Page - 209 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.1 FillingaWaterTank:TwoCases 209 Fig. 8.2 Water volume in a tank as it develops with constantly increasing rate of volume change (r=V).Thenumerical computationsuse piecewise constant ratesasanapproximation to the true rate The similar code structures used for Case 1 and Case 2 suggests that a more general (and improved)codemaybewritten, applicable tobothcases. Aswe move onwith ournextexample,populationgrowth,wewill see howthiscanbedone. 8.1.3 ReformulatingtheProblemsasODEs Typically, the problems we solved in Case 1 and Case 2, would rather have been presented in “morepropermathematical language”asODEs. Case 1 When the rates were piecewise constant, we could have been requested to solve V ′(t)=1Ls−1, 0s<t<1s, V ′(t)=3Ls−1, 1s≤ t <2s, V ′(t)=7Ls−1, 2s≤ t≤3s, with V(0)=1L, whereV(0) isknownasan initial condition.