Houjun Liu

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 \(u(x,y)\)

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


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}


\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