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(-x^{2}/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) |
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