t\) for \( 0 \leq t \leq \pi\).

Exercise 4.4.8: Consider

\[  x''(t)+4x(t)=f(t),\]

where \( f(t)=1\) on \( 0< t<1\). a) Solve for the Dirichlet conditions \( x(0)=0, x(1)=0\). b) Solve for the Neumann conditions .

Exercise 4.4.9: Consider

\[  x''(t)+9x(t)=f(t),\]

for \( f(t)= \sin(2 \pi t)\) on \( 0< t<1\). a) Solve for the Dirichlet conditions \( x(0)=0, x(1)=0\). b) Solve for the Neumann conditions \( x'(0)=0, x'(1)=0\).

Exercise 4.4.10: Consider

\[  x''(t)+3x(t)=f(t),~~~~ x(0)=0,~~~~ x(1)=0, \]

where \( f(t)= \sum_{n=1}^{\infty} b_n \sin(n \pi t)\). Write the solution \(x(t)\) as a Fourier series, where the coefficients are given in terms of \(b_n\).

Exercise 4.4.11: Let \( f(t)=t^2(2-t)\) for \( 0 \leq t \leq 2\). Let \( F(t)\) be the odd periodic extension. Compute \( F(1),F(2),F(3),F(-1),F( \frac{9}{2}),F(101),F(103)\). Note: Do not compute using the sine series.

Exercise 4.4.101: Let \(f(t)= \frac{t}{3}\) on \( 0 \leq t <3\). a) Find the Fourier series of the even periodic extension. b) Find the Fourier series of the odd periodic extension.

Exercise 4.4.102: Let \( f(t)= \cos(2t)\) on \(0 \leq t < \pi\). a) Find the Fourier series of the even periodic extension. b) Find the Fourier series of the odd periodic extension.

Exercise 4.4.103: Let \(f(t)\) be defined on \(0 \leq t < 1\). Now take the average of the two extensions \( g(t)= \frac{F_{odd}(t)+F_{even}(t)}{2}\). a) What is \(g(t)\) if \(0 \leq t < 1\) (Justify!) b) What is \( g(t)\) if  \(-1 < t < 0\) (Justify!)

Exercise 4.4.104: Let \( f(t)= \sum_{n=1}^{\infty} \frac{1}{n^2} \sin(nt)\). Solve \( x'' -x=f(t)\) for the Dirichlet conditions \( x(0)=0\) and \( x(\pi)=0\).

Exercise 4.4.105 (challenging): Let \( f(t)= t+\sum_{n=1}^{\infty} \frac{1}{2^n} \sin(nt)\). Solve \( x'' + \pi x=f(t)\) for the Dirichlet conditions \( x(0)=0\) and \(x(\pi)=1\). Hint: Note that \( \frac{t}{\pi}\) satisfies the given Dirichlet conditions.

4.5: Applications of Fourier series

Exercise 4.5.2: Let \( F(t)= \frac{1}{2}+ \sum^{\infty}_{ {n =1}} \frac{1}{n^2} \cos(n \pi t)\) . Find the steady periodic solution to \(x'' + 2x= F(t)\). Express your solution as a Fourier series.

Exercise 4.5.3: Let \( F(t)=  \sum^{\infty}_{ {n =1}} \frac{1}{n^3} \sin(n \pi t)\). Find the steady periodic solution to \(x'' + x' +x= F(t)\). Express your solution as a Fourier series.

Exercise 4.5.4: Let \( F(t)=  \sum^{\infty}_{ {n =1}} \frac{1}{n^2} \cos(n \pi t)\). Find the steady periodic solution to \(x'' + 4x= F(t)\). Express your solution as a Fourier series.

Exercise 4.5.5: Let \( F(t)=t\) for \(-1<t<1\) and extended periodically. Find the steady periodic solution to \(x'' + x= F(t)\). Express your solution as a series.

Exercise 4.5.6: Let \( F(t)=t\) for \(-1<t<1\) and extended periodically. Find the steady periodic solution to \(x'' + \pi^2 x= F(t)\). Express your solution as a series.

Exercise 4.5.101: Let \(F(t)= \sin(2 \pi t)+0.1 \cos(10 \pi t)\). Find the steady periodic solution to \(x''+\sqrt{2}x=F(t)\). Express your solution as a Fourier series.

