# Math 415 Partial Differential Equation (Fall 2021)

Differential equations represent the most powerful tool humanity has ever created for making sense of the material world.

The Joy of $x$, A Guided Tour of Math, from One to Infinity by Steven Strogatz

## Summary

Introduction to partial differential equations. Classification of PDEs. Basic methods for classical PDEs (transport equation, wave equation, Laplace equation) to model the wave or diffusion phenomena. Separation of variables and (Fourier) series expansions, eigenvalue problems.

## Acknowledgement

This class is mainly adapted from the Partial Differential Equation and Calculus of Variations classes I taught several times at University of California, Irvine.

## Textbook and references:

Partial Differential Equations: An Introduction, by Walter A. Strauss, John Wiley & Sons. Rent an e-copy

## Schedule

This is only tentative, and with a fairly good chance, is subject to change due to COVID-19. Another note is that the lecture numbering may not directly correspond to the actual lecture given. Please refer to the lecture notes posted on Canvas.

## Modules

In each module, we will learn a specific PDE with various methods. Every topic has a list of questions we would like to seek answers to. For actual homework problems please refer to Canvas. Technicalities are needed for exams and homework, but understanding the physical meaning behind would benefit us as scientists or engineers.

### Module 1: Wave equation

Week 1-4.

#### Questions

Why the wave equation can model the phenomena of the elastic wave propagating on a vibrating string? Why the wave can preserve information in the past and propagate information to the future? Why the energy of the wave is conserved?

#### Sample theoretical problems

• Verify that $\displaystyle u(x, t)=\frac{1}{2 c} \int_{0}^{t} \int_{x-c t+c s}^{x+c t-c s} f(y, s) \, \mathrm{d} y \mathrm{d}s$ solves the wave equation $u_{t t}=c^{2} u_{x x}+f$ with homogeneous boundary condition.
• Show that the energy is conserved in an infinite string, i.e., for $\displaystyle E=\frac{1}{2} \int_{-\infty}^{+\infty}\left(\rho u_{t}^{2}+T u_{x}^{2}\right) \mathrm{d}x$, $E$, which is the sum of the kinetic energy and the potential energy, does not change as time goes by.

#### Material

Review of Calculus and ODE; (Strauss Chapter 1-2) Introduction of PDE, terminology, notations and classification of PDEs. Wave equations, transport equations, conservation of energy, causality of waves, domain of dependence/influence, D’Alembert formula, wave reflecting at the boundary, superposition principle.

### Module 2: Diffusion equation

Week 5-7.

#### Questions

What is the diffusion equation modeling? Why is singularity smoothened out in the diffusion equation? Why the energy (and information) is lost in the diffusion phenomenon?

#### Sample theoretical problems

• Verify that $\displaystyle u(x, t)=\frac{1}{\sqrt{4 \pi k t}} \int_{-\infty}^{\infty} e^{-(x-y)^{2} / 4 k t} e^{-y} \, \mathrm{d} y$ solves the diffusion equation on the real line.
• Prove the solution of the diffusion problem is unique by the energy method: $u_{t}-k u_{x x}=f(x, t)$ for $0<x<l, t>0$, initial condition: $u(x, 0)=\phi(x)$, boundary condition: $u_{x}(0, t)=g(t) \quad u_{x}(l, t)=h(t)$.

#### Material

(Strauss Chapter 1-2) Diffusion equations, boundary value problem, solution of the diffusion problem, energy method, diffusion with a source.

### Module 3: Fourier series

Week 7-11.

#### Questions

Why is the Fourier series expansion of a function $f$ of $x$ a good approximation of $f$? Why the Fourier series’ coefficient has a very simple formula? What is orthogonality? Why we are interested in learning the Fourier series?

#### Sample problems

• Solve the eigenvalue problem $X’’ + \lambda X = 0$ in $x\in (0,l)$, with homogeneous Dirichlet boundary condition at $x=0,l$.
• Show $\displaystyle 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots =\sum_{n=1}^{\infty} \frac {1}{n^{2}}= \frac{\pi^2}{6}$.

#### Material

(Strauss Chapter 4-5) Eigenvalue problem, Fourier series expansion, Fourier series of the wave equation, $L^2$-convergence theory of Fourier series.

### Module 4: Laplace equation

Week 11-14.

#### Questions

What is a harmonic function? Why we would like to solve the Laplace equation? Why intuitively harmonic function has nice property such as rotational invariance? How to do separation of variables in the polar coordinate system?

#### Sample problems

• Suppose that $u$ is a harmonic function in the disk $D:=\{r < 2\}$ and $u(r, \theta) = 1+ 3\sin 2\theta$ on $\partial D= \{r=2\}$. Without finding the solution, find (a) Find the maximum value of $u$ in $\overline{D}$, (b) calculate the value of $u$ at the origin.
• Solve $u_{xx} + u_{yy} =0$ in the region $\{\alpha < \theta < \beta, a<r<b\}$ with the boundary condition $u=0$ on the two sides $\theta = \alpha$ and $\theta = \beta$, $u=g(\theta)$ on the arc $r = a$, and $u=h(\theta)$ on the arc $r = b$.

#### Material

(Strauss Chapter 6) Harmonic functions, rotational invariance of Laplacian operator, maximum principle, Laplacian in polar coordinates, separation of variables, Fourier series of the harmonic functions on a disk/wedge/rectangle, Laplace equations, derivation of Poisson formula, mean value formula.