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Write a program that solves this equation by Newton’s method. Use x = 1 as your starting value.

1.A nonlinear algebraic equation f (x) = 0 may be solved in many different ways, and we have met some of these in this chapter. Another, very useful, solution approach is to first re-write the equation into x = φ(x) (this re-write is not unique), and then formulate the iteration xn+1 = φ(xn), n = … Read more

compute the frequencies of a vibrating steel beam with a rectangular cross section having width b = 25 mm and height h = 8 mm.

1.An important engineering problem that arises in a lot of applications is the vibrations of a clamped beam where the other end is free. This problem can be analyzed analytically, but the calculations boil down to solving the following nonlinear algebraic equation: cosh β cos β = −1, where β is related to important beam … Read more

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 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 … Read more

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 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 … Read more

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 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 … Read more

Calculate u1, u2, and u3. Check that the numbers are the same as obtained in a)-c).

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 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 … Read more

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.

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 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 … Read more

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 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 … Read more