Uncategorized
Calculate u1, u2, and u3.
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…
Category: <span>Uncategorized</span>
Uncategorized
compare the three values u1, u2, and u3 with the values obtained in Exercise 8.2.
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…
Uncategorized
find through experimentation the largest value of Δt where the exact solution and the numerical solution by Heun’s method cannot be distinguished visually.
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…
Uncategorized
Calculate u1, u2, and u3. Check that the numbers are the same as obtained in a)-c).
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…
Uncategorized
Make a function test_ode_FE_1()that calls ode_FE to compute three time steps in the problem u = u, u(0) = 1, and compare the three values u1, u2, and u3 with the values obtained in Exercise 8.2.
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…
Uncategorized
For the case in b), find through experimentation the largest value of Δt where the exact solution and the numerical solution by Heun’s method cannot be distinguished visually.
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…
Uncategorized
Write up the complete model, implement it, and rerun the case from Sect. 8.3.8 with various choices of parameters to illustrate various effects.
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…
Uncategorized
Consider the file osc_FE.py implementing the Forward Euler method for the oscillating system model (8.43)–(8.44).
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…
Uncategorized
Add a call to osc_energy in the programs osc_FE.py and osc_EC.py and plot the sum of the kinetic and potential energy.
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…
Uncategorized
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…