Seite - XVI - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

xvi ListofExercises Exercise3.6: Explore roundingerrorswith largenumbers . . . . . . . . . . . . 89 Exercise3.7:Write test functions for R4 0 p xdx . . . . . . . . . . . . . . . . . . . 89 Exercise3.8: Rectanglemethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Exercise3.9:Adaptive integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Exercise3.10: Integratingx raised tox . . . . . . . . . . . . . . . . . . . . . . . . 90 Exercise3.11: Integrateproductsof sine functions . . . . . . . . . . . . . . . . . 91 Exercise3.12:Revisitfit of sines toa function . . . . . . . . . . . . . . . . . . . 91 Exercise3.13:Derive the trapezoidal rule foradouble integral . . . . . . . . . 92 Exercise3.14:Compute theareaofa trianglebyMonteCarlo integration . . . 92 Exercise4.1:Geometricconstructionof theForwardEulermethod . . . . . . . 153 Exercise4.2:Make test functions for theForwardEulermethod . . . . . . . . 153 Exercise4.3: ImplementandevaluateHeun’smethod . . . . . . . . . . . . . . . 153 Exercise4.4: Findanappropriate timestep; logisticmodel . . . . . . . . . . . . 154 Exercise4.5: Findanappropriate timestep;SIRmodel . . . . . . . . . . . . . . 154 Exercise4.6:Modelanadaptivevaccinationcampaign . . . . . . . . . . . . . . 154 Exercise4.7:MakeaSIRVmodelwith time-limitedeffectofvaccination . . . 154 Exercise4.8: Refactoraflat program . . . . . . . . . . . . . . . . . . . . . . . . . 155 Exercise4.9: SimulateoscillationsbyageneralODEsolver . . . . . . . . . . . 155 Exercise4.10:Compute theenergy inoscillations . . . . . . . . . . . . . . . . . 155 Exercise4.11:UseaBackwardEuler schemeforpopulationgrowth . . . . . . 156 Exercise4.12:UseaCrank-Nicolsonschemeforpopulationgrowth . . . . . . 156 Exercise4.13:UnderstandfinitedifferencesviaTaylor series . . . . . . . . . . 157 Exercise4.14:UseaBackwardEuler schemeforoscillations . . . . . . . . . . 158 Exercise4.15:UseHeun’smethodfor theSIRmodel . . . . . . . . . . . . . . . 159 Exercise4.16:UseOdespy to solveasimpleODE . . . . . . . . . . . . . . . . . 159 Exercise4.17: Set upaBackwardEuler schemeforoscillations . . . . . . . . . 159 Exercise 4.18: Set up a ForwardEuler scheme for nonlinear and damped oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Exercise4.19:Discretize an initial condition. . . . . . . . . . . . . . . . . . . . . 160 Exercise5.1: Simulateadiffusionequationbyhand . . . . . . . . . . . . . . . . 177 Exercise5.2: Compute temperaturevariations in theground . . . . . . . . . . . 178 Exercise5.3: Compare implicitmethods . . . . . . . . . . . . . . . . . . . . . . . 178 Exercise5.4: Exploreadaptiveand implicitmethods . . . . . . . . . . . . . . . . 179 Exercise5.5: Investigate the rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Exercise5.6: Compute thediffusionofaGaussianpeak . . . . . . . . . . . . . . 180 Exercise5.7:Vectorizea function forcomputing theareaof apolygon . . . . 181 Exercise5.8: Exploresymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Exercise5.9: Computesolutionsas t !1 . . . . . . . . . . . . . . . . . . . . . 182 Exercise5.10: Solvea two-pointboundaryvalueproblem . . . . . . . . . . . . 183 Exercise6.1:UnderstandwhyNewton’smethodcan fail . . . . . . . . . . . . . 206 Exercise6.2: See if the secantmethodfails . . . . . . . . . . . . . . . . . . . . . 207 Exercise6.3:Understandwhy thebisectionmethodcannot fail . . . . . . . . . 207 Exercise6.4: Combine thebisectionmethodwithNewton’smethod . . . . . . 207 Exercise6.5:Writea test functionforNewton’smethod . . . . . . . . . . . . . 207 Exercise6.6: Solvenonlinearequation foravibratingbeam . . . . . . . . . . . 207