Exercise 4.5.102: Let \( F(t)=  \sum^{\infty}_{ {n =1}} e^{-n} \cos(2nt)\). Find the steady periodic solution to \(x'' + 3x= F(t)\). Express your solution as a Fourier series.

Exercise 4.5.103: Let \( F(t)=|t|\) for \(-1 \leq t \leq 1\) extended periodically. Find the steady periodic solution to \(x'' + \sqrt{3}x= F(t)\). Express your solution as a series.

Exercise 4.5.104: Let \( F(t)=|t|\) for \(-1 \leq t \leq 1\) extended periodically. Find the steady periodic solution to \(x'' + \pi^2x= F(t)\). Express your solution as a series.

4.6: PDEs, Separation of Variables, and the Heat Equation

 

Exercise 4.6.2: Imagine you have a wire of length \(2\), with \(k=0.001\) and an initial temperature distribution of \(u(x,0)=50x\). Suppose that both the ends are embedded in ice (temperature 0). Find the solution as a series.

Exercise 4.6.3: Find a series solution of

\(u_t=u_{xx}, \\ u(0,t)= u(1,t)=0, \\ u(x,0)= 100 ~~~~ {\rm{for~}} 0<x<1.\)

Exercise 4.6.4: Find a series solution of

\(u_t=u_{xx}, \\ u_x(0,t)= u_x( \pi ,t)=0, \\ 3 \cos(x)+ \cos(3x) ~~~~ {\rm{for~}} 0<x< \pi.\)

Exercise 4.6.5: Find a series solution of

\(u_t= \frac{1}{3}u_{xx}, \\ u_x(0,t)= u_x( \pi,t)=0, \\ u(x,0)= \frac{10x}{ \pi} ~~~~ {\rm{for~}} 0<x< \pi.\)

Exercise 4.6.6: Find a series solution of

\(u_t=u_{xx}, \\ u(0,t)=0,~~~ u(1,t)=100, \\ u(x,0)= \sin(\pi x) ~~~~ {\rm{for~}} 0<x<1.\)

Hint: Use the fact that \(u(x,t)=100x\) is a solution satisfying \(u_t=u_{xx}, u(0,t)=0, u(1,t)=100\). Then usesuperposition.

Exercise 4.6.7: Find the steady state temperature solution as a function of \(x\) alone, by letting \(t \rightarrow \infty\) in the solution from exercises 4.6.5 and 4.6.6. Verify that it satisfies the equation \(u_{xx}=0\).

Exercise 4.6.8: Use separation variables to find a nontrivial solution to \(u_{xx}+u_{yy}=0\), where \(u(x,0)=0\) and \(u(0,y)=0\). Hint: Try \(u(x,y) =X(x)Y(y)\).

Exercise 4.6.9 (challenging): Suppose that one end of the wire is insulated (say at \(x=0\)) and the other end is kept at zero temperature. That is, find a series solution of

\(u_t= ku_{xx}, \\ u_x(0,t)= u( L,t)=0, \\ u(x,0)= f(x) ~~~~ {\rm{for~}} 0<x< L.\)

Express any coefficients in the series by integrals of f(x) .

Exercise 4.6.10 (challenging): Suppose that the wire is circular and insulated, so there are no ends. You can think of this as simply connecting the two ends and making sure the solution matches up at the ends. That is, find a series solution of

\(u_t= ku_{xx}, \\ u(0,t)= u( L,t),~~~~u_x(0,t)= u_x( L,t) \\ u(x,0)= f(x) ~~~~ {\rm{for~}} 0<x< L.\)

Express any coefficients in the series by integrals of \(f(x)\).

Exercise 4.6.101: Find a series solution of

\(u_t= 3u_{xx}, \\ u(0,t)= u( \pi,t)=0, \\ u(x,0)= 5 \sin(x)+2 \sin(5x) ~~~~ {\rm{for~}} 0<x< \pi.\)

Exercise 4.6.102: Find a series solution of

\(u_t= 0.1u_{xx}, \\ u_x(0,t)= u_x( \pi,t)=0, \\ u(x,0)= 1+2 \cos(x) ~~~~ {\rm{for~}} 0<x< \pi.\)

Exercise 4.6.103: Use separation of variables to find a nontrivial solution to \(u_{xt}=u_{xx}\).

Exercise 4.6.104: Use separation of variables (Hint: try \(u(x,t)=X(x)+T(t)\)) to find a nontrivial solution to \(u_x+u_t=u\).

