Basic Partial Differential Equations

Canonical partial differential equations in physics: wave equation, heat conduction, Laplace, and Poisson.

What it is

A partial differential equation (PDE) relates a multivariable field $u(x, y, z, \dots, t)$ to its spatial and temporal partial derivatives. For general second-order linear PDEs in two independent variables $u(x, y)$: $A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G(x, y)$ The mathematical classification is governed strictly by the discriminant sign $d = B^2 - 4 A C$:

1. Elliptic ($B^2 - 4 A C < 0$):
   • Prototype: Laplace equation $\nabla^2 u = 0$ and Poisson equation $\nabla^2 u = -\rho/\epsilon_0$.
   • Physics: Time-independent stationary equilibrium profiles, maximum smoothness, satisfying mean-value properties. Requires closed boundary constraints.
2. Parabolic ($B^2 - 4 A C = 0$):
   • Prototype: Heat / Diffusion equation $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$.
   • Physics: Irreversible dissipative transport. Gradients smooth out with infinite propagation speed.
3. Hyperbolic ($B^2 - 4 A C > 0$):
   • Prototype: D'Alembert wave equation $\frac{\partial^2 u}{\partial t^2} = v^2 \nabla^2 u$.
   • Physics: Conservative finite-speed causal signal propagation along characteristic directions. Preserves total mechanical energy.

Why it appears in physics

Physical force fields fill spatial volumes and evolve continuously:

• Electromagnetism: In Lorenz gauge, Maxwell's equations decouple into four decoupled hyperbolic d'Alembert wave PDEs $\Box A^\mu = \mu_0 J^\mu$.
• Gravitational fields: Steady Newtonian gravity satisfies Poisson's elliptic equation $\nabla^2 \Phi = 4 \pi G \rho$, while gravitational radiation propagates hyperbolically $\Box h_{\mu\nu} = 0$.
• Quantum mechanics: The Schrödinger wave equation $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi$ resembles a complex parabolic equation, generating dispersive quantum probability waves.
• Acoustics and seismology: Elastic compression ($P$) and shear ($S$) seismic waves propagate through planetary mantles as hyperbolic PDEs.

Visual or geometric intuition

• Hyperbolic (Waves): In spacetime $(x, t)$, information flows strictly along diagonal characteristic rays $x \pm v t = \text{const}$. Initial state disturbances are bounded strictly within the forward light cone.
• Parabolic (Diffusion): An initial point spike (Dirac delta) spreads out instantly as an expanding Gaussian bell whose spatial variance scales linearly with time: $\sigma^2(t) = 2 \alpha t$.
• Elliptic (Laplace): The potential distribution acts like an elastic rubber sheet clamped along its perimeter. The value at the center of any sphere strictly equals the boundary surface average (mean value property).

Minimal symbolic language

• D'Alembertian wave operator: $\Box = \frac{1}{v^2} \frac{\partial^2}{\partial t^2} - \nabla^2$
• D'Alembert 1D traveling wave formula: $u(x, t) = \frac{1}{2}\left[f(x - v t) + f(x + v t)\right] + \frac{1}{2v} \int_{x - v t}^{x + v t} g(s)\,ds$ where $f(x) = u(x, 0)$ and $g(x) = u_t(x, 0)$.
• Separation of variables (Fourier eigenmode Ansatz): $u(x, t) = X(x) T(t) \implies \frac{X''(x)}{X(x)} = \frac{1}{\alpha} \frac{T'(t)}{T(t)} = -\lambda$ splitting the PDE into two coupled ordinary differential eigenvalue problems.

What the result means

1. Fourier modal superposition: The composite field is represented as an infinite series of orthogonal eigenfunctions: $u(x, t) = \sum_{n=1}^\infty A_n X_n(x) e^{-\alpha k_n^2 t}$ Higher spatial harmonics attenuate drastically faster ($e^{-n^2 t}$), revealing why localized temperature spikes rapidly smooth out.
2. Maximum Principle: Solutions to Laplace and heat equations inside a domain $\Omega$ attain absolute extrema exclusively on the domain boundary $\partial \Omega$.
3. Causal horizon: For wave equations, disturbances from $(x_0, 0)$ cannot influence points outside $x_0 - v t \le x \le x_0 + v t$.

