Green’s Function

First of all， Green’s function is used for solving ODEs with the form $Ly(x) = f(x)$ where y is the unknown variable, f is the applied force/source function, and L is a linear operator.

Linearity is the crux of the idea: if we can decompose the solution into some kind of element solutions, then by combining them together we can get the final solution. Dirac function $$\delta(x)$$ is the “strange” function we define for representing the elemental solutions, usually in the form of $$\delta(x-\xi)$$, where $$\xi$$ is the location of the source and x is the location of influence.

The solution $y(x) = \int f(\xi) G(x;\xi) d\xi$ only applies for homogenous initial/boundary conditions, i.e. each $$G(x;\xi)$$ must satisfy the same initial/boundary condition. This is the key reason why solving an ODE with Green’s function cannot be applied to general problems.

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