Partial Differential Equations

PDEs describe fields varying in space and time. Learn order, linearity and classification (elliptic, parabolic, hyperbolic), the heat, wave and Laplace equations with their physical meaning, initial and boundary conditions (Dirichlet, Neumann, Robin), well-posedness, and solution strategies.

Temperature in an engine block, vibration of a guitar string, stress in a dam, electric potential near a conductor, flow of groundwater — all are governed by PDEs. Recognising the type tells you how information travels and which methods work.

An analogy

Heat PDEs are like a drop of ink spreading in water (smoothing out), wave PDEs like ripples travelling across a pond, and Laplace's equation like a stretched rubber sheet settling into equilibrium.

The big three

Heat equation (parabolic): diffusion, smoothing, needs initial condition + BCs. Wave equation (hyperbolic): propagation at finite speed, needs initial displacement and velocity. Laplace (elliptic): steady state, needs BCs on the whole boundary.

Physical derivation

Heat: energy balance + Fourier's law q = −k∇T → ρc ∂T/∂t = k∇²T. Wave: Newton's law on a string element with tension T and density μ → c = √(T/μ). Laplace: steady heat or incompressible irrotational flow.

Boundary conditions

Fixed temperature (Dirichlet), insulated or prescribed flux (Neumann), convection to air (Robin: −k∂T/∂n = h(T − T∞)). Wrong BCs give wrong answers even with perfect maths.

Solution strategies

Separation of variables (simple geometries), transforms (Fourier, Laplace), similarity solutions, Green's functions, and numerical methods: finite differences, finite volumes, finite elements (M9, M21).

Formulas

• Heat (diffusion): ∂u/∂t = α² ∂²u/∂x²
• Wave: ∂²u/∂t² = c² ∂²u/∂x²
• Laplace / Poisson: ∇²u = 0 / ∇²u = f
• Classification: Au_xx + Bu_xy + Cu_yy + … : B² − 4AC < 0 elliptic, = 0 parabolic, > 0 hyperbolic
• Boundary conditions: Dirichlet u = g; Neumann ∂u/∂n = h; Robin au + b∂u/∂n = k

Worked example: Classify

Given: (a) u_xx + u_yy = 0 (b) u_t = 4u_xx (c) u_tt = 9u_xx

Find: Types and parameters.

• (a) B² − 4AC = −4 < 0 → elliptic
• (b) parabolic, α² = 4
• (c) hyperbolic, c = 3

Answer: Elliptic, parabolic, hyperbolic

Worked example: Guitar string wave speed

Given: Tension 70 N, μ = 1 g/m, L = 0.65 m

Find: c and fundamental.

• c = √(70/0.001) = 264.6 m/s
• f₁ = c/(2L) = 203.5 Hz

Answer: ≈ 204 Hz (close to G3)

Common mistakes

• Wrong: Giving only one initial condition for the wave equation. — Right: Need displacement and velocity.
• Wrong: Mixing up Dirichlet and Neumann. — Right: Value vs normal derivative.

Applications

• Thermal design: Heat equation in components.
• Acoustics & structures: Wave equation in strings, beams, air.
• Electrostatics & seepage: Laplace/Poisson in fields and soils.

Key takeaways

• Heat (parabolic), wave (hyperbolic), Laplace (elliptic).
• Classify with B² − 4AC.
• Dirichlet, Neumann, Robin BCs.
• Type decides required conditions and methods.

Going further

• Characteristics: Information paths for hyperbolic PDEs.
• Well-posedness (Hadamard): Existence, uniqueness, stability.
• Navier–Stokes: Non-linear PDEs of fluid flow.