Past exam of the differential equations 2 course of the University of Oxford / 2015 / 1 / c / Solution

Start with a fundamental solution of the second derivative operator:Here $H$ is the Heaviside step function and the last equality holds in the sense of distributions. The factor $x-\xi$ removes a jump in the function itself, while its first derivative has a unit jump. Adding any homogeneous solution preserves the Dirac delta source.

Choose $y_1(x)=x-2$ and $y_2(x)=x-3$. They are independent homogeneous solutions, with $B_1{y}_1=0$, $B_2{y}_2=0$, $B_1{y}_2=-1$, and $B_2{y}_1=1$. Since $1<\xi <2$, the fundamental solution has $B_1\tilde{g}=1$ and $B_2\tilde{g}=2-\xi$. Imposing the two boundary conditions on the proposed combination therefore gives $-A_2+1=0$ and $A_1+2-\xi =0$. ConsequentlyEquivalently,At $x=\xi$ both expressions equal ${\xi }^2-3\xi +1$ and the right slope minus the left slope is $\xi -(\xi -1)=1$. At the endpoints, $g(1,\xi )=-\xi$ and $g_x(2,\xi )=\xi$, while $g_x(1,\xi )=\xi -1$ and $g(2,\xi )=1-\xi$. These explicitly verify both coupled boundary conditions. The homogeneous equation has only affine solutions; its two boundary conditions give $2a+b=3a+b=0$, hence $a=b=0$, proving uniqueness of the Green function.

The reciprocity defect is $g(x,\xi )-g(\xi ,x)=2(x-\xi )$. It records how the coupled boundary conditions affect self-adjointness, even though the differential expression $D^2$ is formally self-adjoint. The construction is an instance of the boundary-matrix construction of an ordinary differential Green function.