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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 \[ \left\{ \begin{array}{ll} 9u_{xx}=u_{t}, \quad 0 < x < 3, \quad t > 0,&\mbox{The Heat Equation},\\ u_{x}(0,t)=0 \quad \mbox{and} \quad u_{x}(3,t)=0,& \mbox{Boundary conditions},\\ u(x,0)=2\cos(\frac{\pi x}{3}) -4 \cos(\frac{5\pi x}{3}) &\mbox{Initial condition}. \end{array} \right. \]
Solve the given above problems following these steps.
• 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 $.

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