diff options
Diffstat (limited to 'source/know/concept/fundamental-solution')
| -rw-r--r-- | source/know/concept/fundamental-solution/index.md | 22 |
1 files changed, 7 insertions, 15 deletions
diff --git a/source/know/concept/fundamental-solution/index.md b/source/know/concept/fundamental-solution/index.md index 312cc2e..947aada 100644 --- a/source/know/concept/fundamental-solution/index.md +++ b/source/know/concept/fundamental-solution/index.md @@ -42,11 +42,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-solution"/> -<label for="proof-solution">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-solution">Proof.</label> + +{% include proof/start.html id="proof-solution" -%} $$\hat{L}$$ 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, @@ -72,8 +69,8 @@ $$\begin{aligned} By definition, $$\hat{L}$$'s response $$u(x)$$ to $$f(x)$$ satisfies $$\hat{L}\{ u(x) \} = f(x)$$, recognizable here. -</div> -</div> +{% include proof/end.html id="proof-solution" %} + While the impulse response is typically used for initial value problems, the fundamental solution $$G$$ is used for boundary value problems. @@ -117,11 +114,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-reciprocity"/> -<label for="proof-reciprocity">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-reciprocity">Proof.</label> + +{% include proof/start.html id="proof-reciprocity" -%} Consider two parameters $$x_1'$$ and $$x_2'$$. The self-adjointness of $$\hat{L}$$ means that: @@ -135,9 +129,7 @@ $$\begin{aligned} G^*(x_2', x_1') &= G(x_1', x_2') \end{aligned}$$ - -</div> -</div> +{% include proof/end.html id="proof-reciprocity" %} |