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Heat-kernel solution

For initial data

u_{0}

on the real line, Fourier transformation or convolution with the fundamental solution gives

u (x, t) = \int_{- \infty}^{\infty} \frac{e ^{- (x - ξ)^{2} / (4 D t)}}{4 π D t} u_{0} (ξ) d ξ .

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- Heat equation with interval-indicator initial data Heat-kernel solution 13

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