4.7: One dimensional wave equation

 

Exercise 4.7.3: Solve

\[y_{tt}=9y_{xx}, \\ y(0,t)=y(1,t)=0, \\ y(x,0)= \sin(3 \pi x) + \frac{1}{4} \sin(6 \pi x) ~~~~~~~~~ {\rm{for}}~0<x<1, \\ y_t(x,0)=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {\rm{for}}~0<x<1.\]

Exercise 4.7.4: Solve

\[y_{tt}=4y_{xx}, \\ y(0,t)=y(1,t)=0, \\ y(x,0)= \sin(3 \pi x) + \frac{1}{4} \sin(6 \pi x) ~~~~~~~~~ {\rm{for}}~0<x<1, \\ y_t(x,0)= \sin(9 \pi x)~~~~~~~~~~~~~~~~~~~~~~~~~~ {\rm{for}}~0<x<1.\]

Exercise 4.7.5: Derive the solution for a general plucked string of length \(L\), where we raise the string some distance \(b\) at the midpoint and let go, and for any constant \(a\) (in the equation \(y_{tt}=a^2y_{xx}\)).

Exercise 4.7.6: Imagine that a stringed musical instrument falls on the floor. Suppose that the length of the string is 1 and \(a=1\). When the musical instrument hits the ground the string was in rest position and hence \(y(x,0)=0\). However, the string was moving at some velocity at impact \((t=0\))), say \( y_t(x,0)=-1\). Find the solution \(y(x,t)\) for the shape of the string at time \(t\).

Exercise 4.7.7 (challenging): Suppose that you have a vibrating string and that there is air resistance proportional to the velocity. That is, you have

\[y_{tt}=a^2y_{xx}-ky_t, \\ y(0,t)=y(1,t)=0, \\ y(x,0)= f(x)~~~~~~~~~ {\rm{for}}~0<x<1, \\ y_t(x,0)= 0~~~~~~~~~~~~~ {\rm{for}}~0<x<1.\]

Suppose that \(0<k< 2 \pi a\). Derive a series solution to the problem. Any coefficients in the series should be expressed as integrals of \(f(x)\).

Exercise 4.7.101: Solve

\[y_{tt}=y_{xx}, \\ y(0,t)=y( \pi,t)=0, \\ y(x,0)= \sin(x)~~~~~~~~~~~~~~ {\rm{for}}~0<x< \pi, \\ y_t(x,0)= \sin(x)~~~~~~~~~~~~~ {\rm{for}}~0<x< \pi.\]

Exercise 4.7.102: Solve

\[y_{tt}=25y_{xx}, \\ y(0,t)=y( 2,t)=0, \\ y(x,0)= 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {\rm{for}}~0<x< 2, \\ y_t(x,0)= \sin( \pi x) + 0.1 \sin( 2 \pi t)~~~~~~~~~~~~~ {\rm{for}}~0<x< 2.\]

Exercise 4.7.103: Solve

\[y_{tt}=2y_{xx}, \\ y(0,t)=y( \pi ,t)=0, \\ y(x,0)= x~~~~~~~~~~~~~ {\rm{for}}~0<x< \pi, \\ y_t(x,0)= 0~~~~~~~~~~~~~ {\rm{for}}~0<x< \pi.\]

Exercise 4.7.104: Let’s see what happens when \(a=0\). Find a solution to \(y_{tt}=0, y(0,t)=y( \pi,t)=0, y(x,0)= \sin(2x), y_t(x,0)= \sin(x). \)

4.8: D’Alembert solution of the wave equation

Exercise 4.8.2: Using the d’Alembert solution solve \(y_{tt}=4y_{xx}\), \(0<x< \pi , t>0, y(0,t)=y( \pi,t)=0, y(x,0) = \sin x\) , and \(y_t(x,0)= \sin x\). Hint: Note that \( \sin x\) is the odd extension of \(y(x,0)\) and \(y_t(x,0)\).

Exercise 4.8.3: Using the d’Alembert solution solve \(y_{tt}=2y_{xx}\), \(0<x< 1 , t>0, y(0,t)=y( 1,t)=0, y(x,0) = \sin^5( \pi x)\), and \(y_t(x,0)= \sin^3( \pi x) \).

