Page - 95 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4SolvingOrdinaryDifferentialEquations Differential equations constitute one of themost powerful mathematical tools to understand and predict the behavior of dynamical systems in nature, engineering, andsociety.Adynamicalsystemissomesystemwithsomestate,usuallyexpressed by a set of variables, that evolves in time. For example, an oscillating pendulum, thespreadingofadisease, and theweatherareexamplesofdynamical systems.We canusebasic lawsofphysics, or plain intuition, to expressmathematical rules that govern theevolutionof thesystemin time. These rules take the formofdifferential equations.Youareprobablywellexperiencedwithequations,at leastequationslike axCbD0orax2CbxCcD0. Suchequationsareknownasalgebraicequations, and theunknownisanumber. Theunknowninadifferentialequation isa function, andadifferentialequationwillalmostalways involvethis functionandoneormore derivatives of the function. For example, f 0.x/ D f.x/ is a simple differential equation (asking if there is any functionf such that it equals its derivative – you might remember thatex is acandidate). The present chapter starts with explaining how easy it is to solve both single (scalar) first-order ordinarydifferential equations and systemsoffirst-order differ- entialequationsbytheForwardEulermethod.Wedemonstrateall themathematical andprogrammingdetails through twospecificapplications: populationgrowthand spreadingofdiseases. Thenwe turn to a physical application: oscillatingmechanical systems, which arise in awide rangeof engineering situations. Thedifferential equation is nowof secondorder, and theForwardEulermethoddoesnot performwell. This observa- 95©TheAuthor(s)2016 S.Linge,H.P.Langtangen,Programming for Computations –Python, Texts inComputational Science andEngineering15,DOI10.1007/978-3-319-32428-9_4