Page - 213 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.2 PopulationGrowth:AFirstOrderODE 213 Detour:Exactmathematical solution If you have taken a course on mathematical solution methods for differential equations, you may want to recap how an equation like N′ = rN orN′ = r(t)N is solved.Themethodofseparationofvariables is themostconvenient solutionstrategy in thiscase: N′ = rN dN dt = rN dN N = rdt ∫ N N0 dN N = ∫ t 0 rdt lnN− lnN0 = ∫ t 0 r(t)dt N =N0exp( ∫ t 0 r(t)dt), whichforconstantr results inN =N0ert.Note thatexp(t) is thesameaset. As will be described later, r must in more realistic models depend on N. The method of separation of variables then requires to integrate∫N N0 N/r(N)dN, which quickly becomes non-trivial for many choices of r(N). The onlygenerallyapplicable solution approach is thereforea numeri- calmethod. 8.2.2 NumericalSolution:TheForwardEuler(FE)Method There is a huge collection of numerical methods for problems like (8.1), and in general anyequationof the formu′ =f(u,t), whereu(t) is the unknownfunction intheproblem,andf is someknownformulaofuandoptionally t. Inourcasewith populationgrowth, i.e., (8.1),u′(t)correspondstoN′(t),whilef(u,t)corresponds to rN(t). We will first present a simple finite differencemethod solvingu′ =f(u,t). The idea is fourfold: 1. IntroduceNt+1points in time, t0,t1,.. .,tNt , for the relevant timeinterval.We seek the unknownu at these points in time, and introduceun as the numerical approximationtou(tn), see Fig.8.3. 2. Utilize that thedifferentialequation isvalidat themeshpoints. 3. Approximatederivativesbyfinite differences, see Fig.8.4. 4. Formulatea computationalalgorithmthatcan computea newvalueun basedon previouslycomputedvaluesui, i <n.