a) A second-order Runge-Kutta method, also known has Heun’s method, is derived in Sect. 8.4.5. Make a function ode_Heun(f, U_0, dt, T) (as a…
1.In the SIRV model with time-dependent vaccination from Sect. 8.3.9, we want to test the effect of an adaptive vaccination campaign where vaccination is…
1.Consider the file osc_FE.py implementing the Forward Euler method for the oscillating system model (8.43)–(8.44). The osc_FE.py code is what we often refer to…
a) Make a function osc_energy(u, v, omega) for returning the potential and kinetic energy of an oscillating system described by (8.43)–(8.44). The potential energy…
1.We consider the ODE problem N (t) = rN(t), N(0) = N0. At some time, tn = nΔt, we can approximate the derivative N…
1.It is recommended to do Exercise 8.12 prior to the present one. Here we look at the same population growth model N (t) =…
1.A general ODE problem u (t) = f (u, t), u(0) = U0, may be solved numerically by the third order Runge-Kutta method. The…
1.Differing from the single-step methods presented in this chapter, we have the multistep methods, for example the Adams-Bashforth methods. With the single-step methods, un+1…
1.Consider (8.43)–(8.44) modeling an oscillating engineering system. This 2×2 ODE system can be solved by the Backward Euler scheme, which is based on discretizing…
1.Let y be a scalar function of time t and consider the nonlinear ODE y + y = ty3, t ∈ (0, 4), y(0)…
