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…

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 =…

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…

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 computational scheme reads un+1 = un + Δt 6 (k1 + 4k2 + k3) , n = 0, 1,…,Nt − 1, k1 = f (un, tn), k2 = f (un +…

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 is computed by use of the solution from the previous time step, i.e. un. In multi-step methods, the computed solutions from several previous time steps, e.g., un, un−1 and un−2 are…

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 derivatives by collecting information backward in time. More specifically, u (t) is approximated as u (t) ≈ u(t) − u(t − Δt) Δt . A general vector ODE u = f…

1.Let y be a scalar function of time t and consider the nonlinear ODE y + y = ty3, t ∈ (0, 4), y(0) = 1 2 . a) Assume you want to solve this ODE numerically by the Backward Euler method. Derive the computational scheme and show that (contrary to the Forward Euler scheme)…

Select one (1) of the approved topics from the www.procon.org Website and state your position on the issue. From the Procon.org Website, identify three (3) premises (reasons) listed under either the Pro or Con section — whichever section opposes your position. For the three (3) premises (reasons) that oppose your position on the issue, answer…

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