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1.The purpose of this exercise is to make a file test_ode_FE.py that makes use of the ode_FE function in the file ode_FE.py and automatically verifies the implementation of ode_FE. a) The solution computed by hand in Exercise 8.2 can be used as a reference solution. Make a function test_ode_FE_1()that calls ode_FE to compute three time steps in the problem u = u, u(0) = 1, […]

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 counterpart to ode_FE(f, U_0, dt, T) in ode_FE.py) for solving a scalar ODE problem u = f (u, t), u(0) = U0, t ∈ (0, T ], with this method using a time step size Δt. b) Solve the simple ODE […]

1.Section 8.2.4 describes a geometric interpretation of the Forward Euler method. This exercise will demonstrate the geometric construction of the solution in detail. Consider the differential equation u = u with u(0) = 1. We use time steps Δt = 1. a) Start at t = 0 and draw a straight line with slope u (0) = u(0) = 1. Go one time step forward […]

1.The purpose of this exercise is to make a file test_ode_FE.py that makes use of the ode_FE function in the file ode_FE.py and automatically verifies the implementation of ode_FE. a) The solution computed by hand in Exercise 8.2 can be used as a reference solution. b) The test in a) can be made more general using the fact that if f is linear in u and […]

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 counterpart to ode_FE(f, U_0, dt, T) in ode_FE.py) for solving a scalar ODE problem u = f (u, t), u(0) = U0, t ∈ (0, T ], with this method using a time step size Δt. b) Solve the simple ODE […]

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 offered 2.We consider the SIRV model from Sect. 8.3.8, but now the effect of vaccination is time-limited. After a characteristic period of time, π, the vaccination is no more effective and individuals are consequently moved from the V to the S […]

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 as a flat program, meaning that it is just one main program with no functions. Your task is to refactor the code in osc_FE.py according to the specifications below. Refactoring, means to alter the inner structure of the code, while, 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 is taken as 1 2ω2u2 while the kinetic energy is 1 2 v2. (Note that these expressions are not exactly the physical potential and kinetic energy, since these would be 1 2mv2 and 1 2 ku2 for a model mx + […]

1.We consider the ODE problem N (t) = rN(t), N(0) = N0. At some time, tn = nΔt, we can approximate the derivative N (tn) by a backward difference, see Fig. 8.22: N (tn) ≈ N(tn) − N(tn − Δt) Δt = Nn − Nn−1 Δt , which leads to Nn − Nn−1 Δt = rNn , called the Backward Euler scheme. a) Find an […]

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) = rN(t), N(0) = N0. The time derivative N (t) can be approximated by various types of finite differences. Exercise 8.12 considers a backward difference (Fig. 8.22), while Sect. 8.2.2 explained the forward difference (Fig. 8.4). A centered difference is more accurate […]