5 Other important PDEs in science and engineering

1. Poisson’s equation

$\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}=f\left(x,y\right)$ (two-dimensional form)

where $f\left(x,y\right)$ is a given function. This equation arises in electrostatics, elasticity theory and elsewhere.

2. Helmholtz’s equation

$\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+{k}^{2}u=0$ (two dimensional form)

which arises in wave theory.

3. Schrödinger’s equation

$-\frac{{h}^{2}}{8{\pi }^{2}m}\left(\frac{{\partial }^{2}\psi }{\partial {x}^{2}}+\frac{{\partial }^{2}\psi }{\partial {y}^{2}}+\frac{{\partial }^{2}\psi }{\partial {z}^{2}}\right)=E\psi$

which arises in quantum mechanics. ( $h$ is Planck’s constant)

4. Transverse vibrations equation

${a}^{2}\frac{{\partial }^{4}u}{\partial {x}^{4}}+\frac{{\partial }^{2}u}{\partial {t}^{2}}=0$

for a homogeneous rod, where $u\left(x,t\right)$ is the displacement at time $t$ of the cross section through $x$ .

All the PDEs we have discussed are second order (because the highest order derivatives that arise are second order) apart from the last example which is fourth order .