5 Other important PDEs in science and engineering

Poisson’s equation
$\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}=f\left(x,y\right)$ (twodimensional form)
where $f\left(x,y\right)$ is a given function. This equation arises in electrostatics, elasticity theory and elsewhere.

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.

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)

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 .