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 + Δt 2 , tn + Δt 2 ), k3 = […]
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 used to estimate un+1. How many time steps that 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 (u, t), where u and f are vectors, can use […]
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) you have to solve a nonlinear algebraic equation for each […]
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 these “believing” questions suggested by Elbow: What’s interesting or helpful […]
– What is the main stage of the product life cycle? Why? – Give your recommendations about different strategies they have to use to avoid the decline stage. The post Product life cycle first appeared on COMPLIANT PAPERS.
Illustrate the functionality of the following: 1. 8:1 Multiplexor 2. 2:4 Decoder The post Multiplexor first appeared on COMPLIANT PAPERS.
create an environmental journal of current information regarding environmental issues. All submissions must contain the following: 1. Environmental topic (see Course outline) ) 2. Date you collected the information (if the source is tv. or radio) The post Environmental journal on understanding our environment first appeared on COMPLIANT PAPERS.
1.Vectorize the implementation of the function for computing the area of a polygon in Exercise 5.6. Make a test function that compares the scalar implementation in Exercise 5.6 and the new vectorized implementation for the test cases used in Exercise 5.6. 2.One can observe (and also mathematically prove) that the solution u(x, t) of the problem in Exercise 9.6 is symmetric around x = 0: […]
a)The Crank-Nicolson method for ODEs is very popular when combined with diffusion equations. For a linear ODE u = au it reads un+1 − un Δt = 1 2 (aun + aun+1) . Apply the Crank-Nicolson method in time to the ODE system for a onedimensional diffusion equation. Identify the linear system to be solved. b) The Backward Euler, Forward Euler, and Crank-Nicolson methods can […]