When not to use it

• Separation of variables on irregular non-conforming boundaries: Separation requires domain boundaries to lie along constant coordinate curves. Arbitrary shapes demand numerical finite elements (FEM).
• Backward heat diffusion ($t < 0$): Reverse time integration is severely ill-posed and numerically diverges under machine precision noise.
• Imposing Cauchy data on Laplace operators: Initial value prescriptions on elliptic PDEs generate unphysical Hadamard instabilities.

Operational summary

1. Classify the PDE via $B^2 - 4 A C$ (Elliptic, Parabolic, or Hyperbolic).
2. Inspect spatial geometry and boundary conditions (Dirichlet vs. Neumann).
3. Apply separation of variables $u(x, t) = X(x) T(t)$.
4. Solve the spatial Sturm-Liouville boundary eigenvalue problem for $\lambda_n$ and $X_n(x)$.
5. Solve the corresponding temporal ODE $T_n(t)$.
6. Compute Fourier coefficients $A_n$ matching initial data at $t = 0$.

Procedure: Separation of Variables for Basic PDEs

Before calculating

1. Identify PDE classification:
   • Wave (Hyperbolic): $u_{tt} = v^2 u_{xx}$ (requires 2 initial conditions and 2 boundary conditions).
   • Heat (Parabolic): $u_t = \alpha u_{xx}$ (requires 1 initial condition and 2 boundary conditions).
   • Laplace (Elliptic): $u_{xx} + u_{yy} = 0$ (requires 4 closed boundary conditions).
2. Check for boundary homogeneity: If boundaries have non-zero Dirichlet values ($u(0, t) = T_1, u(L, t) = T_2$), isolate the steady-state solution $u_{\text{ss}}(x)$ first and define transient perturbation $\tilde{u} = u - u_{\text{ss}}$.

Step 1: Identify the mathematical object

Write the PDE with its boundary data on domain $x \in [0, L]$:

• Dirichlet boundary data: $u(0, t) = 0$ and $u(L, t) = 0$.
• Neumann boundary data: $u_x(0, t) = 0$ and $u_x(L, t) = 0$.

Step 2: Choose the operation (Separation of variables)

Substitute product trial $u(x, t) = X(x) T(t)$ and divide through:

• Heat equation: $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda$
• Wave equation: $\frac{T''(t)}{v^2 T(t)} = \frac{X''(x)}{X(x)} = -\lambda$

Step 3: Conserve units and signs

• Choose separation constant with negative sign $-\lambda$ to ensure bounded oscillatory spatial eigenfunctions.
• Eigenvalues $\lambda_n$ carry units of $\text{m}^{-2}$ ($k_n^2 = (n\pi/L)^2$).

Step 4: Calculate

1. Solve spatial Sturm-Liouville problem: $X''(x) + \lambda X(x) = 0$
   • Under Dirichlet conditions ($X(0) = 0, X(L) = 0$): $\lambda_n = \left(\frac{n \pi}{L}\right)^2, \quad X_n(x) = \sin\left(\frac{n \pi x}{L}\right), \quad n = 1, 2, \dots$
2. Solve temporal ODE:
   • Heat: $T_n(t) = e^{-\alpha (n\pi/L)^2 t}$.
   • Wave: $T_n(t) = A_n \cos\left(\frac{n\pi v t}{L}\right) + B_n \sin\left(\frac{n\pi v t}{L}\right)$.
3. Assemble Fourier series: $u(x, t) = \sum_{n=1}^\infty A_n \sin\left(\frac{n \pi x}{L}\right) e^{-\alpha (n\pi/L)^2 t}$
4. Compute Fourier coefficients by orthogonality: $A_n = \frac{2}{L} \int_0^L u(x, 0) \sin\left(\frac{n \pi x}{L}\right) dx$

Step 5: Interpret the result

1. Check asymptotic limits: in diffusion, all $n \ge 1$ modes decay to zero as $t \to \infty$.
2. For waves, confirm that total mechanical energy remains conserved over time.

