SU-MATH53 FEB212024

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.