From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
---
source/know/concept/fundamental-solution/index.md | 61 ++++++++++++-----------
1 file changed, 31 insertions(+), 30 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 e8ffda6..312cc2e 100644
--- a/source/know/concept/fundamental-solution/index.md
+++ b/source/know/concept/fundamental-solution/index.md
@@ -8,10 +8,10 @@ categories:
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 $x \in ]a, b[$:
+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 $$x \in ]a, b[$$:
$$\begin{aligned}
\boxed{
@@ -20,17 +20,17 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where $A$ is a constant, usually $1$.
+Where $$A$$ is a constant, usually $$1$$.
Fundamental solutions are often called **Green's functions**,
but are distinct from the (somewhat related)
[Green's functions](/know/concept/greens-functions/)
in many-body quantum theory.
-Note that the definition of $G(x, x')$ generalizes that of
+Note that the definition of $$G(x, x')$$ generalizes that of
the [impulse response](/know/concept/impulse-response/).
And likewise, due to the superposition principle,
-once $G$ is known, $\hat{L}$'s response $u(x)$ to
-*any* forcing function $f(x)$ can easily be found as follows:
+once $$G$$ is known, $$\hat{L}$$'s response $$u(x)$$ to
+*any* forcing function $$f(x)$$ can easily be found as follows:
$$\begin{aligned}
\hat{L} \{ u(x) \}
@@ -47,10 +47,10 @@ $$\begin{aligned}
-$\hat{L}$ only acts on $x$, so $x' \in ]a, b[$ is simply a parameter,
-meaning we are free to multiply the definition of $G$
-by the constant $f(x')$ on both sides,
-and exploit $\hat{L}$'s linearity:
+$$\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')
@@ -58,9 +58,9 @@ $$\begin{aligned}
= \hat{L}\{ f(x') \: G(x, x') \}
\end{aligned}$$
-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 $x$:
+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 $$x$$:
$$\begin{aligned}
A \int_a^b f(x') \: \delta(x - x') \dd{x'}
@@ -70,15 +70,15 @@ $$\begin{aligned}
&= \hat{L} \int_a^b f(x') \: G(x, x') \dd{x'}
\end{aligned}$$
-By definition, $\hat{L}$'s response $u(x)$ to $f(x)$
-satisfies $\hat{L}\{ u(x) \} = f(x)$, recognizable here.
+By definition, $$\hat{L}$$'s response $$u(x)$$ to $$f(x)$$
+satisfies $$\hat{L}\{ u(x) \} = f(x)$$, recognizable here.
While the impulse response is typically used for initial value problems,
-the fundamental solution $G$ is used for boundary value problems.
+the fundamental solution $$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(x)$$ or one of its derivatives is zero at the boundaries.
Then:
$$\begin{aligned}
@@ -95,20 +95,20 @@ $$\begin{aligned}
G_x(a, x') = 0
\end{aligned}$$
-This holds for all $x'$, and analogously for the other boundary $x = b$.
-In other words, the boundary conditions are built into $G$.
+This holds for all $$x'$$, and analogously for the other boundary $$x = b$$.
+In other words, the boundary conditions are built into $$G$$.
What if the boundary conditions are inhomogeneous?
-No problem: thanks to the linearity of $\hat{L}$,
-those conditions can be given to the homogeneous solution $u_h(x)$,
-where $\hat{L}\{ u_h(x) \} = 0$,
-such that the inhomogeneous solution $u_i(x) = u(x) - u_h(x)$
+No problem: thanks to the linearity of $$\hat{L}$$,
+those conditions can be given to the homogeneous solution $$u_h(x)$$,
+where $$\hat{L}\{ u_h(x) \} = 0$$,
+such that the inhomogeneous solution $$u_i(x) = u(x) - u_h(x)$$
has homogeneous boundaries again,
-so we can use $G$ as usual to find $u_i(x)$, and then just add $u_h(x)$.
+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
+If $$\hat{L}$$ is self-adjoint
(see e.g. [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)),
-then the fundamental solution $G(x, x')$
+then the fundamental solution $$G(x, x')$$
has the following **reciprocity** boundary condition:
$$\begin{aligned}
@@ -122,8 +122,8 @@ $$\begin{aligned}
-Consider two parameters $x_1'$ and $x_2'$.
-The self-adjointness of $\hat{L}$ means that:
+Consider two parameters $$x_1'$$ and $$x_2'$$.
+The self-adjointness of $$\hat{L}$$ means that:
$$\begin{aligned}
\int_a^b G^*(x, x_1') \Big( \hat{L} \{ G(x, x_2') \} \Big) \dd{x}
@@ -135,6 +135,7 @@ $$\begin{aligned}
G^*(x_2', x_1')
&= G(x_1', x_2')
\end{aligned}$$
+