Give a flannel board presentation. 1. Set up area with board. 2. Check pieces. 3. Practice. 4. Place pieces in order of appearance. 5. Gather children. 6. Place pieces out….
Find an expression for the Nn in terms of Nn−1 and formulate an algorithm for computing Nn, n = 1, 2,…,Nt .
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 expression for the Nn in terms of Nn−1 and formulate an algorithm for computing Nn, n = 1, 2,…,Nt . b) Implement the algorithm in a) in a function growth_BE(N_0, dt, T) for solving N
= rN, N(0) = N0, t ∈ (0, T ], with time step Δt (dt). c) Implement the Forward Euler scheme in a function growth_FE(N_0, dt, T) as described in b). d) Compare visually the solution produced by the Forward and Backward Euler schemes with the exact solution when r = 1 and T = 6. Make two plots, one with Δt = 0.5 and one with Δt = 0.05.