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, ..., an.

                                     

                                                                              

  

Most common transforms in electrochemistry



f(t)
   
F(s)
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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