[**Back to Previous Page**]
# Partial Differential Equations

Math 3435, Section 001 - Partial Differential Equations, Spring 2018 [Course Syllabus]

**Lectures:** MWF 12:20 - 13:10 at MONT 421.

**Office hours:** Monday at 1:15pm-2:15pm and Wednesday 2pm-3pm at MONT 304.

**Required Text Book:** Basic Partial Differential Equations by David D. Bleecker and George Csordas. ISBN 1-57146-036-5, 2003, International Press of Boston, Inc.

### We will cover following chapters from the text book

- Introduction to PDEs (Chapter 1) Please read Section 1.1 from the book. We will cover Section 1.2 and Section 1.3.
- First Order PDEs (Chapter 2) Section 2.1 and Section 2.2.
- The Heat Equation (Chapter 3) Section 3.1 and Section 3.2.
- Fourier Series (Chapter 4) Section 4.1, Section 4.2, and Section 4.3.
- The Wave Equation (Chapter 5) Section 5.1 and Section 5.2.
- Laplace's Equation (Chapter 6) Section 6.1, Section 6.2, and Section 6.3.
- Fourier Transforms (Chapter 7) if time permits Section 7.1, Section 7.2, Section 7.3, and Section 7.4.
### Exams

- Practice Exam 1 and its solution.
- Exam 1 and its solution.
- Practice Exam 2 and its solution.
- Exam 2 and its solution.
- Practice Final Exam.
- Final Exam and its solution.
### In class presentations(Up to %5 Bonus) -- on Saturday, March 24

- Audrey: Minimal Surface Equation - Wiki - Audrey's talk [PDF]
- Sheryar: Black-Scholes equation - Wiki - Sheryar's talk [PDF]
- Srini: Fisher KPP equation - Wiki - Srini's talk [PDF]
- Hunter: Poisson's equation - Wiki - Hunter's talk [PDF]
- Jhansi: Boussinesq equations - Wiki - Jhansi's talk [PDF]
- Emily: Hunter-Saxton equation - Wiki - Emily's talk [PDF]
- William T.: Diffusion equation - Wiki - William's talk [PDF]
- Krystian: Helmholtz equation - Wiki - Krystian's talk [PDF]
- Rithvik: Navier - Stokes equations - Wiki- Rithvik's talk [PDF]
- Richard: Schrödinger equation - Wiki
- William G.: Maxwell's equations - Wiki- William's talk [PDF]
- Zachariah: Einstein field equations - Wiki- Zachariah's talk [PDF]
### Homework

#### HW1 - Due on Friday, January 26 by the class | Solutions(PDF)

- Exercise 1.2, Page 39, Problems: 1b, 1d, 1f, 2c, 3d, 4d, 5c, 5d.
- Exercise 1.2, Page 40, Problems: 12, 13.
- Exercise 1.3, Page 53, Problems: 1b, 1c.
#### HW2 - Due on Friday, February 2 by the class | Solutions(PDF)

- Exercise 1.3, Page 53, Problems: 2c, 2d.
- Exercise 1.3, Page 54, Problems: 3c, 3d.
- Exercise 1.3, Page 55, Problems: 8b, 9c.
#### HW3 - Due on Friday, February 9 by the class | Solutions(PDF)

- Exercise 2.1, Page 71, Problems: 1c, 1d, 2a, 3.
- Exercise 2.1, Page 72, Problem: 8.
- Exercise 2.2, Page 90, Problems: 1a, 1d, 2a, 2d,
~~3a, 3d~~. #### HW4 - Due on Friday, February 16 by the class | Solutions(PDF)

- Exercise 3.1, Page 136, Problems: 3b, 3d, 6b, 6d.
- Exercise 3.1, Page 137, Problem: 9.
- Solve the following Heat conduction problem \[ \left\{ \begin{array}{ll} 9u_{xx}=u_{t}, \quad 0 < x < 3, \quad t > 0,&\mbox{The Heat Equation},\\ u_{x}(0,t)=0 \quad \mbox{and} \quad u_{x}(3,t)=0,& \mbox{Boundary conditions},\\ u(x,0)=2\cos(\frac{\pi x}{3}) -4 \cos(\frac{5\pi x}{3}) &\mbox{Initial condition}. \end{array} \right. \]
- By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation

