2.1: Examples of PDE

Classification of partial differential equations.

Examples of PDE

Partial differential equations occur in many different areas of physics, chemistry and engineering. Let me give a few examples, with their physical context. Here, as is common practice, I shall write \({\vect{\nabla}}^2\) to denote the sum \[{\vect{\nabla}}^2 = \pdxx{~}+\pdyy{~}+\ldots\]

Second order PDE

Second order P.D.E. are usually divided into three types. Let me show this for two-dimensional PDE’s: \[a\pdxx{u} +2c\pdxy{u} + b \pdyy{u} + d\pdx{u}+e\pdy{u}+f u+g=0\] where \(a,\ldots,g\) can either be constants or given functions of \(x,y\). If \(g\) is 0 the system is called homogeneous, otherwise it is called inhomogeneous. Now the differential equation is said to be \[\left. \begin{array}{r} \text{elliptic}\\ \text{hyperbolic}\\ \text{parabolic} \end{array}\right\} \text{ if } \Delta(x,y) = ab-c^2 {\rm~is~} \left\{ \begin{array}{l} \text{positive}\\ \text{negative}\\ \text{zero} \end{array} \right.\]

Reason behind names

Why do we use these names? The idea is most easily explained for a case with constant coefficients, and correspond to a classification of the associated quadratic form (replace derivative w.r.t. \(x\) and \(y\) with \(\xi\) and \(\eta\)) \[a \xi^2 + b\eta^2 +2c\xi\eta+f=0.\] We neglect \(d\) and \(e\) since they only describe a shift of the origin. Such a quadratic equation can describe any of the geometrical figures discussed above. Let me show an example, \(a=3\), \(b=3\), \(c=1\) and \(f=-3\). Since \(ab-c^2=8\), this should describe an ellipse. We can write \[3 \xi^2+3\eta^2+2\xi\eta = 4(\frac{\xi+\eta}{\sqrt{2}})^2+ 2(\frac{\xi-\eta}{\sqrt{2}})^2=3,\label{eq:II:ellipse}\] which is indeed the equation of an ellipse, with rotated axes, as can be seen in Fig. [fig:II:ellipse],

The ellipse corresponding to Eq. ([eq:II:ellipse])

We should also realise that Eq. ([eq:II:ellipse]) can be written in the vector-matrix-vector form \[(\xi,\eta) \left(\begin{array}{cc} 3 & 1 \\ 1 & 3 \end{array} \right) \left(\begin{array}{c} \xi \\ \eta \end{array}\right) = 3.\] We now recognise that \(\Delta\) is nothing more than the determinant of this matrix, and it is positive if both eigenvalues are equal, negative if they differ in sign, and zero if one of them is zero. (Note: the simplest ellipse corresponds to \(x^2+y^2=1\), a parabola to \(y=x^2\), and a hyperbola to \(x^2-y^2=1\))

Quiz

More than 2D

In more than two dimensions we use a similar definition, based on the fact that all eigenvalues of the coefficient matrix have the same sign (for an elliptic equation), have different signs (hyperbolic) or one of them is zero (parabolic). This has to do with the behaviour along the characteristics, as discussed below.

Let me give a slightly more complex example \[x^2 \pdxx{u} + y^2\pdyy{u} + z^2\pdzz{u}+2 xy\pdxy{u}+2 xz\pdxz{u}+2 yz\pdyz{u}=0.\] The matrix associated with this equation is \[\left(\begin{array}{lll} x^2 & xy & xz \\ xy & y^2 & yz \\ xz & yz & z^2 \end{array}\right)\] If we evaluate its characteristic polynomial we find that it is \[\lambda^2 (x^2-y^2+z^2-\lambda)=0.\] Since this has always (for all \(x,y,z\)) two zero eigenvalues this is a parabolic differential equation.

Characteristics and classification A key point for classifying equations this way is not that we like the conic sections so much, but that the equations behave in very different ways if we look at the three different cases. Pick the simplest representative case for each class, and look at the lines of propagation.

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