Microfluidics: Modeling, Mechanics and Mathematics

27.3.4 Solving the System of Coupled ODEs Using the Forward Euler Method

As an illustration, we will solve the system of coupled ODEs given by Eq. 27.43 using the Euler method. For the sake of simplicity, we assume unitless numbers for the variables using

D = 1

k = 1

m = 5

S = 1

h = 0.5

As we can see from Fig. 27.12, this is an underdamped oscillation. We start with the initial values F^(…,0) at t = 0 which are given by

F^(s,0) = 1

F^(v,0) = 1

from which we can calculate the initial values X^(…,0) at t = 0 using Eq. 27.43a and Eq. 27.43b to find

$\begin{array}{lll}{X}^{\left(s,0\right)}\hfill & =v\left(0\right)\hfill & =F^{\left(v,0\right)}\hfill \\ X^{\left(v,0\right)}\hfill & =-\frac{1}{m}\left(k{F}^{\left(v,0\right)}+D{F}^{\left(s,0\right)}\right)=-\frac{1}{5}\left(0+1\right)\hfill & =-0.2\hfill \end{array}$

Calculating the First Step. We now apply the forward Euler method ...