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Partial Differential Equations

★★★★☆Undergraduate

📖Definition

A partial differential equation (PDE) involves partial derivatives of an unknown function with respect to multiple independent variables. Describes waves, heat transfer, quantum mechanics.

📐Formulas

(∂² u)/(∂ t²) = c² ∇² u

Wave equation

(∂ u)/(∂ t) = α ∇² u

Heat equation (diffusion)

∇² u = 0

Laplace equation

i\hbar (∂ Ψ)/(∂ t) = \hatHΨ

Schrödinger equation

✏️Examples

예제 1

Explain the physical meaning of 1D heat equation.

📜History

Discovered by: Daniel Bernoulli, d'Alembert (18th century)

PDE theory developed while solving string vibration and heat conduction problems.

⚡Applications

Physics

Waves, thermodynamics, quantum mechanics

Engineering

Fluid dynamics, structural analysis

Finance

Option pricing (Black-Scholes)

🔗Related Documents

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