Previous
Next

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:   \( \mathcal{L}\{f(t-a)H_s(t-a)\}=e^{-as}F(s), a>0 \)

 \( H_s(t-a)=\begin{cases}0,t<\\1, t>a\end{cases} \)  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)
----------------------------------------------------------------------------

-------------------------
\(1\)


\(\displaystyle \frac{1}{s} \)
\( t \)

   \(\displaystyle \frac{1}{s^2} \)
 \(\displaystyle \frac{t^{n-1}}{(n-1)!} \)

    \(\displaystyle \frac{1}{s^n} \)
\(\displaystyle\frac{1}{\sqrt{\pi t}}\)

   \( \displaystyle\frac{1}{\sqrt{s}} \)
\(\displaystyle2\sqrt{\frac{t}{\pi}}\)

   \(\displaystyle \frac{1}{s\sqrt{s}} \)
\(\displaystyle e^{-at} \)

   \(\displaystyle \frac{1}{s+a} \)
\( \sin(\omega t) \)

   \(\displaystyle \frac{\omega}{s^2+\omega^2} \)
\( \cos(\omega t) \)

   \(\displaystyle \frac{s}{s^2+\omega^2} \)
\( \displaystyle\frac{1}{\sqrt{\pi t}}-ae^{a^2t}\text{erfc}(a\sqrt{t}) \)

   \(\displaystyle \frac{1}{\sqrt{s}+a} \)
\( \displaystyle\frac{1}{\sqrt{\pi t}}+ae^{a^2t}\text{erf}(a\sqrt{t}) \)

   \(\displaystyle \frac{\sqrt{s}}{s-a^2} \)
\( \displaystyle e^{a^2t}\text{erf}({a\sqrt{t}}) \)

   \(\displaystyle \frac{a}{\sqrt{s}(s-a^2)} \)
\( \displaystyle1-e^{a^2t}\text{erfc}({a\sqrt{t}}) \)

   \(\displaystyle \frac{a}{s(\sqrt{s}+a)} \)
\( \displaystyle e^{a^2t}\text{erfc}({a\sqrt{t}}) \)

   \(\displaystyle \frac{1}{\sqrt{s}(\sqrt{s}+a)} \)
\( \displaystyle\frac{a}{2\sqrt{\pi t^3}}\exp\left(-\frac{a^2}{4t}\right) \)

   \(\displaystyle e^{-a\sqrt{s}},a \geq0 \)

\(\displaystyle \frac{1}{\sqrt{\pi t}} \text{exp}\left(-\frac{a^2}{4t}\right) \)

    \(\displaystyle \frac{1}{\sqrt{s}}e^{-a\sqrt{s}}, a \geq0 \)
\( \displaystyle\text{erfc}\left(\frac{a}{2\sqrt{t}}\right) \)

   \(\displaystyle \frac{1}{s}e^{-a\sqrt{s}}, a \geq0 \)
\(\displaystyle 2\sqrt{\frac{t}{\pi}}\exp\left(-\frac{a^2}{4t}\right)-a\text{ erfc}\left(\frac{a}{2\sqrt{t}}\right) \)

   \(\displaystyle \frac{1}{s\sqrt{s}}e^{-a\sqrt{s}}, a \geq0 \)
\(\displaystyle \frac{1}{\sqrt{\pi t}}\exp\left(-\frac{a^2}{4t}\right)-be^{ab+b^2t}\text{erfc}\left(\frac{a}{2\sqrt{t}}+b\sqrt{t}\right) \)
   \(\displaystyle \frac{e^{-a\sqrt{s}}}{\sqrt{s}+b}, a \geq0 \)

                             

 



Previous
Next