Abstract
In Chap. 2, Sect. 2.1.1, we considered one method, variation of parameters (or variation of constants ), for solving the linear inhomogeneous differential equation. In the method considered here, rather than determining the solution to the differential equation with the inhomogeneous term defined at each point of the interval, we consider the equation in which the inhomogeneous term is the Dirac delta function, \(\delta (x-\xi )\), which is zero except at the point \(x=\xi \) within the interval. Its solution is called the Green’s function \(G(x,\xi )\). In considering the Green’s function for difference equations we follow closely the analysis presented for differential equations.
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Notes
- 1.
See [36, Sect. 1.17(i)].
- 2.
Named for the British mathematician George Green.
- 3.
Note that L is a function of x and hence can be taken outside of the integral over \(\xi \).
- 4.
To illustrate the similarity of the analyses, we give in italics the equation number in the corresponding derivation for differential equations.
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© 2016 Springer International Publishing Switzerland
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Maximon, L.C. (2016). Green’s Function. In: Differential and Difference Equations. Springer, Cham. https://doi.org/10.1007/978-3-319-29736-1_6
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DOI: https://doi.org/10.1007/978-3-319-29736-1_6
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