- #1

evinda

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I want to show using the Fourier transform that the fundamental solution of $\frac{\partial{E}}{\partial{t}}-a^2 \Delta{E}=\delta(t,x), x \in \mathbb{R}^n$, is given by $E(t,x)=\frac{H(t)}{(2 a \sqrt{\pi t})^n} e^{-\frac{|x|^2}{4a^2 t}}$.

$H$ is the Heaviside function.

We have:$$\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{\phi(\xi)} e^{i x \xi} d \xi=\phi(x)=\frac{\partial{E}}{\partial{t}}-a^2 \Delta E=\left( \frac{\partial}{\partial t}-a^2 \Delta \right)E=\left( \frac{\partial}{\partial t}-a^2 \Delta \right) \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \hat{E}(\xi) e^{ix \xi} d \xi$$How can we continue?