Green's function for Sturm-Louiville ODE

In summary, the Green's function for this problem does not exist due to the unboundedness of the solution at x = 1. Instead, the problem can be solved using an eigenvalue approach with a suitable weight function and self-adjoint boundary condition. This approach will yield a discrete spectrum of eigenvalues and eigenfunctions, which can be used to find a particular solution and a complimentary function that will satisfy the given boundary conditions.
  • #1
member 428835
Hi PF!

Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function?

This is the associated Legendre ODE, which admits two linearly independent solutions: $$y_1 = P_1^1, \,\,\, y_1 = Q_1^1$$ where ##P_1^1## and ##Q_1^1## are associated Legendre polynomials first and second kind respectively. Things now get a little murky: notice ##y_1## automatically satisfies ##y_1'(a) + \sec(a)\tan(a)y_1(a) = 0##, but ##y_2(b) = \infty##. If ##y_2(b)## was bounded, the Green's function would be very simple to calculate. But it's not bounded, so I'm confused how to proceed?

Any help is greatly appreciated.
 
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  • #2
If you want a solution bounded at 1, then you are limited to scalar multiples of [itex]P_1^1[/itex]: [tex]
G(x;x_0) = \begin{cases}
AP_1^1(x) & x \leq x_0 \\
BP_1^1(x) & x > x_0\end{cases}.[/tex] The conditions you need to satisfy are [tex]
\begin{split}
(A - B)P_1^1(x_0) &= 0 \\
(A - B){P_1^1}'(x_0) &= \frac{1}{p(x_0)}\end{split}[/tex] which cannot be done unless [itex]P_1^1(x_0) = 0[/itex] or you are prepared to relax the constraint that [itex]G(\cdot;x_0)[/itex] be continuous at [itex]x_0[/itex].
 
  • #3
I appreciate your reply. So it looks like a Green's function doesn't exist for this problem, right?
 
  • #4
joshmccraney said:
I appreciate your reply. So it looks like a Green's function doesn't exist for this problem, right?

Yes.

The way to solve the non-homogenous boundary value problem [tex]
L(y) = (py')' + qy = -g[/tex] on the interval [itex][a,b][/itex] subject to arbitrary boundary conditions on [itex]y[/itex] is to first solve the eigenvalue problem [tex]
L(\phi) = -\lambda w \phi[/tex] for a convenient weight function [itex]w[/itex] which is strictly positive on [itex](a,b)[/itex] with [itex]\phi[/itex] subject to a convenient self-adjoint boundary condition. This gives you a discrete spectrum of eigenvalues [itex]\lambda_n \neq 0[/itex] with corresponding eigenfunctions [itex]\phi_n[/itex] which are orthogonal with respect to the inner product [itex](f,g)_w = \int_a^b f(x)g(x) w(x)\,dx[/itex].

(In this case, choosing a suitable boundary condition and setting [itex]w = 1[/itex] will give [itex]\phi_n = P_n^1[/itex] with [itex]\lambda_n = n(n+1) - 2[/itex] for [itex]n \geq 2[/itex].)

Next, look for a particular solution [tex]y_P = \sum_n a_n \phi_n[/tex] so that [tex]
L(y_P) = -\sum_na_n\lambda_n w \phi_n = -g[/tex] and then multiplying by [itex]\phi_m[/itex] and integrating you should find[tex]
a_m = \frac{1}{\lambda_m \|\phi_m\|_w^2}\int_a^b g(x)\phi_m(x)\,dx.[/tex] A complimentary function [itex]y_C \in \ker L[/itex] can then be added to [itex]y_P[/itex] in order to satisfy the boundary conditions on [itex]y[/itex].
 

What is the Green's function for Sturm-Louiville ODE?

The Green's function for Sturm-Louiville ODE is a mathematical tool used to solve certain types of ordinary differential equations (ODEs). It represents the solution to a specific type of boundary value problem.

What is the purpose of the Green's function for Sturm-Louiville ODE?

The Green's function allows us to solve a boundary value problem by breaking it down into simpler problems. It is particularly useful for solving problems with non-homogeneous boundary conditions.

How is the Green's function for Sturm-Louiville ODE calculated?

The Green's function is calculated by solving the ODE with specific boundary conditions and then taking the derivative of the solution with respect to the boundary conditions. This process results in a function that can be used to solve the original boundary value problem.

What are the applications of the Green's function for Sturm-Louiville ODE?

The Green's function has many applications in physics, engineering, and other scientific fields. It is commonly used to solve problems related to heat transfer, wave propagation, and diffusion processes. It is also used in the study of quantum mechanics and other areas of physics.

What are the limitations of the Green's function for Sturm-Louiville ODE?

The Green's function can only be used for linear boundary value problems. It also requires specific boundary conditions to be known, which may not always be the case. Additionally, the calculation of the Green's function can be complex and time-consuming for certain types of ODEs.

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