Self-check checklist

• Are the boundary conditions on $X(x)$ homogeneous?
• Did I choose $-\lambda$ with correct negative sign?
• Did I use orthogonality factor $L/2$ when evaluating Fourier integrals?
• Are physical units for diffusivity $\alpha$ ($\text{m}^2/\text{s}$) or speed $v$ ($\text{m}/\text{s}$) consistent?
• Does evaluating $u(x, 0)$ match initial profile $f(x)$?

Worked Examples: Basic Partial Differential Equations

Step-by-step guided example

Problem

A homogeneous thin metal rod of length $L = 1\,\text{m}$ and thermal diffusivity $\alpha = 10^{-4}\,\text{m}^2/\text{s}$ has its ends plunged in ice water: $u(0, t) = 0^\circ\text{C}$ and $u(1, t) = 0^\circ\text{C}$ for $t > 0$. At $t = 0$, the initial temperature profile across the rod is: $u(x, 0) = 50 \sin(\pi x) - 20 \sin(3 \pi x)\,\left(^\circ\text{C}\right)$

1. State the governing PDE and boundary/initial data.
2. Find the exact analytic solution $u(x, t)$ via separation of variables.
3. Compute the temperature at midpoint $x = 0.5\,\text{m}$ at $t = 1000\,\text{seconds}$.

Detailed solution

Step 1: Governing formulation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}, \quad x \in (0, 1), \quad t > 0$ Boundary conditions: $u(0, t) = 0, u(1, t) = 0$. Initial condition: $u(x, 0) = 50 \sin(\pi x) - 20 \sin(3\pi x)$.

Step 2: Separation of variables Assuming $u(x, t) = X(x) T(t)$: $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda$ Spatial Sturm-Liouville problem: $X''(x) + \lambda X(x) = 0, \quad X(0) = 0, \quad X(1) = 0$ Eigenvalues and eigenfunctions: $\lambda_n = (n\pi)^2, \quad X_n(x) = \sin(n\pi x), \quad n = 1, 2, \dots$ Temporal ODE: $T_n'(t) + \alpha (n\pi)^2 T_n(t) = 0 \implies T_n(t) = e^{-\alpha n^2 \pi^2 t}$ General Fourier series: $u(x, t) = \sum_{n=1}^\infty A_n \sin(n\pi x) e^{-\alpha n^2 \pi^2 t}$

Step 3: Matching initial data At $t = 0$: $u(x, 0) = \sum_{n=1}^\infty A_n \sin(n\pi x) = 50 \sin(\pi x) - 20 \sin(3\pi x)$ By orthogonality: $A_1 = 50\,^\circ\text{C}, \quad A_3 = -20\,^\circ\text{C}, \quad A_{n \neq 1,3} = 0$ The exact closed solution is: $u(x, t) = 50 \sin(\pi x) e^{-\alpha \pi^2 t} - 20 \sin(3\pi x) e^{-9 \alpha \pi^2 t}$

Step 4: Midpoint evaluation at $t = 1000\,\text{s}$ Parameters: $\alpha \pi^2 t = 10^{-4} \times \pi^2 \times 1000 = 0.1 \pi^2 \approx 0.98696$. $e^{-0.98696} \approx 0.3727, \quad e^{-9(0.98696)} = e^{-8.8826} \approx 0.000139$ At $x = 0.5$: $\sin(\pi/2) = 1, \sin(3\pi/2) = -1$. $u(0.5, 1000) = 50(1)(0.3727) - 20(-1)(0.000139) \approx 18.64\,^\circ\text{C}$ The third harmonic is practically wiped out ($99.99\%$ decay).

Inverse example

Problem

An acoustic pressure disturbance inside a 1D loss-free waveguide follows: $u(x, t) = \frac{1}{1 + (x - 340 t)^2} + \frac{1}{1 + (x + 340 t)^2}$

1. Prove this satisfies D'Alembert's wave PDE and determine wave speed $v$.
2. Reconstruct the initial spatial shape $u(x, 0)$ and initial velocity $\partial_t u(x, 0)$.

Detailed solution

Step 1: Testing D'Alembert's form The expression matches D'Alembert's traveling wave structure $u(x, t) = F(x - vt) + G(x + vt)$ with $v = 340\,\text{m}/\text{s}$. Differentiating confirms: $\frac{\partial^2 u}{\partial t^2} = (340)^2 \frac{\partial^2 u}{\partial x^2}$ verifying the wave equation identically.

