PDEs – Comprehensive Reference Guide

Classification

General 2nd Order: $Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$

Discriminant: $\Delta = B^2 - 4AC$

Elliptic: $\Delta < 0$ (Laplace, Poisson)

Parabolic: $\Delta = 0$ (Heat)

Hyperbolic: $\Delta > 0$ (Wave)

Characteristics

Lines along which info propagates

$\frac{dy}{dx} = \frac{B \pm \sqrt{\Delta}}{2A}$

Physical interpretation: wave speed, diffusion, equilibrium

First-Order PDEs

Linear: $a(x,y)u_x + b(x,y)u_y = c(x,y)u + d(x,y)$

Method of Characteristics

Solve: $\frac{dx}{a} = \frac{dy}{b} = \frac{du}{cu+d}$

Quasi-Linear: $a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$

Characteristic equations:

$\frac{dx}{a} = \frac{dy}{b} = \frac{du}{c}$

Transport Eq

$u_t + cu_x = 0$

Solution: $u(x,t) = u_0(x-ct)$

Characteristics: straight lines with slope $c$

Sturm-Liouville Theory

Regular S-L Problem:

$-\frac{d}{dx}\left(p\frac{d\phi}{dx}\right) + q\phi = \lambda w\phi$

on $[a,b]$ with BCs at endpoints

Properties

Eigenvalues $\lambda_n$ real, simple, ordered

Eigenfunctions $\phi_n$ orthogonal w.r.t. weight $w$

$\int_a^b \phi_m \phi_n w\,dx = 0$ ($m \neq n$)

Orthogonality

$\langle \phi_m, \phi_n \rangle_w = \delta_{mn} \|\phi_n\|_w^2$

Basis for Fourier series generalizations

Heat Equation

PDE: $u_t = k u_{xx}$ (on $[0,L] \times [0,T]$)

Fundamental Solution

$G(x,t) = \frac{1}{\sqrt{4\pi kt}} e^{-x^2/(4kt)}$

Gaussian spreading over time

Separation Solution

Dirichlet BC: $u(0,t)=u(L,t)=0$

$u(x,t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n\pi x}{L}\right) e^{-k\lambda_n t}$

where $\lambda_n = (n\pi/L)^2$

Maximum Principle

$\max u = \max\{u(\text{boundary}), u(\text{init})\}$

No interior maxima

Properties

Smoothing effect: infinite regularity for $t>0$

Irreversible in time

Energy dissipation: $\|u\|_2$ decreases

Wave Equation

PDE: $u_{tt} = c^2 u_{xx}$ (Wave speed $c$)

d'Alembert Formula (1D)

$u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)]$

$+ \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$

Separation Solution

$u(x,t) = \sum_{n=1}^{\infty} \left(A_n \cos(\omega_n t) + B_n \sin(\omega_n t)\right) \sin\left(\frac{n\pi x}{L}\right)$

where $\omega_n = \frac{n\pi c}{L}$

Domain of Dependence

Value at $(x,t)$ depends only on $[x-ct, x+ct]$ at $t=0$

Finite propagation speed: $c$

Energy Conservation

$E = \frac{1}{2}\int_0^L (u_t^2 + c^2 u_x^2)\,dx = \text{const}$

Reversible: solution extends backward in time

Laplace/Poisson Eqs

Laplace: $\nabla^2 u = 0$

Poisson: $\nabla^2 u = f$

Harmonic Functions

Solutions to Laplace equation

Mean value property: $u(x) = \text{avg}(u \text{ on sphere})$

No local max/min in interior (max principle)

Rectangle (Dirichlet)

$\nabla^2 u = 0$ on $[0,a] \times [0,b]$

Solution: double Fourier series

$u(x,y) = \sum_{m,n} A_{mn} \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right)$

Circular Domain

Polar coords: $u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\theta\theta} = 0$

Solution via eigenfunction expansion

Uniqueness

Dirichlet BC: unique solution

Neumann BC: unique up to constant

Poisson: add particular solution to homogeneous

Separation of Variables

Technique

1. Assume $u(x,t) = X(x)T(t)$

2. Substitute into PDE

3. Separate: $\frac{1}{X}\mathcal{L}_x[X] = \frac{1}{T}\mathcal{L}_t[T]$

4. Each side = constant $-\lambda$

Eigenvalue Problems

Spatial ODE: $X'' + \lambda X = 0$

with BC on boundaries

Gives eigenvalues $\lambda_n$ and eigenfunctions $X_n$

Temporal ODE

Heat: $T' + k\lambda T = 0 \Rightarrow T_n(t) = e^{-k\lambda_n t}$

Wave: $T'' + c^2\lambda T = 0 \Rightarrow T_n(t) = A_n\cos(\omega_n t) + B_n\sin(\omega_n t)$

General Solution

$u(x,t) = \sum_n c_n(t) X_n(x)$

Determine $c_n$ from initial/boundary data

Works when domain, BC, and PDE are compatible

Boundary Conditions

Dirichlet (1st Type)

$u = g$ on $\partial\Omega$

Specifies solution values

Most common for elliptic/steady-state

Neumann (2nd Type)

$u_n = g$ on $\partial\Omega$

Specifies normal derivative (flux)

For Laplace: unique up to constant

Robin (Mixed, 3rd Type)

$\alpha u + \beta u_n = g$ on $\partial\Omega$

Linear combination of value & flux

Physical: Newton's cooling law

Initial Conditions

Heat: $u(x,0) = f(x)$

Wave: $u(x,0) = f(x)$, $u_t(x,0) = g(x)$

Must be compatible with BC at $t=0$!

