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 .