Partial Differential Equations

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The solution to a Partial Differential Equations (PDEs) is a function of more than one variable (e.g. $u(x,y)$). PDEs include more than one dependent variable (e.g. $x,y$) and the derivative of the unknown function (e.g. $u(x,y)$) with respect to the dependent variables. The best known PDEs are in one spatial dimension

\begin{align} \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}&=0\\ u_x+v u_y&=0 \end{align}

  • the diffusion or heat equation

\begin{align} \frac{\partial^2 u}{\partial x^2} &= k \frac{\partial u}{\partial t}\\ u_{xx}&=k u_{t} \end{align}

\begin{align} \frac{\partial^2 u}{\partial x^2} &= c^2 \frac{\partial^2 u}{\partial t^2}\\ u_{xx}&=c^2 u_{tt} \end{align}

\begin{align} \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}&=0\\ u_{xx}+ u_{yy}&=0 \end{align}

  • the Schrödinger equation with $i=\sqrt{-1}$

\begin{align} i \frac{\partial^2 u}{\partial x^2} &= \frac{\partial u}{\partial t}\\ iu_{xx}&= u_{t} \end{align}

The order of a PDE is determined by the highest partial derivative occuring in the equation (just like in the case of ODEs), as far as the highest partial derivative of the equation can't be reduced to a smaller one. For example the equation $u_xx+u_xy=0$ is can be reduced to a first order equation by making the substitution $w=u_x$. This would yield the transport equation $u_x+u_y=0$ and the ODE $w=u_x$.

We can define an operator of a PDE, e.g. of the wave equation $\frac{\partial^2 u}{\partial x^2} = c^2 \frac{\partial^2 u}{\partial t^2}$ as $\mathcal{L} = \frac{\partial^2}{\partial x^2} - c^2 \frac{\partial^2}{\partial t^2}$ and write the wave equation as \begin{align} \left( \frac{\partial^2}{\partial x^2} - c^2 \frac{\partial^2}{\partial t^2} \right) u = \mathcal{L}[ u] = 0 \end{align}

A linear operator of a PDE satisfys the equation $\mathcal{L}(u)+\mathcal{L}(v)=\mathcal{L}(u+v)$, were $u$ and $v$ are solutions to the PDE. Are the PDEs presented above linear? The equation $u_{tt}-u_{xx}+u^2$ is nonlinear, because $(u+v)^2=u^2+2uv+v^2 \not = u^2+v^2$.

An inhomogenious PDE contains a constant or a function that is independent of the unknown function, if it doesn't the PDE is called homogenious.

A general second order linear PDE has the form

\begin{align} au_{xx}+2bu_{xy}+cu_{yy}+du_{x}+eu_y+fu=g \end{align}

if $g=0$ the equation is homogenious, otherwise not. In compact notation a second order PDE in $n$ dimensions can be written as

\begin{align} \sum_{i,j=1}^n a_{ij} u_{x_ix_j} + \sum_{i=1}^n b_iu_{xi} + cu = d \end{align} The term $\sum_{i,j=1}^n a_{ij} u_{x_ix_j}$ is called the principle part of the PDE. In two dimensions ($n=2$) the matrix $a_{ij}$ is symmetric ($\mathbf{A}=\mathbf{A}^T$) with

\begin{align} a_{ij}= \mathbf{A} =\begin{pmatrix} a &b \\ b &c \end{pmatrix} \end{align} if $u(x,y)$ (and its derivatives) is continuously differentiable in $x$ and $y$ (Schwarz theorem). Such PDEs can be devided into three classes according accroding to the determinant of the matrix $\mathbf{A}$.

det($\mathbf{A}$) for $a>0$ PDE type example
$ac-b^2<0$ hyperbolic wave equation
$ac-b^2>0$ elliptic Laplace equation
$ac-b^2=0$ parabolic heat equation

A geometric interpretation of the three PDE types is given in the article about Positive Definiteness.