Well-Posedness (Hadamard)

1. Solution exists

2. Solution is unique

3. Solution depends continuously on data

Check these for every new problem

Green's Functions

Fundamental Solution: $\mathcal{L}[G] = \delta(x-\xi)$

1D Green's Function

$\mathcal{L}[u] = f$ on $(a,b)$

$u(x) = \int_a^b G(x,\xi)f(\xi)\,d\xi + \text{BC terms}$

Properties

$G(x,\xi)$ symmetric: $G(x,\xi) = G(\xi,x)$

Vanishes on boundary (Dirichlet)

Singular at $(x,\xi)$ (contains $\delta$)

Construction

1. Solve $\mathcal{L}[u] = 0$ on each side

2. Apply continuity + jump at $\xi$

3. Enforce boundary conditions

Reduces PDE to integral equation

Transform Methods

Fourier Transform

$\hat{f}(k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}\,dx$

$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(k)e^{ikx}\,dk$

Use on $(-\infty,\infty)$ with decay conditions

Fourier Series

Periodic: $f(x+L) = f(x)$

$f(x) = \frac{a_0}{2} + \sum_n(a_n\cos(n\pi x/L) + b_n\sin(n\pi x/L))$

$a_n = \frac{2}{L}\int_0^L f(x)\cos(n\pi x/L)\,dx$

Laplace Transform

$F(s) = \int_0^{\infty} f(t)e^{-st}\,dt$

Converts $u_t \to sU - u(0,x)$

Good for time-dependent problems

Benefits

PDE becomes algebraic in transform space

Invert to find physical solution

Powerful for unbounded domains

Numerical Methods

Finite Differences

Grid: $x_i = ih$, $t_n = n\tau$

Forward difference: $u_x \approx \frac{u_{i+1} - u_i}{h}$

Central difference: $u_x \approx \frac{u_{i+1} - u_{i-1}}{2h}$

$u_{xx} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{h^2}$

Heat Equation Schemes

Explicit (FTCS): $\frac{u_i^{n+1} - u_i^n}{\tau} = k\frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{h^2}$

Stability: $r := k\tau/h^2 \leq 1/2$

Implicit (backward Euler): unconditionally stable

Wave Equation Schemes

Explicit: $u_i^{n+1} = 2u_i^n - u_i^{n-1} + c^2 r^2 (u_{i+1}^n - 2u_i^n + u_{i-1}^n)$

Stability: CFL condition $cr = c\tau/h \leq 1$

Consistency & Convergence

Lax Equivalence: Consistency + Stability $\Rightarrow$ Convergence

Truncation error: local discretization error

Always check CFL/stability conditions!

Advanced Topics & Tips

Fourier Series Tricks

Odd extension for symmetric functions

Even extension for antisymmetric

Gibbs phenomenon at discontinuities

Inhomogeneous Problems

$u = u_h + u_p$ (homogeneous + particular)

Duhamel's principle for time-inhomog

$u(x,t) = \int_0^t G(x,s;t) f(x,s)\,ds$

Multidimensional Problems

Cartesian: separate each variable

Polar/Cylindrical: use adapted coordinates

Bessel/Legendre functions arise naturally

Perturbation Methods

Expand solution in small parameter $\epsilon$

Solve at each order: $u = u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots$

Check: Substitute back into original PDE

Verify: Initial and boundary conditions satisfied

Compare: Known solutions or limiting cases

Key reminder: Always verify uniqueness via max principle or energy estimates before solving!

Energy Methods & Analysis

Energy Estimates

Heat: Multiply by $u$, integrate, use Gronwall

Wave: $E = \frac{1}{2}\int(u_t^2 + c^2u_x^2)\,dx$

Gives stability and decay rates

Sobolev Spaces

$H^1(\Omega) = \{u : u, u_x \in L^2\}$

$\|u\|_{H^1} = \sqrt{\|u\|_2^2 + \|u_x\|_2^2}$

Weak solutions via variational formulation

Variational Methods

Minimize functional: $I[u] = \int \frac{1}{2}|\nabla u|^2 - fu\,dx$

Euler-Lagrange $\Rightarrow$ PDE $-\nabla^2u = f$

Contraction Mapping

Picard iteration for existence/uniqueness

Fixed point of $T[u]$ on Banach space

Weak solutions may not be classical (differentiable)

Modern approach via functional analysis

Quick Formula Reference

Heat: $u_t = ku_{xx}$
Wave: $u_{tt} = c^2u_{xx}$
Laplace: $\Delta u = 0$

Common Eigenvalues

String $[0,L]$ Dirichlet: $\lambda_n = (n\pi/L)^2$

Disk radius $a$: $\lambda_{n,m} = (\mu_{n,m}/a)^2$

Orthogonality

$\int_0^L \sin(n\pi x/L)\sin(m\pi x/L)\,dx = \frac{L}{2}\delta_{nm}$

Cosines, Bessel, Legendre: similar

d'Alembert

$u(x,t) = \frac{1}{2}[f(x{-}ct){+}f(x{+}ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$

Fundamental Solutions

Heat (1D): $G = (4\pi kt)^{-1/2}e^{-x^2/4kt}$

Laplace (2D): $G = -\frac{1}{2\pi}\ln r$

Wave (3D): $G = \delta(t - r/c)/(4\pi r)$

Memory aid: Check dimensions & singularities!

PARTIAL DIFF EQ