## Superposition
Consider the PDE $L\phi=f$
- $L$ is linear if and only if $L(a\phi_{1}+b\phi_{2})=aL\phi_{1}+bL\phi_{2}$
- $L$ is homogeneous if and only if $f=0$
- Homogenous: Any linear combinations of solutions is also a solution
- Inhomogenous: Solved by any particular solution plus a homogenous solution
## Key Equations
| Name | Equation | Notes |
| --------- | --------------------------------------------------------------------------- | --------------------------------------------------------------------------------- |
| Laplace | $\nabla^{2}\psi=0$ | source-free potentials |
| Poisson | $\nabla^{2}\psi=-f$ | potentials with known sources $f$ |
| Diffusion | $\nabla^{2}\psi=\frac{1}{D}\frac{ \partial \psi }{ \partial t }$ | $D=$ diffusivity |
| Wave | $\nabla^{2}\psi=\frac{1}{c^{2}}\frac{ \partial^{2}\psi }{ \partial t^{2} }$ | $c =$ wave speed |
| Helmholtz | $\nabla^{2}\psi+k^{2}\psi=0$ | wave equation for constant frequency $\omega
lt;br>wavenumber: $k=\frac{\omega}{c}$ |
## Solving Approaches
### Green's Functions
Define a field $G(\vec{r},\vec{r}')$ produced by a point source at $\vec{r}'$, then integrate over the real source distribution.
$
\phi = \iiint_{V'}G(\vec{r},\vec{r}')f(\vec{r}')dV'
$
| |$G(\vec{r},\vec{r}')$|
|---|---|
|Poisson (3D)|$\frac{1}{4\pi\lvert \vec{r}-\vec{r}' \rvert}$|
|Poisson (2D)|$-\frac{1}{2\pi}\ln\lvert \vec{r}-\vec{r}' \rvert$|
|Helmholtz (3D)|$\frac{e^{-jk\lvert \vec{r}-\vec{r}' \rvert}}{4\pi\lvert \vec{r}-\vec{r}' \rvert}$|
Used for superposition integrals and Biot-Savart in [[quasistatics]].
### Laplace Equation $\nabla^{2}\psi=0$
Solutions:
$1, x, y, z, xy, yz, xz, xyz$
## 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 |
### 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?