Consider the following non-linear ordinary differential equation (ODE) \'(t) _ a
ID: 2262595 • Letter: C
Question
Consider the following non-linear ordinary differential equation (ODE) '(t) _ a(t)(1-u(t)) = _ 101e-100t-c-200t du(t (0) = 1. Here, u,(t) = T). Take a domain of t-o. 1] with initial condition at t-0 as u Use forward Euler method and find the solution u(t) with 1001 discretization points (i.e. you divide the interval [0,1] into 1000 equal subintervals). Compare it with the true solution e-100r tin a plot. (10 points) ·Examine the stability of forward Euler method for the following number of discretiza- tion points l 1.51. 101.501 (i.e. you divide the interval 0.11 into 10. 50. 100, 500 equal subintervals). By comparing with true solution mentioned above, discuss whether it's stable or not. If 111 numerical(t = 1)-teract (t the solution is termed as unstable and use this as a criteria for detecting stability or instability in each case. (10 points) » Use backward Euler method to discretize the above equation. Then move all the terms to the left hand side so that we have the following equation for the nth time step: 7l 7t 100t Let us define 7n Tt 200t e z00r + 101e-1000 so that the equation for the nth time step now reads F(u") = 0Explanation / Answer
The forward euler table is as follows
x y=f(x)
0 0
0.02 0
0.04 0
0.06 0
0.08 0
0.1 0
0.12 0
0.14 0
0.16 0
0.18 0
0.2 0
0.22 0
0.24 0
0.26 0
0.28 0
0.3 0
0.32 0
0.34 0
0.36 0
0.38 0
0.4 0
0.42 0