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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.

• ### In class presentations(Up to %5 Bonus) -- Deadline: February 23 by the class

• Audrey: Minimal Surface Equation - Wiki
• Sheryar: Black-Scholes equation - Wiki
• Srini: Fisher KPP equation - Wiki
• Hunter: Poisson's equation - Wiki
• Jhansi: Boussinesq equations - Wiki
• Emily: Hunter-Saxton equation - Wiki
• William: Diffusion equation - Wiki
• Krystian: Helmholtz equation - Wiki
• Rithvik: Navier - Stokes equations - Wiki
• Richard: Schrödinger equation - Wiki
• ### 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

• Exercise 3.3, Page 169, Problems: 3, 4.
• Exercise 3.4, Page 184, Problems: 3, 4.
• Exercise 3.4, Page 185, Problems: 7, 8.