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

• MHB
• evinda
In summary, to find the fundamental solution using the Fourier transform, we can use the properties of the transform and the Leibniz rule to solve for the solution in the form of a Gaussian function.
evinda
Gold Member
MHB
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?

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:$$E(t,x)=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \hat{E}(t,\xi) e^{-i x \xi} d \xi= \frac{H(t)}{(2 a \sqrt{\pi t})^n} e^{-\frac{|x|^2}{4a^2 t}}$$

Hello! It looks like you are trying to use the Fourier transform to show that the fundamental solution for the given partial differential equation is in the form of a Gaussian function. To continue, you can use the fact that the Fourier transform of the Dirac delta function is 1, and the Fourier transform of the Gaussian function is also a Gaussian function. By equating the two expressions for $\phi(x)$ and using the properties of the Fourier transform, you can solve for $\hat{E}(\xi)$ and then take the inverse Fourier transform to get the desired solution for $E(t,x)$. I hope this helps! Let me know if you have any other questions.

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