Partial Differential Equations

Source lecture(s): PHY621 Lec6

Intuition

PDEs describe fields varying in space and time. Classification tells you how information propagates.

Formal Definition

Second-order linear PDE: \(Au_{xx}+2Bu_{xy}+Cu_{yy}+\text{lower order}=0\). Discriminant \(B^2-AC\) determines the type.

Mathematical Formulation

Elliptic (\(B^2-AC<0\)): Laplace \(\nabla^2 u=0\). Parabolic (\(B^2-AC=0\)): Heat \(u_t=\kappa\nabla^2 u\). Hyperbolic (\(B^2-AC>0\)): Wave \(u_{tt}=c^2\nabla^2 u\).

Derivation

Try \(u=X(x)Y(y)\); separation yields \(X''/X + Y''/Y = 0\) for Laplace, so each equals a separation constant.

Worked Example

Rectangle Laplace problem gives \(\sinh\) and \(\sin\) eigenfunctions matched to boundary temperatures.

Common Mistakes

• Treating hyperbolic PDEs like parabolic (diffusion vs wave).
• Forgetting boundary conditions.

Related Concepts

• Ordinary Differential Equations
• Green's Functions

Quiz

Q1. Which PDE class has no time evolution?

Answer

Elliptic.