Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 2 / 6B (Differential Equations)

Apply the chain rule to $U(s,y)=u(s,y+s)$. Holding $y$ fixed while differentiating in $s$ gives $U_s=u_t+u_x$, while $U_y=u_x$ and $U_{yy}=u_{xx}$. Thus the advection-diffusion equation becomesThe moving coordinate follows the unit-speed advection, leaving the heat equation with a transformed source.

For $y<0$, the Heaviside function vanishes, so $w=Ay+B$. For $y>0$, $w^{\prime \prime }=-1$, giving $w=-y^2/2+Cy+D$. Continuity of $w$ and $w^{\prime }$ at zero requires $D=B$ and $C=A$. Consequently the general continuously differentiable profile isIts second derivative jumps from zero to minus one. The equation is satisfied classically away from zero and in the weak, or distributional, sense across it; no globally twice-differentiable function can have the prescribed step as its second derivative. There is no Dirac delta contribution because $w^{\prime }$ is continuous. The value assigned to the Heaviside function at a single point does not change this weak solution.

Under the moving-coordinate transformation, the forcing is $H(y)$. Choose a stationary solution $U(s,y)=w(y)$, which requires precisely $w^{\prime \prime }+H=0$. The left spatial limit of $Ay+B$ is zero only if $A=B=0$. Therefore one solution is the transported step forcing in an advection-diffusion equation profileIt has $u_t=(x-t{)}_{+}$ and $u_x=-(x-t{)}_{+}$, so their sum vanishes; almost everywhere $u_{xx}=-H(x-t)$, canceling the source. The first derivatives are continuous on the moving interface. For each fixed $t$, the profile is zero far to the left and tends to minus infinity to the right, as required. The PDE holds pointwise off that interface and weakly across it, consistently with the requested continuously differentiable regularity.