## PDE Classification $A \frac{ \partial^{2}u }{ \partial x^{2} } + B \frac{ \partial^{2}u }{ \partial x \partial y } + C \frac{ \partial^{2}u }{ \partial y^{2} } + D = 0$ | | Discriminant | Example PDE | Description | Domain of Dependence | | ---------- | ------------------- | ---------------------------------------------------------------------------------------------- | ------------ | -------------------- | | Elliptic | ${} B^{2}-4AC<0 {}$ | $\frac{ \partial^{2}T }{ \partial x^{2} } + \frac{ \partial^{2} T }{ \partial y^{2} }=0$ | Steady state | Global | | Parabolic | $B^{2}-4AC=0$ | $\frac{ \partial T }{ \partial t } = k' \frac{ \partial^{2}T }{ \partial^{2}x }$ | Diffusion | | | Hyperbolic | $B^{2}-4AC>0$ | $\frac{ \partial^{2} u }{ \partial t^{2} } - c^{2} \frac{ \partial^{2}u }{ \partial x^{2} }=0$ | Waves | Cone | https://www.youtube.com/watch?v=ptw-IDzlhRo ### Hyperbolic PDEs - Allows non-smooth solutions - Information travels along characteristics (streamlines for example) You can convert 1D wave equations into 2D hyperbolic wave equations ### Domain of Dependence Which points affect other points in the PDE?