Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 15B (Methods)

Use the Fourier transform $\tilde{f}(k)={\int }_{\mathbb{R}}f(x)e^{-ikx}dx$. For a rapidly decaying smooth function, integration by parts gives $\widetilde{f^{\prime }}=ik\tilde{f}$, and differentiation under the integral gives $\widetilde{xf}=id\tilde{f}/dk$. Both are justified here by Gaussian decay. Since $f^{\prime }=-2a^{2}xf$,The Gaussian integral supplied in the hint gives $\tilde{f}(0)=\sqrt{\pi }/a$. Solving this first-order equation yieldsThe calculation states the transform normalization and the two differentiation properties explicitly.

Taking the spatial Fourier transform of the diffusion equation gives ${\tilde{\theta }}_t=-D{k}^2\tilde{\theta }$ and $\tilde{\theta }(k,0)=\tilde{g}(k)$. Hence $\tilde{\theta }=e^{-D{k}^{2}t}\tilde{g}$. Fourier inversion and the Gaussian transform from part (a), or direct evaluation of the inverse Gaussian integral, give the heat kernelSubstituting the integral for $\tilde{g}$ and interchanging absolutely integrable terms produces the convolution solutionFor the stated nonnegative compactly supported data, understood to be nonzero on a set of positive measure, the kernel is strictly positive everywhere. Therefore $\theta (x,t)>0$ for every real $x$ and every $t>0$. Property P holds: diffusion has infinite propagation speed. The distant value can be extremely small, but is not zero. For continuous initial data, any nontrivial nonnegative $g$ automatically has the required positive-measure set.

For the wave equation, introduce characteristic coordinates $\xi =x-ct$, $\eta =x+ct$ when $c\ne 0$. The equation becomes $u_{\xi \eta }=0$, so $u=F(x-ct)+G(x+ct)$. The initial conditions imply $F+G=\varphi$ and $-c{F}^{\prime }+c{G}^{\prime }=0$. Constants can be absorbed into $F,G$, giving the D'Alembert formulaIf $\varphi$ vanishes outside $[-\alpha ,\alpha ]$, then $u(x,t)=0$ whenever $|x|>\alpha +|c|t$. Thus Property P fails: the wave equation has finite propagation speed $|c|$. No arbitrarily short time can transmit the disturbance arbitrarily far. If $c=0$, the zero initial velocity instead gives $u(x,t)=\varphi (x)$, and the conclusion still holds.