- #1

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[itex]LG(x,s)=δ(x-s) [/itex]

the definition of a right inverse of a function f is:

[itex]h(y)=x,f(x)=y→f°h=y[/itex]

how does it add up?

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- Thread starter ENgez
- Start date

In summary, a green's function is a function that acts as the identity with respect to convolution. This means that when convolved with another function, it results in the original function. A right inverse of a function f is another function h that, when composed with f, gives the identity function. The green's function can be used to solve differential equations by convolving it with the desired function.

- #1

- 75

- 0

[itex]LG(x,s)=δ(x-s) [/itex]

the definition of a right inverse of a function f is:

[itex]h(y)=x,f(x)=y→f°h=y[/itex]

how does it add up?

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- #2

- 1,773

- 130

You want to say that the delta function is the identity with respect to the convolution.

In the discrete version of this, L would be a difference operator, which could be represented by a matrix. The delta function would be replace by Kronecker delta, which has the same components as the identity matrix.

- #3

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you mean ∫G(x,s)δ(x-s)ds=1, with integration over all domain of s?

- #4

- 1,773

- 130

you mean ∫G(x,s)δ(x-s)ds=1, with integration over all domain of s?

No. What I mean is that L * G (* means convolution) is equal to the identity with respect to convolution, which is the delta function. The delta function is the identity because convolving it with a function just gives you the function.

- #5

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Example: Solve y"+y=cos(x).

Solution: The operator here is L=D^2+1, where D=d/dx. The wikipedia entry on greens function tells you that the Green's function of D^2+1 is sin(x). So y=sin(x)*cos(x) would solve the above example.

- #6

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Here the convolution is the one used in Laplace transforms.

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