Why the green function is useful?

[wdlang]

as a student in physics, i cannot see the usefulness of green function

to me, the definition of a green function is ugly and singular

we have to deal with functions that are not smooth, e.g., the derivative is not continuous at some point.

How these functions can be useful in math and physics?

 

[elibj123]

The Green Function contains all the information you need to know about a system. If a differential equation is a step forward from the definition of the system, and pure mathematical description- then the Green Function is even a step farther, and it contains as much (usually) information you need to know about the system and its behaviour, and it is also a great tool to achieve a solution given an input- if not an analytical closed-form solution, then at least an integral form which can be approximated.

As to singularities, many times in system analysis, we speak of responses to singular signals which seem more natural to us. But what's better about a step function rather than an impulse function, when one is simply the derivative of the other?

 

[wdlang]

The Green Function contains all the information you need to know about a system. If a differential equation is a step forward from the definition of the system, and pure mathematical description- then the Green Function is even a step farther, and it contains as much (usually) information you need to know about the system and its behaviour, and it is also a great tool to achieve a solution given an input- if not an analytical closed-form solution, then at least an integral form which can be approximated.

As to singularities, many times in system analysis, we speak of responses to singular signals which seem more natural to us. But what's better about a step function rather than an impulse function, when one is simply the derivative of the other?

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if not an analytical closed-form solution, then at least an integral form which can be approximated.
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the problem is that, we also get an integral equation

the solution is not given explicitly but it depends on itself!

it seems that we can gain nothing by expressing something in terms of itself.