Question

4. please help with both parts a and b

4. Consider the pendulum with friction modeled by the second order ODE: where θ is the angle the pendulum makes with the vertical axis, α is a friction coefficient and w is the pendulum natural frequency. (a) Turn (4) into a first order system. (b) Use Euler method to find an approximation to the solution in [0,5] with initial conditions θ(0)-1 and θ'(0)-0. Take α-0.2 and w-2. Verify the expected order of convergence and plot the approximation for a sufficiently small k

Answer 1

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%%Matlab code for Euler's forward clear all close all %all parameter value w-2; alpha 0.2; function for Euler equation solution fl-d (thl,th2,t) th2; f2- (th1,th2, t -alpha th2-w 2*thl; all step size 加0.001; %Initial conditions th 10-1 ; %theta ( 0 ) 1-0 th 20=0 ; %dtheta/dt | t-o-o . 1 initial t t0-0; %t end values tend-5; tn-t0:h:tend; Euler steps th1_result (1)-th10; th2_result (1) h20 tresult ( 1 ) 0: - %loop for Euler step for i= 1:leng th ( tn)-1 for finding the solution t_result(i+1) t_result (i)+h; th1 result(i +1 )-thi-result (i) +hs double ( f1 ( th 1-result ( İ ) ,th 2-result ( İ ) , tre sult ( İ ) ) ) ; th 2 result (i +1 )th 2-result (i)th* double ( f2 (th 1-result ( í), th2-result ( İ ) , t-result ( İ ) ) ) ; Plotting numerical solution figure(1) plot(t_result,thlresult, 'linewidth',2) xlabel(t' ylabel ('theta(t) title( 'theta(t) vs. t plot') figure (2) plot (t_result,th2_result, 'linewidth',2) xlabelt) ylabel"dtheta (t)/dt' title( 'dtheta(t)/dt vs. t plot') 888888888888888888 End of Code 옹8888888888888888888

thetalt) vs. t plot 0.8 0.6 0.4 0.2 0.2 -0.4 -0.6 0.8 dtheta(t)/dt vs. t plot 1.5 0.5 も 0.5

%%Matlab code for Euler's forward
clear all
close all
%all parameter value
w=2; alpha=0.2;

%function for Euler equation solution
f1=@(th1,th2,t) th2;
f2=@(th1,th2,t) -alpha*th2-w^2*th1;

%all step size
h=0.001;
%Initial conditions
th10=1;   %theta(0)|=0
th20=0;   %dtheta/dt|t=0=0.1
%initial t
t0=0;
%t end values
tend=5;
tn=t0:h:tend;

% Euler steps
th1_result(1)=th10;
th2_result(1)=th20;
t_result(1)=t0;

%loop for Euler step for finding the solution
for i=1:length(tn)-1
  
    t_result(i+1)= t_result(i)+h;
    th1_result(i+1)=th1_result(i)+h*double(f1(th1_result(i),th2_result(i),t_result(i)));
    th2_result(i+1)=th2_result(i)+h*double(f2(th1_result(i),th2_result(i),t_result(i)));
  
end

%Plotting numerical solution
figure(1)
plot(t_result,th1_result,'linewidth',2)
xlabel('t')
ylabel('theta(t)')
title('theta(t) vs. t plot')

figure(2)
plot(t_result,th2_result,'linewidth',2)
xlabel('t')
ylabel('dtheta(t)/dt')
title('dtheta(t)/dt vs. t plot')

%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%