A few questions about Green's Functions....

[vancouver_water]

Hi, I am having some trouble understanding exactly when a modified green's function is needed. Here is the general problem:

$$Lu = (p(x)u'(x))' + q(x)u(x) = f(x), x_0 \leq x \leq x_1, p(x) > 0,\\ \alpha_0 u(0) + \beta_0 u'(0) = 0, \alpha_1 u(1) + \beta_1 u'(1) = 0$$

In my notes it says if the corresponding homogeneous problem with the same boundary conditions has a non-zero solution $u^*(x)$, then we can construct a standard Green's function, with the solvability condition $\int_{x_0}^{x_1} u^*(x)f(x)dx = 0$. Then later in my notes it says if $u^*(x)$ is a non-zero solution, we need to use a modified Green's function. Which one is it?

Another question I have is regarding the formula involving the Wronskian, if we have to linearly independent solutions,

$$G(x,z) = -\dfrac{u_1(x)u_2(z)}{p(x)W(x)}, x_0 \leq z \leq x,\\ -\dfrac{u_1(z)u_2(x)}{p(x)W(x)}, x \leq z \leq x_1$$

When exactly is this solution valid?

Thank you.

 

[gtoguy97]

The modified Green's function is needed if the corresponding homogeneous problem has a non-zero solution u*(x) and the solvability condition \int_{x_0}^{x_1} u^*(x)f(x)dx \neq 0. The modified Green's function is then given by G(x,z) = -\dfrac{u_1(x)u_2(z)}{p(x)W(x)} - \dfrac{u_1(z)u_2(x)}{p(x)W(x)} + \dfrac{u^*(x)u^*(z)}{p(x)W(x)}, where u_1(x) and u_2(x) are two linearly independent solutions of the homogeneous equation and W(x) is the Wronskian of these two functions. This solution is valid for all x_0 \leq z \leq x_1.