in terms of two ordinary differential equations in $X$ and $T$ (take arbitrary constant as $c$). - Rewrite the boundary values in terms of $X$ and $T$.
- Now choose the boundary values which will not give a non-trivial solution and write the ordinary differential equation corresponding to $X$.
- By considering $c=0, \lambda^2=c>0, -\lambda^2=c<0$, solve the two-point boundary value problem corresponding to $X$. Find all eigenvalues $\lambda_n$ and eigenfunctions $X_n$.
- For each eigenvalue $\lambda_n$ you found, rewrite and solve the ordinary differential equation corresponding to $T_n$.
- Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
- Using the given initial value and the general solution you found in, find the particular solution.
#### HW5 - Due on Friday, March 2 by the class | Solutions(PDF)

- Exercise 3.3, Page 169, Problems: 3, 4.
- Exercise 3.4, Page 184, Problems: 3, 4(optional).
- Exercise 3.4, Page 185, Problems: 7, 8(optional).
#### HW6 - Due on Friday, March 9 by the class | Solutions(PDF)

- Exercise 4.1, Page 205, Problems: 2, 4.
- Let $f(x)$ be given as
\[
f(x)=\left\{
\begin{array}{ll}
0 &\mbox{when}\, \, -\pi\leq x \leq 0, \\
x & \mbox{when}\, \, 0\leq x\leq \pi.
\end{array}
\right.
\]
- Find the Fourier series $\mathfrak{F}(x)$ of $f(x)$ on $-\pi\leq x \leq \pi$.
- Using the first part verify that \[ \frac{\pi}{4}=\sum\limits_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)}. \]
- Using the first part verify also that
\[
\frac{\pi^2}{8}=\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)^{2}}.
\]
**Hint: You can assume that $\mathfrak{F}(x)=f(x)$ when $-\pi < x < \pi$.**

- Let $f(x)$ be given as
\[
f(x)=\left\{
\begin{array}{ll}
0 &\mbox{when}\, \, -\pi\leq x < 0, \\
1 & \mbox{when}\, \, 0\leq x<\pi.
\end{array}
\right.
\]
- Find the Fourier series $\mathfrak{F}(x)$ of $f(x)$ on $-\pi\leq x \leq \pi$.
- Using the first part verify that
\[
\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}=\frac{\pi}{4}.
\]
**Hint: You can assume that $\mathfrak{F}(x)=f(x)$ when $-\pi < x < 0$ and $0 < x< \pi$**.

#### HW7 - Due on Wednesday, April 4 by the class | Solutions(PDF)

- Exercise 5.1, Page 295, Problems: 1a, 1c.
- Exercise 5.1, Page 296, Problem: 5a,
~~5b~~. - Exercise 5.1, Page 298, Problem: 9.
#### HW8 - Due on Wednesday, April 11 by the class | Solutions(PDF)

- Exercise 5.2, Page 317, Problems: 1a, 1c, 1d, 1e
- Exercise 5.2, Page 318, Problem: 6, 7, .
- Exercise 5.3, Page 336, Problem: 2.
- Exercise 5.3, Page 337, Problem: 6,7,10.
#### HW9 - Due on Wednesday, April 18 by the class The solution is in Section 6.2 of the book.

- Exercise 6.1, Page 349, Problem: 3
#### HW10 - Due on Wednesday, April 25 by the class | Solutions(PDF)

- Consider the following Dirichlet problem
$$
\left\{
\begin{array}{ll}
U_{rr}+\frac{1}{r}U_r+\frac{1}{r^2}U_{\theta\theta}=0 & \mbox{in} \, \, r < 2,\\
U(2,\theta)=1+3\sin(2\theta). &
\end{array}
\right.
$$
Without finding the solution, answer the following questions.

1. Find the maximum value of $U$ on disk with radius $2$.

2. Calculate the value of $U$ at the origin.

3. Using the Poisson's integral formula write down solution to the Dirichlet problem in the disk with radius $ 0 < 2 $. ### Important Links

- Please log-on HuskyCT for the course announcements and grades.
- Academic Calendar for this semester https://registrar.uconn.edu/academic-calendar/spring-2018/.
- Exam rescheduling https://guide.uconn.edu/student-interactions/rescheduling-exams/.
- Academic Integrity Policy https://community.uconn.edu/academic-integrity-faculty-faq/.