MATH529: Partial Differetial Equations Review

Overview

• Linear PDEs

• Sturm-Liouville problem

• Separation of variables

• Curvilinear coordinates

• Eigenfunction expansions

Canonical linear PDEs of physics

• Diffusion equation $u_t=\alpha {\nabla }^{2}u$, $\alpha$ - diffusivity, $\left[\alpha \right]=$m${}^2/$s, unresolved transport.

  • Thermal diffusion. Empirical Fourier heat flux: $𝒒=-k\nabla u$, $\left[𝒒\right]=$J/s/m${}^2$

    $u$ - temperature $\left[u\right]=$K, $k$ - conductivity $\left[k\right]=$J/s/m/K, W=J/s

  • Mass diffusion. Empirical Fick flux: $𝒋=-\alpha \nabla u$, $\left[𝒋\right]=$mole/s/m${}^2$

    $u$ - number density $\left[u\right]=$mole/m${}^3$

  empirical flux Conservation $u$

• Wave equation $u_{tt}=c^2{\nabla }^{2}u$