List of Exercises
xxiii
Exercise 8.12: Use a Backward Euler Scheme for Population Growth . . . . . . . 276
Exercise 8.13: Use a Crank-Nicolson Scheme for Population Growth . . . . . . . 277
Exercise 8.14: Understand Finite Differences via Taylor Series . . . . . . . . . . . . 277
Exercise 8.15: The Leapfrog Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Exercise 8.16: The Runge-Kutta Third Order Method . . . . . . . . . . . . . . . . . . . . 280
Exercise 8.17: The Two-Step Adams-Bashforth Method . . . . . . . . . . . . . . . . . . 280
Exercise 8.18: The Three-Step Adams-Bashforth Method . . . . . . . . . . . . . . . . . 282
Exercise 8.19: Use a Backward Euler Scheme for Oscillations . . . . . . . . . . . . . 282
Exercise 8.20: Use Heun’s Method for the SIR Model . . . . . . . . . . . . . . . . . . . . 283
Exercise 8.21: Use Odespy to Solve a Simple ODE . . . . . . . . . . . . . . . . . . . . . . 283
Exercise 8.22: Set up a Backward Euler Scheme for Oscillations . . . . . . . . . . . 284
Exercise 8.23: Set up a Forward Euler Scheme for Nonlinear and Damped
Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Exercise 8.24: Solving a Nonlinear ODE with Backward Euler . . . . . . . . . . . . 285
Exercise 8.25: Discretize an Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Exercise 9.1: Simulate a Diffusion Equation by Hand . . . . . . . . . . . . . . . . . . . . 303
Exercise 9.2: Compute Temperature Variations in the Ground . . . . . . . . . . . . . . 303
Exercise 9.3: Compare Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Exercise 9.4: Explore Adaptive and Implicit Methods . . . . . . . . . . . . . . . . . . . . 305
Exercise 9.5: Investigate the θ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Exercise 9.6: Compute the Diffusion of a Gaussian Peak . . . . . . . . . . . . . . . . . . 306
Exercise 9.7: Vectorize a Function for Computing the Area of a Polygon . . . . 307
Exercise 9.8: Explore Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Exercise 9.9: Compute Solutions as t → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Exercise 9.10: Solve a Two-Point Boundary Value Problem . . . . . . . . . . . . . . . 309
xxiii
Exercise 8.12: Use a Backward Euler Scheme for Population Growth . . . . . . . 276
Exercise 8.13: Use a Crank-Nicolson Scheme for Population Growth . . . . . . . 277
Exercise 8.14: Understand Finite Differences via Taylor Series . . . . . . . . . . . . 277
Exercise 8.15: The Leapfrog Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Exercise 8.16: The Runge-Kutta Third Order Method . . . . . . . . . . . . . . . . . . . . 280
Exercise 8.17: The Two-Step Adams-Bashforth Method . . . . . . . . . . . . . . . . . . 280
Exercise 8.18: The Three-Step Adams-Bashforth Method . . . . . . . . . . . . . . . . . 282
Exercise 8.19: Use a Backward Euler Scheme for Oscillations . . . . . . . . . . . . . 282
Exercise 8.20: Use Heun’s Method for the SIR Model . . . . . . . . . . . . . . . . . . . . 283
Exercise 8.21: Use Odespy to Solve a Simple ODE . . . . . . . . . . . . . . . . . . . . . . 283
Exercise 8.22: Set up a Backward Euler Scheme for Oscillations . . . . . . . . . . . 284
Exercise 8.23: Set up a Forward Euler Scheme for Nonlinear and Damped
Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Exercise 8.24: Solving a Nonlinear ODE with Backward Euler . . . . . . . . . . . . 285
Exercise 8.25: Discretize an Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Exercise 9.1: Simulate a Diffusion Equation by Hand . . . . . . . . . . . . . . . . . . . . 303
Exercise 9.2: Compute Temperature Variations in the Ground . . . . . . . . . . . . . . 303
Exercise 9.3: Compare Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Exercise 9.4: Explore Adaptive and Implicit Methods . . . . . . . . . . . . . . . . . . . . 305
Exercise 9.5: Investigate the θ Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Exercise 9.6: Compute the Diffusion of a Gaussian Peak . . . . . . . . . . . . . . . . . . 306
Exercise 9.7: Vectorize a Function for Computing the Area of a Polygon . . . . 307
Exercise 9.8: Explore Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Exercise 9.9: Compute Solutions as t → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Exercise 9.10: Solve a Two-Point Boundary Value Problem . . . . . . . . . . . . . . . 309