partial differential equation: Nonlinear Function
Created: June 07, 2024
Modified: June 07, 2024

partial differential equation

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

References for PDEs:

Fundamental PDEs

wave equation

utt=c2uxxu_{tt} = c^2 u_{xx}

Laplace equation

uxx+uyy=0u_{xx} + u_{yy} = 0

heat equation

The heat equation in 1D is written

ut=uxxu_t = u_{xx}

and more generally

∂u∂t=a(∂2u∂x2+∂2u∂y2+∂2u∂z2)=a⋅div  ∇u=a⋅∇2u\begin{align*} \frac{\partial u}{\partial t} &= a\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)\\ & = a \cdot \text{div}\; \nabla u\\ & = a \cdot \nabla^2 u\\ \end{align*}

in terms of the vector divergence or (even more compactly) the Laplacian operator ∇2=∇⋅∇=div  ∇\nabla^2 = \nabla \cdot \nabla = \text{div}\; \nabla.

transport equation

ut+cux=0u_t + c u_x = 0

this implies that directional derivatives along the vector (x,t)=(c,1)(x, t) = (c, 1) are zero, so the function value at time t+1t + 1 and position x+cx + c is equal to the previous timestep value at position xx, i.e., everything is transported along the xx axis at rate cc.

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