Exercise 4.8.4: Take \(y_{tt}=4y_{xx}\), \(0<x< \pi , t>0, y(0,t)=y( \pi,t)=0, y(x,0) = x( \pi -x)\), and \(y_t(x,0)=0\). a) Solve using the d’Alembert formula. Hint: You can use the sine series for \(y(x,0)\). b) Find the solution as a function of \(x\) for a fixed \(t=0.5,t=1,\) and \(t=2\). Do not use the sine series here.

Exercise 4.8.5: Derive the d’Alembert solution for \(y_{tt}=a^2y_{xx}\), \(0<x< \pi , t>0, y(0,t)=y( \pi,t)=0, y(x,0) = f(x)\), and \(y_t(x,0)=0\), using the Fourier series solution of the wave equation, by applying an appropriate trigonometric identity.

Exercise 4.8.6: The d’Alembert solution still works if there are no boundary conditions and the initial condition is defined on the whole real line. Suppose that \(y_{tt}=y_{xx}\) (for all \(x\) on the real line and \(t \geq 0\)), \(y(x,0)=f(x)\), and \(y_t(x,0)\), where

\( f(x) = \left\{ \begin{array}{ccc} 0 & {\rm{if}} & x<-1, \\ x+1 & {\rm{if}} & -1 \leq x < 0, \\ -x+1& {\rm{if}} & 0 \leq x < 1 \\ 0 & {\rm{if}} & x>1. \end{array} \right.\) 

Solve using the d’Alembert solution. That is, write down a piecewise definition for the solution. Then sketch the solution for \(t=0,t= 1/2, t=1\), and \(t=2\).

Exercise 4.8.101: 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 \(y_t(x,0)= \sin(3 \pi x)\).

Exercise 4.8.102: 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.

Exercise 4.8.103: 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) \(y(1,1)\), b) \(y(4,3)\), c) \(y(3,9)\).

4.9: Steady state temperature and the Laplacian

 

Exercise 4.9.1: Let \(R\) be the region described by \(0<x< \pi\) and \(0<y< \pi\). Solve the problem

\( \Delta u=0, ~~~~ u(x,0)= \sin x,~~~~ u(x, \pi)= 0, ~~~~ u(0, y)= 0,~~~~ u( \pi, y)= 0.\)

Exercise 4.9.2: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 1\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = \sin( \pi x)- \sin(2 \pi x), ~~~ u(x,1)=0, \\ u(0,y)=0,~~~ u(1,y)=0.\)

Exercise 4.9.3: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 1\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = u(x,1)=u(0,y)=u(1,y)=C.\)

for some constant \(C\). Hint: Guess, then check your intuition.

Exercise 4.9.4: Let \(R\) be the region described by \(0<x< \pi\) and \(0<y< \pi\). Solve

\( \Delta u=0,~~~ u(x,0) = 0,~~~ u(x, \pi) = \pi ,~~~ u(0,y) = y,~~~ u( \pi,y) = y.\)

Hint: Try a solution of the form \(u(x,y)=X(x)+Y(y)\) (different separation of variables).

Exercise 4.9.5: Use the solution of Exercise 4.9.4 to solve

\( \Delta u=0,~~~ u(x,0) = \sin x,~~~ u(x, \pi) = \pi ,~~~ u(0,y) = y,~~~ u( \pi,y) = y.\)

Hint: Use superposition.

Exercise 4.9.6: Let \(R\) be the region described by \(0<x< w\) and \(0<y< h\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = 0, ~~~ u(x,h)=f(x), \\ u(0,y)=0,~~~ u(w,y)=0.\)

The solution should be in series form using the Fourier series coefficients of \(f(x)\).

Exercise 4.9.7: Let \(R\) be the region described by \(0<x< w\) and \(0<y< h\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = 0, ~~~ u(x,h)=0, \\ u(0,y)=f(y),~~~ u(w,y)=0.\)

The solution should be in series form using the Fourier series coefficients of \(f(y)\).

Exercise 4.9.8: Let \(R\) be the region described by \(0<x< w\) and \(0<y< h\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = 0, ~~~ u(x,h)=0, \\ u(0,y)=0,~~~ u(w,y)=f(y).\)

The solution should be in series form using the Fourier series coefficients of \(f(y)\).

Exercise 4.9.9: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 1\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = \sin(9 \pi x), ~~~ u(x,1)= \sin(2 \pi x), \\ u(0,y)=0,~~~ u(1,y)=0.\)

Hint: Use superposition.

