Question

where is says use euler2, for that please create a function file for euler method and use that! please help out with this! please! screenahot the outputs and code! thanks!!!

The van der Pol equation is a 2nd-order ODE that describes self-sustaining oscillations in which energy is withdrawn from large oscilations and fed into the small oscillations. This equation typically models electronic circuits containing vacuum tubes. The van der Pol equation is: dy dt where y represents the position coordinate, t is time, and μ is a damping coefficient. The 2*-order ODE can be solved as a set of 1st-order ODEs, as shown below. Here, z is a 'dummy variable. dy dt dt The two 1 order ODEs above are coupled and must be solved simultaneously, ie, the solutions are dependent on each other and therefore cannot be solved individually. The first iteration will solve for yi and z1 using initial conditions yo and z. The second iteration will solve for y2 and z using y and z, and so forth. You are strongly encouraged to complete a few iterations by hand to fully understand the process. Q3a In the euler2,m file, complete the function file to perform Euller's method to solve two 1st-order ODE equations "You should still have three figure windows by the end of this task Q3b In the 03b m file, consider initial conditions y (O) 1 and z(0)-1, and 1. Solve the van der Pol equation using the euler2) function written in Q2a with time steps of 0.25, 0.125 and 0.0625 from t0 to 30s. In figure(4), plot the following in 2-by-1 subplot arrangement using the colours variable provided in the m- file to represent time steps of 0.25, 0.125 and 0.0625, respectively. [Top panel] y against t solved using Euler's method for each time step (Bottom panl against t solved using Euler's method for each time step It is suspected that the solutions obtained are not highly accurate. Use fprintfi) to print a statement describing how the accuracy of the solutions can be improved. "You should have four figure windows by the end of this task ENG1060 Assignment Page 8 of a3c in thea3c.mfile, use the euler20 function to solve the van der Pol equation for μ using a time step of 0.01. Use the same initial conditions and time span as in Q3b. 0.01,0. 1,0.5, 1, 2, 3.5 In figure(S)/. prodp f az against y for each value of p as solid lines. Use the RGB values defined in the 'colourmap' variable as provided in the m-file to color solutions from the smallest μ to the largest μ (top row to the bottom row). The axis has been set to square. Position the legend in the northwest corner

Answer 1

File 'euler2.m':

function [y, z] = euler2(dydt, dzdt, ti, tf, h, y0, z0)
n = (tf-ti)/h;
t = ti:h:tf;
y(1) = y0; z(1) = z0;
for i=1:n
m = dydt(t(i), y(i), z(i));
n = dzdt(t(i), y(i), z(i));
  
y(i+1) = y(i) + h*m;
z(i+1) = z(i) + h*n;
end
end

File 'Q3b.m':

close all
clear
clc

mu = 1;
dydt = @(t,y,z) z;
dzdt = @(t,y,z) mu*(1 - y^2)*z - y;
H = [0.25 0.125 0.0625];
ti = 0; tf = 30; y0 = 1; z0 = 1;
figure(4)
for i = 1:length(H)
h = H(i);
[y, z] = euler2(dydt, dzdt, ti, tf, h, y0, z0);
subplot(211), plot(ti:h:tf, y), hold on
subplot(212), plot(ti:h:tf, z), hold on
end
subplot(211), xlabel('t'), ylabel('y')
legend('h = 0.25', 'h = 0.125', 'h = 0.0625', 'Location', 'northwest'), hold off
subplot(212), xlabel('t'), ylabel('dy/dt')
legend('h = 0.25', 'h = 0.125', 'h = 0.0625', 'Location', 'northwest'), hold off

Output Figure (4):

3 h 0.25 h0.125 h 0.0625 0 10 15 25 30 20 h 025 h 0.125 h 0.0625 -1 -3 10 15 20 25 30

File 'Q3c.m':

close all
clear
clc

MU = [0.01 0.1 0.5 1 2 3 5];
h = 0.01;
ti = 0; tf = 30; y0 = 1; z0 = 1;
figure(5)
colourmap = [1 0 0; 0 1 0; 0 0 1; 1 1 0; 1 0 1; 0 1 1; 0 0 0];
for i = 1:length(MU)
mu = MU(i);
dydt = @(t,y,z) z;
dzdt = @(t,y,z) mu*(1 - y^2)*z - y;
[y, z] = euler2(dydt, dzdt, ti, tf, h, y0, z0);
plot(z, y, 'Color', colourmap(i,:)), hold on
end
xlabel('y'), ylabel('dy/dt')
legend('\mu = 0.01', '\mu = 0.1', '\mu = 0.5', '\mu = 1', '\mu = 2', '\mu = 3', '\mu = 5', 'Location', 'northwest')
axis('square')
hold off

Output Figure (5):

8 6 4 0 015 0001235 -6 -8 2 0 pAp