What is Wrong with the Generalized Green Function Equation?

In summary, the generalized Green function $G_g(x, x')$ satisfies the equation $(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')$ and the condition that $\int_0^{\ell}u_m(x)G_g(x, x')dx = 0$.
  • #1
Dustinsfl
2,281
5
The generalized Green function is
\[
G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.
\]
Show \(G_g\) satisfies the equation
\[
(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')
\]
where \(\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\)
and the condition that
\[
\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.
\]
I found
\[
u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).
\]
I then end up with
\begin{gather}
(\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\
\sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')
\end{gather}
What is going wrong?
 
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  • #2
It appears that you have not evaluated the left-hand side of the equation correctly. The expression $\mathcal{L} + k_m^2$ should be evaluated first, before applying it to $G_g(x, x')$. This can be done using the definition of $\mathcal{L}$ provided in the problem statement:\[(\mathcal{L} + k_m^2)G_g(x, x') = \sum_{n\neq m}\frac{-k_n^2u_n(x)u_n(x')}{k_m^2 - k_n^2} + k_m^2\sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.\]The right-hand side of the equation should then be evaluated using the definition of $\delta(x-x')$ provided in the problem statement:\[\delta(x - x') - u_m(x)u_m(x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x').\]Once these expressions are evaluated, they should match, giving you a proof that $G_g$ satisfies the equation.Finally, the condition that $\int_0^{\ell}u_m(x)G_g(x, x')dx = 0$ can be satisfied by noting that $u_m(x)$ is an eigenfunction of $\mathcal{L}$ with eigenvalue $k_m^2$, and so $u_m(x)$ is orthogonal to all other eigenfunctions (which appear in the sum of $G_g(x, x')$). Therefore, the integral of $u_m(x)G_g(x, x')$ over the interval from 0 to $\ell$ is zero.
 

What is a Generalized Green function?

A Generalized Green function is a mathematical tool used in physics and engineering to solve partial differential equations. It represents the solution to a given differential equation with a specific set of boundary conditions.

How is a Generalized Green function different from a regular Green function?

A Generalized Green function takes into account non-homogeneous boundary conditions, while a regular Green function only considers homogeneous boundary conditions. This makes the Generalized Green function more versatile and applicable to a wider range of problems.

What is the importance of Generalized Green functions in physics?

Generalized Green functions are important in physics because they allow us to solve complex differential equations that arise in many physical systems. They are also useful in studying the behavior of waves, heat transfer, and other physical phenomena.

How are Generalized Green functions used in engineering?

In engineering, Generalized Green functions are used to model and analyze the behavior of structures and systems. They are particularly useful in solving problems related to heat transfer, fluid flow, and electrical circuits.

What are some applications of Generalized Green functions?

Generalized Green functions have a wide range of applications in various fields such as electromagnetics, acoustics, and quantum mechanics. They are also used in image processing, signal processing, and medical imaging.

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