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Diffstat (limited to 'source/know/concept/fundamental-solution/index.md')
| -rw-r--r-- | source/know/concept/fundamental-solution/index.md | 34 |
1 files changed, 22 insertions, 12 deletions
diff --git a/source/know/concept/fundamental-solution/index.md b/source/know/concept/fundamental-solution/index.md index 947aada..4728c6f 100644 --- a/source/know/concept/fundamental-solution/index.md +++ b/source/know/concept/fundamental-solution/index.md @@ -11,7 +11,7 @@ layout: "concept" Given a linear operator $$\hat{L}$$ acting on $$x \in [a, b]$$, its **fundamental solution** $$G(x, x')$$ is defined as the response of $$\hat{L}$$ to a [Dirac delta function](/know/concept/dirac-delta-function/) -$$\delta(x - x')$$ for $$x \in ]a, b[$$: +$$\delta(x - x')$$ located at $$x' \in \: ]a, b[$$: $$\begin{aligned} \boxed{ @@ -24,7 +24,7 @@ Where $$A$$ is a constant, usually $$1$$. Fundamental solutions are often called **Green's functions**, but are distinct from the (somewhat related) [Green's functions](/know/concept/greens-functions/) -in many-body quantum theory. +in quantum mechanics. Note that the definition of $$G(x, x')$$ generalizes that of the [impulse response](/know/concept/impulse-response/). @@ -44,20 +44,20 @@ $$\begin{aligned} {% include proof/start.html id="proof-solution" -%} -$$\hat{L}$$ only acts on $$x$$, so $$x' \in ]a, b[$$ is simply a parameter, +$$\hat{L}$$ only acts on $$x$$, so $$x' \in \: ]a, b[$$ is simply a parameter, meaning we are free to multiply the definition of $$G$$ by the constant $$f(x')$$ on both sides, and exploit $$\hat{L}$$'s linearity: $$\begin{aligned} A f(x') \: \delta(x - x') - = f(x') \hat{L}\{ G(x, x') \} + = f(x') \: \hat{L}\{ G(x, x') \} = \hat{L}\{ f(x') \: G(x, x') \} \end{aligned}$$ We then integrate both sides over $$x'$$ in the interval $$[a, b]$$, allowing us to consume $$\delta(x \!-\! x')$$. -Note that $$\int \dd{x'}$$ commutes with $$\hat{L}$$ acting on $$x$$: +Note that integration commutes with $$\hat{L}$$'s action: $$\begin{aligned} A \int_a^b f(x') \: \delta(x - x') \dd{x'} @@ -72,27 +72,37 @@ satisfies $$\hat{L}\{ u(x) \} = f(x)$$, recognizable here. {% include proof/end.html id="proof-solution" %} +In practice, $$G$$ usually only depends on the difference $$x - x'$$, +in which case the integral shown above becomes a convolution: + +$$\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}$$ + While the impulse response is typically used for initial value problems, the fundamental solution $$G$$ is used for boundary value problems. Suppose those boundary conditions are homogeneous, -i.e. $$u(x)$$ or one of its derivatives is zero at the boundaries. +i.e. $$u$$ or its derivative $$\dot{u}$$ is zero at the boundaries. Then: $$\begin{aligned} 0 &= u(a) = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'} - \qquad \implies \quad + \quad \implies \quad G(a, x') = 0 \\ 0 - &= u_x(a) - = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'} + &= \dot{u}(a) + = \frac{1}{A} \int_a^b f(x') \: \dot{G}(a, x') \dd{x'} \quad \implies \quad - G_x(a, x') = 0 + \dot{G}(a, x') = 0 \end{aligned}$$ -This holds for all $$x'$$, and analogously for the other boundary $$x = b$$. +Where $$\dot{G}$$ is the derivative of $$G$$ with respect to its first argument. +This holds for all $$x'$$, and also at the other boundary $$x = b$$. In other words, the boundary conditions are built into $$G$$. What if the boundary conditions are inhomogeneous? @@ -104,7 +114,7 @@ has homogeneous boundaries again, so we can use $$G$$ as usual to find $$u_i(x)$$, and then just add $$u_h(x)$$. If $$\hat{L}$$ is self-adjoint -(see e.g. [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)), +(see [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)), then the fundamental solution $$G(x, x')$$ has the following **reciprocity** boundary condition: |