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(x)=\; \{$0 if x < 0; 1 if x ≥ 0 Solution by Gaussian Convolution: $u(x,t)\; =\; G$t*h(x)

Initial conditions: $f(x)\; =\; h(x+2)\; -\; h(x-2)$ Solution by Gaussian Convolution: $u(x,t)\; =\; G$t*f(x)

Initial conditions: $f(x)\; =\; h(x+2)\; +h(x)\; -\; 2\; h(x-2)$ Solution by Gaussian Convolution: $u(x,t)\; =\; G$t*f(x)

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

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