Partial Differential Equations
Classification of PDEs
Classification
Heat Equation
Models diffusion of heat. Solution by separation of variables or Fourier series.
Wave Equation
Models vibrating strings. Solution by d'Alembert's formula:
Laplace Equation
Solutions are harmonic functions. Models steady-state temperature, electrostatics, potential flow.
Separation of Variables
Example (Heat Equation on ): , ,
,
Fourier Series
Boundary Conditions
Maximum Principle
Applications
- Heat conduction in solids
- Wave propagation (sound, light, water waves)
- Electrostatics and magnetostatics
- Fluid dynamics
- Quantum mechanics (SchrΓΆdinger equation)