Exercise 4.9.10: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 1\). Solve the problem

\( u_{xx}+u_{yy}=0, \\ u(x,0) = \sin( \pi x), ~~~ u(x,1)= \sin( \pi x), \\ u(0,y)= \sin( \pi y),~~~ u(1,y)= \sin( \pi y).\)

Hint: Use superposition.

Exercise 4.9.11 (challenging): Using only your intuition find \(u(1/2,1/2)\), for the problem \( \Delta u=0\), where \(u(0,y)=u(1,y)=100\) for \(0<y<1\), and \(u(x,0)=u(x,1)=0\) for \(0<x<1\). Explain.

Exercise 4.9.101: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 1\). Solve the problem

\( \Delta u=0,~~~ u(x,0) = \sum_{n=1}^{ \infty}\sin(n \pi x),~~~ u(x, 1) = 0,~~~ u(0,y) = 0,~~~ u(1,y) = 0.\)

Exercise 4.9.102: Let \(R\) be the region described by \(0<x< 1\) and \(0<y< 2\). Solve the problem

\( \Delta u=0,~~~ u(x,0) = 0.1 \sin ( \pi x),~~~ u(x, 2) = 0 ,~~~ u(0,y) = 0,~~~ u( 1,y) = 0.\)

4.10: Dirichlet problem in the circle and the Poisson kernel

Exercise 4.10.2: Using series solve \(\Delta u=0, u(1,\theta)= |\theta|\) for \( -\pi< \theta \leq \pi\).

Exercise 4.10.3: Using series solve \(\Delta u=0, u(1,\theta)=g(\theta)\)  for the following data. Hint: trig identities.

\( a) ~~g(\theta)=1/2 + 3\sin(\theta)+\cos(3\theta)~~~~~~~~b)~~g(\theta)=\cos(3\theta) +3\sin(3\theta)+\sin(9\theta) \\ c) ~~g(\theta)=2\cos(\theta+1)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d)~~g(\theta)=\sin^2(\theta)\)

Exercise 4.10.4: Using the Poisson kernel, give the solution to \(\Delta u=0\), where \(u(1,\theta)\) is zero for \(\theta\) outside the interval \([-\pi/4, \pi/4]\) and \(u(1,\theta)\) is \(1\) for \(\theta\) on the interval \([-\pi/4, \pi/4]\).

Exercise 4.10.5: a) Draw a graph for the Poisson kernel as a function of \(\alpha\) when \(r=1/2\) and \(\theta=0\). b) Describe what happens to the graph when you make \(r\) bigger (as it approaches 1). c) Knowing that the solution \(u(r,\theta)\) is the weighted average of \(g(\theta)\) with Poisson kernel as the weight, explain what your answer to part b means.

Exercise 4.10.6: Take the function \(g(\theta)\) to be the function \(xy=\cos(\theta) \sin(\theta)\) on the boundary. Use the series solution to find a solution to the Dirichlet problem \(\Delta u=0, u(1,\theta)=g(\theta)\). Now convert the solution to Cartesian coordinates \(x\) and \(y\). Is this solution surprising? Hint: use your trig identities.

Exercise 4.10.7: Carry out the computation we needed in the separation of variables and solve \(r^2R''+rR'-n^2R=0\), for \(n=0,1,2,3,...\).

Exercise 4.10.8 (challenging): Derive the series solution to the Dirichlet problem if the region is a circle of radius \(\rho\) rather than \(1\). That is, solve \(\Delta u=0, u(\rho,\theta)=g(\theta)\).

Exercise 4.10.101: Using series solve \(\Delta u=0, u(1,\theta)=1+ \sum_{n=1}^{\infty} \frac{1}{n^2}\sin(n \theta)\).

Exercise 4.10.102: Using the series solution find the solution to \(\Delta u=0, u(1,\theta)=1- \cos( \theta)\). Express the solution in Cartesian coordinates (that is, using \(x\) and \(y\)).

Exercise 4.10.103: a) Try and guess a solution to \(\Delta u=-1, u(1,\theta)=0\). Hint: try a solution that only depends on \(r\). Also first, don’t worry about the boundary condition. b) Now solve \(\Delta u=-1, u(1,\theta)= \sin( 2\theta)\) using superposition.

Exercise 4.10.104 (challenging): Derive the Poisson kernel solution if the region is a circle of radius \(\rho\) rather than \(1\). That is, solve \(\Delta u=0, u(\rho,\theta)=g( \theta)\).