I am trying to replicate a solution that has been given (via semi analytical methods) in literature, for the following first order differential equation:
$\frac{dS(t)}{dt}$ + $\frac{S(t)}{\tau(t)}$ = $2G\frac{de}{dt}$.$~~~~~~~~~~~~~$ (1)
The authors in https://doi.org/10.1002/pen.10256 construct an incremental solution by considering a discrete number of time-steps $t_n \in [0,T]$. Where the time is updated via:
$t_{n+1} = t_{n} + \Delta t. $$~~~~~~~~~~~~~$ (2)
The authors specify that both $\tau(t)$ and $\frac{de}{dt} = \frac{\Delta e}{\Delta t}$ are constant over each time step $t_{n} \rightarrow t_{n+1}$. And then through applying analytical integration to (1) the authors obtain that the increment $\Delta S$ over the time step is given by:
$\Delta S = S(t_{n+1}) - S(t_{n}) = \bigg(2\tau G \frac{\Delta e}{\Delta t} - S(t_n) \bigg)\bigg[1 - \exp\bigg(\frac{-\Delta t}{\tau}\bigg) \bigg] $.$~~~~~~~~~~~~~$ (3)
But I just cannot reproduce this solution, my attempt was:
(1) write the differential equation (1) in the integration factor form:
$\frac{d}{dt}\big(e^\frac{t}{\tau} S(t) \big) = 2G e^\frac{t}{\tau} \frac{de}{dt}$,$~~~~~~~~~~~~~$ (4)
which upon integrating, the solution at $n+1$ is given by:
$S(t_{n+1}) = e^{-\frac{t_{n+1}}{\tau}} \int_0^{t_{n+1}}{2Ge^{\frac{t_{n+1}}{\tau} } \frac{de}{dt} dt}$.$~~~~~~~~~~~~~$ (5)
Using the property of the exponential function $e^{t_{n+1}/\tau} = e^{t_{n}/\tau}e^{\Delta t/\tau}$, which inserted into (5), gives:
$S(t_{n+1}) = e^{-\frac{t_{n}}{\tau}} e^{-\frac{\Delta t}{\tau}}\int_0^{t_{n+1}}{2G e^{\frac{t_{n}}{\tau}} e^{\frac{\Delta t}{\tau}}\frac{de}{dt} dt}$,$~~~~~~~~~~~~~$ (6)
and then I split the integral such that:
$S(t_{n+1}) = e^{-\frac{t_{n}}{\tau}} e^{-\frac{\Delta t}{\tau}}\bigg (\int_0^{t_{n}}{2G e^{\frac{t_{n}}{\tau}} e^{\frac{\Delta t}{\tau}}\frac{de}{dt} dt} + \int_{t_n}^{t_{n+1}}{2G e^{\frac{t_{n}}{\tau}} e^{\frac{\Delta t}{\tau}}\frac{de}{dt} dt}\bigg ) $,$~~~~~~~~~~~~~$ (7)
But I feel this is getting no where, and through various manipulations I cannot reach the form (3). So I just want to know if this is along the right lines?
Any help would be much appreciated!
Regards,
Stephen