Green's Function

Let

$$\tilde L = \tilde D^n + a_{n-1}(t)\tilde D^{n-1} + \ldots + a_1(t)\tilde D + a_0(t)$$ (1)

be a differential Operator in 1-D, with $a_i(t)$ Continuous for $i=0$, 1, ..., $n-1$ on the interval $I$, and assume we wish to find the solution $y(t)$ to the equation

$$\tilde L y(t) = h(t),$$ (2)

where $h(t)$ is a given Continuous on $I$. To solve equation (2), we look for a function $g: C^n(I)\mapsto C(I)$ such that $\tilde L(g(h)) = h$, where

$$y(t)=g(h(t)).$$ (3)

This is a Convolution equation of the form

$$y=g*h,$$ (4)

so the solution is

$$y(t) = \int^t_{t_0} g(t-x)h(x)\,dx,$$ (5)

where the function $g(t)$ is called the Green's function for $\tilde L$ on $I$.

Now, note that if we take $h(t)=\delta(t)$, then

$$y(t)=\int_{t_0}^t g(t-x)\delta(x)\,dx = g(t),$$ (6)

so the Green's function can be defined by

$$\tilde Lg(t)=\delta(t).$$ (7)

However, the Green's function can be uniquely determined only if some initial or boundary conditions are given.

For an arbitrary linear differential operator $\tilde L$ in 3-D, the Green's function $G({\bf r},{\bf r}')$ is defined by analogy with the 1-D case by

$$\tilde LG({\bf r},{\bf r}') = \delta({\bf r}-{\bf r}').$$ (8)

The solution to $\tilde L\phi = f$ is then

$$\phi({\bf r}) = \int G({\bf r},{\bf r}')f({\bf r}')\,d^3{\bf r}'.$$ (9)

Explicit expressions for $G({\bf r},{\bf r}')$ can often be found in terms of a basis of given eigenfunctions $\phi_n({\bf r}_1)$ by expanding the Green's function

$$G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1)$$ (10)

and Delta Function,

$$\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty b_n\phi_n({\bf r}_1).$$ (11)

Multiplying both sides by $\phi_m({\bf r}_2)$ and integrating over ${\bf r}_1$ space,

$$\int\phi_m({\bf r}_2)\delta^3({\bf r}_1-{\bf r}_2)\,d^3{\bf ... ...\infty b_n\int\phi_m({\bf r}_2)\phi_n({\bf r}_1)\,d^3{\bf r}_1$$ (12)

$$\phi_m({\bf r}_2)=\sum_{n=0}^\infty b_n \delta_{nm} = b_m,$$ (13)

so

$$\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).$$ (14)

By plugging in the differential operator, solving for the $a_n$s, and substituting into $G$, the original nonhomogeneous equation then can be solved.

References

Arfken, G. ``Nonhomogeneous Equation--Green's Function,'' ``Green's Functions--One Dimension,'' and ``Green's Functions--Two and Three Dimensions.'' §8.7 and §16.5-16.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 480-491 and 897-924, 1985.