A Partial Differential Equation is a Differential Equation which has more than one **independent variable**: $u(x,y), u(t,x,y), …$

For instance:

\begin{equation} \pdv{U}{t} = \alpha \pdv[2]{U}{x} \end{equation}

## Key Intuition

- PDEs may have no solutions (unlike Uniqueness and Existance for ODEs)
- yet, usually, there are too many solutions—so… how do you describe all solutions?
- usually, there are no explicit formulas

## Laplacian of \(u(x,y)\)

## Examples

### Heat Equation

See Heat Equation

### Wave Equation

see Wave Equation

### Transport Equation

\begin{equation} \pdv{u}{t} = \pdv{u}{x} \end{equation}

generally any \(u = w(x+t)\) should solve this

### Schrodinger Equation

We have some:

\begin{equation} u(x,t) \end{equation}

and its a complex-valued function:

\begin{equation} i \pdv{u}{t} = \pdv[2]{u}{x} \end{equation}

which results in a superposition in linear equations

### Nonlinear Example

\begin{equation} \pdv{u}{t} = \pdv[2]{u}{x} + u(1-u) \end{equation}

this is a PDE variant of the logistic equation: this is **non-linear**

### Monge-Ampere Equations

\begin{equation} u(x,y) \end{equation}

#### Hessian

\begin{equation} Hess(u) = \mqty(\pdv[2]{u}{x} & \frac{\partial^{2} u}{\partial x \partial y} \\ \frac{\partial^{2} u}{\partial x \partial y} & \pdv[2]{u}{y}) \end{equation}

If we take its determinant, we obtain:

\begin{equation} \pdv[2]{u}{x} \pdv[2]{u}{y} - \qty(\frac{\partial^{2} u}{\partial x \partial y})^{2} \end{equation}

## Traveling Wave

For two-variable PDEs, it is called a Traveling Wave if solutions to \(u\) takes on the form:

\begin{equation} u(t,x) = w(x-ct) \end{equation}

for some constant \(c\), and where \(w(x)\) is a function which depends on only one of the two variables.

## Bell Curves

See also Bell Curves