diff options
Diffstat (limited to 'source/know/concept/fundamental-solution')
| -rw-r--r-- | source/know/concept/fundamental-solution/index.md | 61 |
1 files changed, 31 insertions, 30 deletions
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 @@ $$\begin{aligned} <label for="proof-solution">Proof</label> <div class="hidden" markdown="1"> <label for="proof-solution">Proof.</label> -$\hat{L}$ only acts on $x$, so $x' \in ]a, b[$ is simply a parameter, -meaning we are free to multiply the definition of $G$ -by the constant $f(x')$ on both sides, -and exploit $\hat{L}$'s linearity: +$$\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)$$, recognizable here. </div> </div> While the impulse response is typically used for initial value problems, -the fundamental solution $G$ is used for boundary value problems. +the fundamental solution $$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} <label for="proof-reciprocity">Proof</label> <div class="hidden" markdown="1"> <label for="proof-reciprocity">Proof.</label> -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}$$ + </div> </div> |