Homework 6 Solution

Description

Please submit to CANVAS a .zip le that includes the following Matlab functions:

tridiag solver.m

AB3.m

solve ODE system.m

Exercise 1 Write a Matlab function that implements the Thomas algorithm to solve tridiagonal linear systems of equations in the form

Heun scheme. The function should be of the form

function [y,t] = AB3(fun,y0,T,DT,IOSTEP)

Input:

fun: function handle representing f(y; t)

y0: column vector representing the initial condition

T: period of integration

DT: time step

IOSTEP: Input/output step. The solution is saved in the output matrix y every IOSTEP steps.

Output:

y: n S matrix collecting the time snapshots of the solution to (2). Note that the total number of snapshots S (including the initial condition) is floor(T/(IOSTEP*DT))+1.

t: vector collecting the time instants at which the solution is saved in the output matrix y.

Exercise 3 Consider the following nonlinear dynamical system

>

8	dy1(t)	= y1(t) + y2(t)y3(t)
	dt
>	dy2(t)	= y2(t) + (y3(t) 2)y1(t)	(3)
>
>	dt
>
>
<
>
>
>	dy3(t)	= 1 y1(t)y2(t)
	dt

>

:

It is known that the solution to (3) is chaotic in time and it settles on a strange attractor. By using the function AB3.m you coded in Exercise 2, compute the numerical solution to (3). To this end, set NSTEPS=1000000, IOSTEP=50, DT= 1e-3, y0=[1 2 3]^T , and write a function

function [y,t]=solve ODE system()

Output:

y: 3 S matrix collecting S time snapshots of the solution to (3). Here, S=floor(NSTEPS/IOSTEP)+1.

t: vector collecting the time instants at which the solution is saved in the output matrix y.

The function solve ODE system should also return the following items:

1. The plots of the trajectories y₁(t), y₂(t) and y₃(t) versus time in figure(1).

2. A three-dimensional plot of the curve (y₁(t); y₂(t); y₃(t)) in figure(2) (use the command plot3()).

2