Categories: Mathematics, Physics.

Fundamental solution

Given a linear operator L^\hat{L} acting on x∈[a,b]x \in [a, b], its fundamental solution G(x,x′)G(x, x') is defined as the response of L^\hat{L} to a Dirac delta function δ(x−x′)\delta(x - x') located at x′∈ ]a,b[x' \in \: ]a, b[:

L^{G(x,x′)}=Aδ(x−x′)\begin{aligned} \boxed{ \hat{L}\{ G(x, x') \} = A \delta(x - x') } \end{aligned}

Where AA is a constant, usually 11. Fundamental solutions are often called Green’s functions, but are distinct from the (somewhat related) Green’s functions in quantum mechanics.

Note that the definition of G(x,x′)G(x, x') generalizes that of the impulse response. And likewise, due to the superposition principle, once GG is known, L^\hat{L}’s response u(x)u(x) to any forcing function f(x)f(x) can easily be found as follows:

L^{u(x)}=f(x)  ⟹  u(x)=1A∫abf(x′) G(x,x′)dx′\begin{aligned} \hat{L} \{ u(x) \} = f(x) \quad \implies \quad \boxed{ u(x) = \frac{1}{A} \int_a^b f(x') \: G(x, x') \dd{x'} } \end{aligned}

L^\hat{L} only acts on xx, so x′∈ ]a,b[x' \in \: ]a, b[ is simply a parameter, meaning we are free to multiply the definition of GG by the constant f(x′)f(x') on both sides, and exploit L^\hat{L}’s linearity:

Af(x′) δ(x−x′)=f(x′) L^{G(x,x′)}=L^{f(x′) G(x,x′)}\begin{aligned} A f(x') \: \delta(x - x') = f(x') \: \hat{L}\{ G(x, x') \} = \hat{L}\{ f(x') \: G(x, x') \} \end{aligned}

We then integrate both sides over x′x' in the interval [a,b][a, b], allowing us to consume δ(x ⁣− ⁣x′)\delta(x \!-\! x'). Note that integration commutes with L^\hat{L}’s action:

A∫abf(x′) δ(x−x′)dx′=∫abL^{f(x′) G(x,x′)}dx′Af(x)=L^∫abf(x′) G(x,x′)dx′\begin{aligned} A \int_a^b f(x') \: \delta(x - x') \dd{x'} &= \int_a^b \hat{L}\{ f(x') \: G(x, x') \} \dd{x'} \\ A f(x) &= \hat{L} \int_a^b f(x') \: G(x, x') \dd{x'} \end{aligned}

By definition, L^\hat{L}’s response u(x)u(x) to f(x)f(x) satisfies L^{u(x)}=f(x)\hat{L}\{ u(x) \} = f(x), recognizable here.

In practice, GG usually only depends on the difference x−x′x - x', in which case the integral shown above becomes a convolution:

u(x)=1A∫abf(x′) G(x−x′)dx′=1A(f∗G)(x)\begin{aligned} u(x) = \frac{1}{A} \int_a^b f(x') \: G(x - x') \dd{x'} = \frac{1}{A} (f * G)(x) \end{aligned}

While the impulse response is typically used for initial value problems, the fundamental solution GG is used for boundary value problems. Suppose those boundary conditions are homogeneous, i.e. uu or its derivative u˙\dot{u} is zero at the boundaries. Then:

0=u(a)=1A∫abf(x′) G(a,x′)dx′  ⟹  G(a,x′)=00=u˙(a)=1A∫abf(x′) G˙(a,x′)dx′  ⟹  G˙(a,x′)=0\begin{aligned} 0 &= u(a) = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'} \quad \implies \quad G(a, x') = 0 \\ 0 &= \dot{u}(a) = \frac{1}{A} \int_a^b f(x') \: \dot{G}(a, x') \dd{x'} \quad \implies \quad \dot{G}(a, x') = 0 \end{aligned}

Where G˙\dot{G} is the derivative of GG with respect to its first argument. This holds for all x′x', and also at the other boundary x=bx = b. In other words, the boundary conditions are built into GG.

What if the boundary conditions are inhomogeneous? No problem: thanks to the linearity of L^\hat{L}, those conditions can be given to the homogeneous solution uh(x)u_h(x), where L^{uh(x)}=0\hat{L}\{ u_h(x) \} = 0, such that the inhomogeneous solution ui(x)=u(x)−uh(x)u_i(x) = u(x) - u_h(x) has homogeneous boundaries again, so we can use GG as usual to find ui(x)u_i(x), and then just add uh(x)u_h(x).

If L^\hat{L} is self-adjoint (see Sturm-Liouville theory), then the fundamental solution G(x,x′)G(x, x') has the following reciprocity boundary condition:

G(x,x′)=G∗(x′,x)\begin{aligned} \boxed{ G(x, x') = G^*(x', x) } \end{aligned}

Consider two parameters x1′x_1' and x2′x_2'. The self-adjointness of L^\hat{L} means that:

∫abG∗(x,x1′)(L^{G(x,x2′)})dx=∫ab(L^{G(x,x1′)})∗G(x,x2′)dx∫abG∗(x,x1′) δ(x−x2′)dx=∫abδ∗(x−x1′) G(x,x2′)dxG∗(x2′,x1′)=G(x1′,x2′)\begin{aligned} \int_a^b G^*(x, x_1') \Big( \hat{L} \{ G(x, x_2') \} \Big) \dd{x} &= \int_a^b \Big( \hat{L} \{ G(x, x_1') \} \Big)^* G(x, x_2') \dd{x} \\ \int_a^b G^*(x, x_1') \: \delta(x - x_2') \dd{x} &= \int_a^b \delta^*(x - x_1') \: G(x, x_2') \dd{x} \\ G^*(x_2', x_1') &= G(x_1', x_2') \end{aligned}

References

  1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.