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diff --git a/source/know/concept/fundamental-solution/index.md b/source/know/concept/fundamental-solution/index.md new file mode 100644 index 0000000..f5a51d5 --- /dev/null +++ b/source/know/concept/fundamental-solution/index.md @@ -0,0 +1,145 @@ +--- +title: "Fundamental solution" +date: 2021-11-02 +categories: +- Mathematics +- Physics +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 $x \in ]a, b[$: + +$$\begin{aligned} + \boxed{ + \hat{L}\{ G(x, x') \} + = A \delta(x - x') + } +\end{aligned}$$ + +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 +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: + +$$\begin{aligned} + \hat{L} \{ u(x) \} + = f(x) + \quad \implies \quad + \boxed{ + u(x) + = \frac{1}{A} \int_a^b f(x') \: G(x, x') \dd{x'} + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-solution"/> +<label for="proof-solution">Proof</label> +<div class="hidden"> +<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: + +$$\begin{aligned} + A f(x') \: \delta(x - x') + = f(x') \hat{L}\{ G(x, x') \} + = \hat{L}\{ f(x') \: G(x, x') \} +\end{aligned}$$ + +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 $x$: + +$$\begin{aligned} + A \int_a^b f(x') \: \delta(x - x') \dd{x'} + &= \int_a^b \hat{L}\{ f(x') \: G(x, x') \} \dd{x'} + \\ + A f(x) + &= \hat{L} \int_a^b f(x') \: G(x, x') \dd{x'} +\end{aligned}$$ + +By definition, $\hat{L}$'s response $u(x)$ to $f(x)$ +satisfies $\hat{L}\{ u(x) \} = f(x)$, recognizable here. +</div> +</div> + +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. +Then: + +$$\begin{aligned} + 0 + &= u(a) + = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'} + \qquad \implies \quad + G(a, x') = 0 + \\ + 0 + &= u_x(a) + = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'} + \quad \implies \quad + G_x(a, x') = 0 +\end{aligned}$$ + +This holds for all $x'$, and analogously for the other boundary $x = b$. +In other words, the boundary conditions are built into $G$. + +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 $u_i(x) = u(x) - u_h(x)$ +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/)), +then the fundamental solution $G(x, x')$ +has the following **reciprocity** boundary condition: + +$$\begin{aligned} + \boxed{ + G(x, x') = G^*(x', x) + } +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-reciprocity"/> +<label for="proof-reciprocity">Proof</label> +<div class="hidden"> +<label for="proof-reciprocity">Proof.</label> +Consider two parameters $x_1'$ and $x_2'$. +The self-adjointness of $\hat{L}$ means that: + +$$\begin{aligned} + \int_a^b G^*(x, x_1') \Big( \hat{L} \{ G(x, x_2') \} \Big) \dd{x} + &= \int_a^b \Big( \hat{L} \{ G(x, x_1') \} \Big)^* G(x, x_2') \dd{x} + \\ + \int_a^b G^*(x, x_1') \: \delta(x - x_2') \dd{x} + &= \int_a^b \delta^*(x - x_1') \: G(x, x_2') \dd{x} + \\ + G^*(x_2', x_1') + &= G(x_1', x_2') +\end{aligned}$$ +</div> +</div> + + + +## References +1. O. Bang, + *Applied mathematics for physicists: lecture notes*, 2019, + unpublished. |