What're the condition for a green function to exist?

[Karlisbad]

What're the condition for a "green function to exist?

That's my question,let's suppose i define the functions:

$$G(x,s)=exp(x-s)^{2}$$ and $$R(x,s)=(e^{st}-1)^{-1}$$

My question is, could G and R satisfy the condition (for a linear operator L)

$$LG(x,s)=\delta (x-s)$$ ?.

My interest lies on converting Integral equation with Symmetric Kernel:

$$\int_{a}^{b} K(x,s)f(x)=g(s)+f(s)$$ Into ODE's ...in order to solve

them.

 

[gtoguy97]

A Green's Function is a function that satisfies the conditions LG(x,s) = δ(x - s), where L is a linear differential operator and δ is the Dirac delta function. This is also known as the homogeneous boundary condition.

 

[tacman]

The conditions for a Green function to exist depend on the specific problem and the linear operator involved. In general, a Green function must satisfy the following conditions:

1. The Green function must be a solution to the homogeneous version of the differential equation associated with the linear operator.

2. The Green function must satisfy the boundary conditions of the problem.

3. The Green function must be continuous at all points except for the point where it is evaluated.

4. The Green function must have a finite limit as it approaches the point where it is evaluated.

For your specific functions, G(x,s) and R(x,s), it is possible for them to satisfy the condition for a Green function to exist, depending on the specific linear operator L and the boundary conditions of the problem. However, it is not possible to determine this without more information about the problem. Additionally, the conversion of an integral equation with a symmetric kernel into ODEs is not always possible and may require further assumptions or conditions.