Page - 106 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

106 4 SolvingOrdinaryDifferentialEquations from numpy import linspace, zeros, exp import matplotlib.pyplot as plt def ode_FE(f, U_0, dt, T): N_t = int(round(float(T)/dt)) u = zeros(N_t+1) t = linspace(0, N_t*dt, len(u)) u[0] = U_0 for n in range(N_t): u[n+1] = u[n] + dt*f(u[n], t[n]) return u, t def demo_population_growth(): """Test case: u’=r*u, u(0)=100.""" def f(u, t): return 0.1*u u, t = ode_FE(f=f, U_0=100, dt=0.5, T=20) plt.plot(t, u, t, 100*exp(0.1*t)) plt.show() if __name__ == ’__main__’: demo_population_growth() Thisprogramfile, calledode_FE.py, is a reusablepieceof codewith ageneral ode_FE function that can solve anydifferential equationu0 Df.u;t/ and ademo function for the special caseu0 D 0:1u,u.0/ D 100. Observe that the call to the demo function is placed in a test block. This implies that the call is not active if ode_FE is importedasamodule inanotherprogram,butactive ifode_FE.py is run asaprogram. Thesolutionshouldbeidentical towhatthegrowth1.pyprogramproduceswith the sameparameter settings (r D0:1,N0 D100). This featurecaneasilybe tested byinsertingaprintstatement,butamuchbetter,automatedverificationissuggested inExercise4.1.Youare stronglyencouraged to takea“break”anddo that exercise now. Remarkontheuseofuasvariable In theode_FEprogram, the variableu is used in different contexts. Inside the ode_FE function,u is anarray,but in thef(u,t) function,asexemplified in the demo_population_growth function, the argument u is a number. Typically, we callf (inode_FE)with theu argument as one element of the arrayu in the ode_FE function:u[n]. 4.1.6 MakingthePopulationGrowthModelMoreRealistic Exponential growthofapopulationaccording themodelN 0 D rN ,with exponen- tial solutionN DN0ert, isunrealistic in the longrunbecause theresourcesneeded to feed thepopulationarefinite. At somepoint therewill not beenough resources and the growthwill decline. A commonmodel taking this effect into account as-