What is the Green's function for this specific problem?

[member 428835]

Homework Statement

Find Green's function of $$K(\phi(s)) = \phi''(s)+\cot(s)\phi'(s)+\left(2-\frac{1}{\sin(s)^2}\right)\phi(s):s\in[0,\alpha]$$
subject to boundary conditions: $$\phi|_{s=0} < \infty\\
\phi|_{s=\alpha} = 0.$$

Homework Equations

Green's function $G$ is found via variation of parameters: $$
G_1(s,y) = \frac{v_1(s)v_2(y)}{W(s)}:0<s<y<\alpha\\
G_2(s,y) = \frac{v_1(y)v_2(s)}{W(s)}:0<y<s<\alpha$$
where $W$ is the Wronskian of $v_1,v_2$ and $v_i$ is a linearly independent solution satisfying a boundary condition.

The Attempt at a Solution

Legendre polynomials of first and second kind ($P_1^1(\cos(s)),Q_1^1(\cos(s))$) solve this ODE, so we construct $v_1$ and $v_2$ from these. To satisfy the finite domain at $s=0$ we take $v_1 = P_1^1(\cos(s))$ and to satisfy $\phi(\alpha)=0$ we take the solution $$v_2(s) = \frac{\tau_2}{\tau_1}(P_1^1(\cos(s))-Q_1^1(\cos(s))):\\
\tau_1 = P_1^1(\cos(\alpha));\,\,\tau_2 = Q_1^1(\cos(\alpha)).$$
Then we find $W(s) = -2\csc(s)$. Then we know the Green's function from above. However, when I substitute these results into the proposed method, Mathematica is suggesting the ODE is not solved. Any help is much appreciated!

Edit: Evidently the problem is solved when I take solutions of the form $$
G_1(s,y) = \frac{v_1(s)v_2(y)}{W(y)}:0<s<y<\alpha\\
G_2(s,y) = \frac{v_1(y)v_2(s)}{W(y)}:0<y<s<\alpha
$$
since now the boundaries are satisfied $G1(s=0,y)<\infty;\, G2(s=\alpha,y)=0$. Also, $G$ is continuous: $G2(s,y)|_{s=y}-G1(s,y)|_{s=y} = 0$, and $G$ exhibits the jump discontinuity at $s=y$: $G2'(s,y)|_{s=y}-G1'(s,y)|_{s=y} = 1$.

However, I should note that $G(s,y)$ satisfies the above conditions, but $G(y,s)$ does not. Can anyone comment here?

 

[jasonRF]

Your problem isn't self adjoint, so in general $G(s,y)\neq G(y,s)$. By the way, given that your problem is not self adjoint, are you sure that your formulae,

$$
G_1(s,y) = \frac{v_1(s)v_2(y)}{W(s)}:0<s<y<\alpha\\
G_2(s,y) = \frac{v_1(y)v_2(s)}{W(s)}:0<y<s<\alpha$$

are correct?

jason

 

[member 428835]

Your problem isn't self adjoint, so in general $G(s,y)\neq G(y,s)$. By the way, given that your problem is not self adjoint, are you sure that your formulae,are correct?

jason

Thanks for replying Jason! Good to know that fact about Green's functions. Regarding the formula, I think it's slightly incorrect. I in fact think if you evaluate the Wronskian at $y$ instead of $s$ you get the correct solution. Then I'm pretty sure the edited solution in post 1 is valid fro this problem.

Taking the proposed $v$'s from post 1 and their Wronskian, the Green's function suggested in post 1 edit satisfies $K(G) = 0$ for $s\neq y$, $G$ is continuous, $G'$ has a unit jump at $s=y$, and boundary conditions are satisfied by $G1$ and $G2$. What do you think?