Step 2: Reconstructing initial conditions

• Initial profile at $t = 0$: $u(x, 0) = \frac{2}{1 + x^2}$
• Initial velocity at $t = 0$: $\left.\frac{\partial u}{\partial t}\right|_{t=0} = \frac{2vx}{(1+x^2)^2} - \frac{2vx}{(1+x^2)^2} = 0$ The initial configuration was a stationary symmetric pressure hump released from rest.

Common Errors: Basic Partial Differential Equations

Error 1: Using Dirichlet sine bases for Neumann boundary problems

Applying $\sin(n\pi x/L)$ when boundaries are thermally insulated or free-ended.

• Correction: Zero-flux / free-end conditions demand derivative vanishing $\partial_x u = 0$ (Neumann), requiring cosine modes $\cos(n\pi x/L)$ and the $n = 0$ constant zero mode.

Error 2: Dropping the $n = 0$ average mode in Neumann diffusion

Omitting constant term $A_0$ in Fourier expansions for insulated bars.

• Consequence: Predicts that insulated thermal energy drops to zero, violating global energy conservation.

Error 3: Attempting separation of variables with inhomogeneous boundary conditions

Testing $u = X T$ directly when boundary values are non-zero (e.g. $u(0, t) = 100$).

• Correction: Subtract the steady-state baseline $u_{\text{ss}}(x)$ first to homogenize boundary data.

Error 4: Conflating temporal operators between heat and wave equations

Assigning harmonic sines/cosines to diffusion or exponential decay to lossless waves.

• Rule: $u_t$ yields dissipative irreversible decay $e^{-\lambda t}$; $u_{tt}$ yields conservative oscillations $\cos(\omega t)$.

Error 5: Forgetting the initial velocity condition in wave problems

Fitting initial displacement $u(x, 0)$ while ignoring initial velocity $\partial_t u(x, 0)$.

• Correction: Second-order time derivatives demand two initial conditions to resolve both modal amplitudes and velocity phases.

Quick control rule

Evaluate your final series at $t = 0$: it must reproduce the initial profile $u(x, 0)$ identically and satisfy spatial boundary constraints at $x = 0$ and $x = L$.

Physical Applications: Basic Partial Differential Equations

1. Heat conduction and metallurgical quenching

Steel heat treatment and cryo-tempering in industrial metallurgy follow anisotropic heat diffusion: $\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T)$ Simulating thermal gradients prevents residual mechanical stresses and avoids microfractures in aerospace turbine blades.

2. Architectural acoustics and auditorium sound field design

Acoustic pressure propagation across concert halls satisfies the Helmholtz frequency-domain PDE: $\nabla^2 p + \frac{\omega^2}{c^2} p = 0$ Resolving room eigenmodes allows acoustic engineers to eliminate destructive standing wave flutter echoes via Schroeder diffusers.

3. Magnetohydrodynamics and Tokamak plasma confinement

Magnetic fusion reactors (e.g. ITER) model magnetohydrodynamic plasma equilibrium via the Grad-Shafranov elliptic PDE: $\Delta^* \psi = -\mu_0 R^2 p'(\psi) - F F'(\psi)$ where poloidal magnetic flux $\psi$ shapes closed toroidal magnetic confinement surfaces.

4. Atmospheric dynamics and synoptic meteorology

Synoptic-scale weather system propagation is governed by the quasi-geostrophic potential vorticity equation: $\frac{\partial}{\partial t} (\nabla^2 \psi) + J(\psi, \nabla^2 \psi + f) = 0$ where $\psi$ denotes the streamfunction and $f$ the Coriolis parameter, forecasting jet stream meanders and cyclone development.

5. Ultrafast photonics and optical solitons in fiber telecommunications

Femtosecond light pulse propagation along optical fibers balances group velocity dispersion and optical Kerr nonlinearity via the Non-Linear Schrödinger Equation (NLSE): $i \frac{\partial A}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + \gamma |A|^2 A = 0$ This balance generates stable optical solitons transmitting petabits of data through undersea transoceanic cables without pulse broadening.

