# The 1D Heat Equation on the Real Line

## The Gauss-Weierstrass Kernel

 The Gauss-Weierstrass Kernel $G$t(x) = 1/(2 π t)1/2 exp(-x2/2t) Initial conditions: Heaviside step function: $h\left(x\right)= \left\{0 if x < 0;$ 1 if x ≥ 0 Solution by Gaussian Convolution: $u\left(x,t\right) = G$t*h(x) Initial conditions: $f\left(x\right) = h\left(x+2\right) - h\left(x-2\right)$ Solution by Gaussian Convolution: $u\left(x,t\right) = G$t*f(x) Initial conditions: $f\left(x\right) = h\left(x+2\right) +h\left(x\right) - 2 h\left(x-2\right)$ Solution by Gaussian Convolution: $u\left(x,t\right) = G$t*f(x)

These animations were generated using Waterloo `MAPLE`, and were optimized using the GIMP image manipulation program.