Seite - 285 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

8.6 Exercises 285 Filename:osc_FE_general.py. Exercise8.24:SolvingaNonlinearODEwithBackward Euler Lety bea scalar functionof time t andconsider thenonlinearODE y′+y= ty3, t∈ (0,4), y(0)= 1 2 . a) Assumeyouwant tosolvethisODEnumericallybytheBackwardEulermethod. Derive the computational scheme and show that (contrary to the Forward Euler scheme) you have to solve a nonlinear algebraic equation for each time step whenusing this scheme. b) Implement the scheme in a program that also solves the ODE by a Forward Eulermethod.WithBackwardEuler,useNewton’smethodtosolvethealgebraic equation.Asyour initial guess,youhaveonegoodalternative,whichone? Let your program plot the two numerical solutions together with the exact solution,which isknown(e.g., fromWolframAlpha) tobe y(t)= √ 2√ 7e2t+2t+1 . Filename:nonlinBE.py. Exercise8.25:DiscretizeanInitial Condition Assume that the initial condition on u′ is nonzero in the finite difference method fromSect. 8.4.12:u′(0)=V0. Derive thespecial formulaforu1 in thiscase. Filename:ic_with_V_0.pdf. Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,distributionandreproduction inanymediumor format, as longasyougiveappropriate credit to the original author(s) and thesource, provide a link to theCreative Commons licenceand indicate ifchanges were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons licence and your intended use is not permitted by statutory regulation orexceeds thepermitted use, you willneed toobtain permission directly from thecopyright holder.