From fda947364c33ea7f6273a7f3ad1e8898edbe1754 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 29 Sep 2024 22:15:59 +0200
Subject: Improve knowledge base

---
 source/know/concept/fundamental-solution/index.md | 34 +++++++++++++++--------
 1 file changed, 22 insertions(+), 12 deletions(-)

(limited to 'source/know/concept/fundamental-solution')

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:
 
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