MHB Can the Fourier Transform Prove the Fundamental Solution for a Heat Equation?

[evinda]

Hello! (Wave)

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?

 

[Jhoneine]

We can apply the operator $\left( \frac{\partial}{\partial t}-a^2 \Delta \right)$ to the integral and use the Leibniz rule:$$\left( \frac{\partial}{\partial t}-a^2 \Delta \right) \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \hat{E}(\xi) e^{ix \xi} d \xi= \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \left( \frac{\partial}{\partial t}-a^2 \Delta \right) \hat{E}(\xi) e^{ix \xi} d \xi= \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \delta(t,x) e^{ix \xi} d \xi$$Using the Fourier inversion theorem, we have:$$\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \delta(t,x) e^{ix \xi} d \xi = H(t) \delta^n (x)$$where $\delta^n(x)$ is the n-dimensional Dirac delta function.Now, using the Fourier transform of the fundamental solution, we have:$$\hat{E}(t,\xi)=\frac{H(t)}{(2 a \sqrt{\pi t})^n} e^{-\frac{|\xi|^2}{4a^2 t}}$$Finally, we can apply the inverse Fourier transform to get the desired result:$$E(t,x)=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \hat{E}(t,\xi) e^{-i x \xi} d \xi= \frac{H(t)}{(2 a \sqrt{\pi t})^n} e^{-\frac{|x|^2}{4a^2 t}}$$