Interdisciplinary connection

PDEs form the mathematical core of oil exploration seismology (reverse-time seismic wave migration), cardiac electrophysiology (Hodgkin-Huxley wave propagation), and relativistic numerical astrophysics (gravitational waveform modeling from binary black hole mergers).

Guided Practice: Basic Partial Differential Equations

Exercise 1: Plucked guitar string released from a triangular pluck

A string clamped at $x = 0$ and $x = L$ is plucked upward to height $h$ at center and released from rest: $u(x, 0) = \begin{cases} \frac{2 h}{L} x, & 0 \le x \le \frac{L}{2} \\ \frac{2 h}{L}(L - x), & \frac{L}{2} \le x \le L \end{cases}, \quad u_t(x, 0) = 0$

1. Formulate the Fourier series solution for $u_{tt} = v^2 u_{xx}$.
2. Compute Fourier coefficients $A_n$.
3. Explain why all even harmonics vanish.

Solution

1. Wave equation series: $u(x, t) = \sum_{n=1}^\infty A_n \sin\left(\frac{n \pi x}{L}\right) \cos\left(\frac{n \pi v t}{L}\right)$
2. Integrating: $A_n = \frac{8 h}{\pi^2 n^2} \sin\left(\frac{n \pi}{2}\right)$
3. Even harmonics vanish because $\sin(n\pi/2) = 0$ for all even $n$, reflecting mirror symmetry of the initial pluck around midpoint $x = L/2$. $u(x, t) = \frac{8 h}{\pi^2} \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^2} \sin\left(\frac{(2m+1)\pi x}{L}\right) \cos\left(\frac{(2m+1)\pi v t}{L}\right)$

Exercise 2: Thermal relaxation in an insulated rod (Neumann conditions)

A heat conductor of length $L$ has insulated ends: $u_x(0, t) = 0$ and $u_x(L, t) = 0$. Initial temperature: $u(x, 0) = T_0 \cos^2\left(\frac{\pi x}{L}\right)$.

1. State the spatial eigenfunctions for homogeneous Neumann boundaries.
2. Find the analytic temperature profile $u(x, t)$.
3. Determine the equilibrium temperature as $t \to \infty$.

Solution

1. Eigenfunctions: $\lambda_n = (n\pi/L)^2, X_n(x) = \cos(n\pi x / L)$ for $n = 0, 1, 2, \dots$.
2. Rewriting initial data: $u(x, 0) = \frac{T_0}{2} + \frac{T_0}{2} \cos\left(\frac{2 \pi x}{L}\right)$ Matching coefficients: $A_0 = T_0/2$, $A_2 = T_0/2$, other $A_n = 0$. $u(x, t) = \frac{T_0}{2} + \frac{T_0}{2} \cos\left(\frac{2 \pi x}{L}\right) e^{-4 \alpha (\pi/L)^2 t}$
3. As $t \to \infty$, transient decays: $\lim_{t \to \infty} u(x, t) = \frac{T_0}{2}$ Thermal energy redistributes uniformly without boundary heat loss.

Exercise 3: Electrostatic potential in a square plate (Laplace equation)

A 2D conducting sheet $[0, a] \times [0, a]$ satisfies $\nabla^2 V = 0$. Sides $x = 0, x = a$, and $y = 0$ are grounded ($V = 0$), while top edge $y = a$ is held at potential $V(x, a) = V_0$. Solve for interior potential $V(x, y)$.

Solution

1. Separating $V(x, y) = X(x) Y(y)$: $X_n(x) = \sin\left(\frac{n \pi x}{a}\right), \quad Y_n(y) = \sinh\left(\frac{n \pi y}{a}\right)$
2. Series form: $V(x, y) = \sum_{n=1}^\infty C_n \sin\left(\frac{n \pi x}{a}\right) \sinh\left(\frac{n \pi y}{a}\right)$
3. Matching $V(x, a) = V_0$: $C_n = \frac{4 V_0}{n \pi \sinh(n \pi)} \quad (n \text{ odd})$ $V(x, y) = \frac{4 V_0}{\pi} \sum_{m=0}^\infty \frac{1}{2m+1} \frac{\sinh\left(\frac{(2m+1)\pi y}{a}\right)}{\sinh((2m+1)\pi)} \sin\left(\frac{(2m+1)\pi x}{a}\right)$