Kirja

11. Appendices

11.4. Laplace transforms


Definition:			$\mathcal{L}\{f(t)\}=F(s)=\int_0^{\infty}{e^{-st}}f(t)dt$
Derivation:			$\mathcal{L}\{f'(t)\}=sF(s)-f(0)$
Integration			$\mathcal{L}\left\{\int_{0}^{t}{f(u)du}\right\}=\frac{1}{s}F(s)$ $\mathcal{L}\left\{ \int_{0}^{t}\int_{0}^{u}{f(v)dvdu} \right\}=\frac{1}{s^2}F(s)$
Linearity:			$\mathcal{L}\{Af(t)+Bg(t)\}=AF(s)+BG(s)$
Transfer in Laplace domain:			$\mathcal{L}\{e^{at}f(t)\}=F(s-a)$
Transfer in time domain:			$\mathcal{L}\{f(t-a)H_s(t-a)\}=e^{-as}F(s), a>0$ $H_s(t-a)=\begin{cases}0,ta\end{cases}$ Heaviside step function
Convolution:			$\mathcal{L}\left\{\int_{0}^{t}{f(t-u)g(u)du}\right\}=F(s)G(s)$
Heaviside theory:			$\mathcal{L}\left\{\sum\limits_{n=1}^m{\frac{p(a_n)}{q'(a_n)}e^{a_nt}}\right\}=\frac{P(s)}{Q(s)}$ , where roots of Q(s) are: a₁, a₂, ..., a_n.

Most common transforms in electrochemistry


f(t)			F(s)
----------------------------------------------------------------------------			-------------------------
$1$			$\displaystyle \frac{1}{s}$
$t$			$\displaystyle \frac{1}{s^2}$
$\displaystyle \frac{t^{n-1}}{(n-1)!}$			$\displaystyle \frac{1}{s^n}$
$\displaystyle\frac{1}{\sqrt{\pi t}}$			$\displaystyle\frac{1}{\sqrt{s}}$
$\displaystyle2\sqrt{\frac{t}{\pi}}$			$\displaystyle \frac{1}{s\sqrt{s}}$
$\displaystyle e^{-at}$			$\displaystyle \frac{1}{s+a}$
$\sin(\omega t)$			$\displaystyle \frac{\omega}{s^2+\omega^2}$
$\cos(\omega t)$			$\displaystyle \frac{s}{s^2+\omega^2}$
$\displaystyle\frac{1}{\sqrt{\pi t}}-ae^{a^2t}\text{erfc}(a\sqrt{t})$			$\displaystyle \frac{1}{\sqrt{s}+a}$
$\displaystyle\frac{1}{\sqrt{\pi t}}+ae^{a^2t}\text{erf}(a\sqrt{t})$			$\displaystyle \frac{\sqrt{s}}{s-a^2}$
$\displaystyle e^{a^2t}\text{erf}({a\sqrt{t}})$			$\displaystyle \frac{a}{\sqrt{s}(s-a^2)}$
$\displaystyle1-e^{a^2t}\text{erfc}({a\sqrt{t}})$			$\displaystyle \frac{a}{s(\sqrt{s}+a)}$
$\displaystyle e^{a^2t}\text{erfc}({a\sqrt{t}})$			$\displaystyle \frac{1}{\sqrt{s}(\sqrt{s}+a)}$
$\displaystyle\frac{a}{2\sqrt{\pi t^3}}\exp\left(-\frac{a^2}{4t}\right)$			$\displaystyle e^{-a\sqrt{s}},a \geq0$
$\displaystyle \frac{1}{\sqrt{\pi t}} \text{exp}\left(-\frac{a^2}{4t}\right)$			$\displaystyle \frac{1}{\sqrt{s}}e^{-a\sqrt{s}}, a \geq0$
$\displaystyle\text{erfc}\left(\frac{a}{2\sqrt{t}}\right)$			$\displaystyle \frac{1}{s}e^{-a\sqrt{s}}, a \geq0$
$\displaystyle 2\sqrt{\frac{t}{\pi}}\exp\left(-\frac{a^2}{4t}\right)-a\text{ erfc}\left(\frac{a}{2\sqrt{t}}\right)$			$\displaystyle \frac{1}{s\sqrt{s}}e^{-a\sqrt{s}}, a \geq0$
$\displaystyle \frac{1}{\sqrt{\pi t}}\exp\left(-\frac{a^2}{4t}\right)-be^{ab+b^2t}\text{erfc}\left(\frac{a}{2\sqrt{t}}+b\sqrt{t}\right)$			$\displaystyle \frac{e^{-a\sqrt{s}}}{\sqrt{s}+b}, a \geq0$