Past exam of the differential equations 2 course of the University of Oxford / 2017 / 1 / b

Interpret the Dirac delta coefficient as a point interaction in a second-order differential equation. The solution is continuous at $x=1$ and is twice differentiable on either side, where $y^{\prime \prime }=f$. Its transmission conditions for a delta coefficient areIndeed, the distributional second derivative across a jump contributes $[y^{\prime }{]}_1\delta (x-1)$, which must cancel the delta coefficient because $f$ has no point source.

Put $s=y^{\prime }(0)$ and $F(x)={\int }_0^x(x-t)f(t)dt$. Integrating the ordinary equation and inserting the required derivative jump giveswhere $(x-1{)}_{+}=\max (x-1,0)$. This formula is continuous, has the stated jump, satisfies the equation, and parametrizes every solution with the left endpoint value. Imposing the right endpoint value gives the scalar equationConsequently a unique $s$, and hence a unique solution for every $f,c_1,c_2$, exists precisely whenAt the exceptional value, the nonzero homogeneous solution with zero endpoint values isIt has derivative jump $-\pi /(\pi -1)=-{\alpha }_{\ast }\varphi (1)$. Thus uniqueness fails. More precisely, there are infinitely many solutions ifand none otherwise. This follows either from the scalar equation or by integration by parts against $\varphi$. It is the compatibility obstruction at Dirichlet resonance for a point interaction.

Use a Green function satisfying $L_{x}G(x,\xi )=\delta (x-\xi )$ with homogeneous Dirichlet boundary conditions $G(0,\xi )=G(\pi ,\xi )=0$, for $0<\xi <\pi$. Since $G_{xx}=0$ away from $x=1$ and $x=\xi$, it is continuous and piecewise affine, with possible derivative jumps at those two locations.

For $\xi \ne 1$, the transmission conditions for a delta coefficient and the unit source condition areIf $\xi =1$, there is a single junction, and the correct combined condition isFor example, a suitable piecewise form when $0<\xi <1$ iswhere the coefficients depend on $\xi$. The two continuity conditions, $B-A=1$ and $-D-B=-G(1,\xi )$ determine them. When $1<\xi <\pi$, use the same affine form on the intervals cut at $1,\xi$, with $B-A=-G(1,\xi )$ and $-D-B=1$ instead. This gives all requested boundary and jump conditions without needing the coefficients.

An explicit compact kernel also makes the construction transparent. The left and right homogeneous solutions at $\alpha =1$ areThey obey the interaction jump and have Wronskian $u{v}^{\prime }-u^{\prime }v=-1$ on either side. ThereforeDifferentiating its two branches at the source gives a unit derivative jump whenever $\xi \ne 1$. At $\xi =1$, $G(1,1)=-(\pi -1)$ and $[G_x{]}_1=\pi$, so the combined condition is also satisfied. This is the Dirichlet Green function for a point interaction specialized to the present parameters.

The Green kernel itself has zero endpoint values. The prescribed nonzero endpoint values enter through the homogeneous lift:Indeed, $v(0)=u(\pi )=1$ and $u(0)=v(\pi )=0$. The kernel accounts for the forcing; the lift accounts for the boundary data. Uniqueness follows from part (i), since $1\ne \pi /(\pi -1)$.