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I have a few problems which need to be solved using Fourier's transform. My problem is that I don't know how should I start this type of exercise (I just begin learning differentional equations). Could someone help me solved it step by step? One of them:

$$ u_t(t,x) = \Delta_x u(t,x) $$ $$ u(0,x) = f(x) $$ by using $$ (2\pi)^{-\frac{n}{2}} \int\limits_{R^n}e^{ix\xi-t|\xi|^2} d\xi = (2t)^{-\frac{n}{2}}e^{-\frac{|x|^2}{4t}} * $$

Edit:

I've managed to reach the stage

$$ u(t,x)=F^{-1}(e^{-t|\xi|^2}F(f)) $$

But how can I solve it more specific, by using * ?

pupilx
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1 Answers1

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\begin{align} u(t,x) & = \mathcal{F}^{-1}(e^{-t|\xi|^2}\mathcal{F}(f)) \\ & = \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^n}e^{i\xi\cdot x}e^{-t|\xi|^2}\left(\int_{\mathbb{R}^{n}}f(y)e^{-iy\cdot\xi}dy\right)d\xi \\ & = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}f(y)\left(\int_{\mathbb{R}^{n}}e^{i\xi\cdot(x-y)}e^{-t|\xi|^2}d\xi\right)dy \\ & = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n}f(y)(2\pi)^{n/2}(2t)^{-n/2}e^{-|x-y|^2/4t}dy \\ & = \frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^n}f(y)e^{-|x-y|^2/4t}dy \end{align}

Disintegrating By Parts
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  • I have one more question, maybe basic, but why can I change order of integrals (I mean in 3 line) ? – pupilx Feb 21 '16 at 12:13
  • @pupilx : Assume, for example, that $f \in L^1(\mathbb{R})$. Then the double integral is absolutely convergent and Fubini's Theorem applies. That's a typical scenario because $f$ is the initial distribution of heat, and the total heat should be finite. – Disintegrating By Parts Feb 21 '16 at 14:42