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$u_t-u_{xx}=0,$ $x\in\mathbb R,$ $t>0$ and $u(x,0)=e^{-x^2}.$

By applying Fourier transform on $t$ I have $\dfrac{\partial }{\partial t}F(u)+{{\omega }^{2}}F(u)=0,$ the solution of the latter equation is $F(u)(\omega,t)=ce^{-\omega^2t},$ now by applying the initial condition I have $F(u)(x,0)=c=e^{-x^2},$ so $F(u)(\omega,t)=e^{-x^2}e^{-\omega^2t}.$ So I use the convolution property to get $u(x,t)=\dfrac1{\sqrt{2\pi}}(e^{-x^2}*e^{-\omega^2t}).$

Is this correct?

Thanks!

By applying Fourier transform on $t$ I have $\dfrac{\partial }{\partial t}F(u)+{{\omega }^{2}}F(u)=0,$ the solution of the latter equation is $F(u)(\omega,t)=ce^{-\omega^2t},$ now by applying the initial condition I have $F(u)(x,0)=c=e^{-x^2},$ so $F(u)(\omega,t)=e^{-x^2}e^{-\omega^2t}.$ So I use the convolution property to get $u(x,t)=\dfrac1{\sqrt{2\pi}}(e^{-x^2}*e^{-\omega^2t}).$

Is this correct?

Thanks!

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