Question

How to do question B 2,3,4,5?

3. a) Find the solution v ote ordinary diferetinl equation with the initial coditions: b) i) Recast our third ord ODE into a system af first order ODEs af the formA.v, where v' = dv/dz f(v) and v = (y, y,y")". You should show all working to find the corresponding matrix A. Do not solve the system. 4 mark Solve it only by hand and show your complete work. Do not use a calculator or any symbolic caleu lations). 8 marks i) Show your full work to find the initial condition v(0) to “mplete the initial value problem for the system of first order ODEs. 2 marks ii) Now that you find a first order system of ODEs, you can solve it using the Huerns method ac- cording to the material you covered in computational module 2. A template of the code which will help you to get started is also available. Attach the complete code as a pdf page to your assignment. 14ark assignment. [1 mark iv) Explain the inluence of the initial conditions on the behaviour of yt). Hint: You can change some of initial conditions in the code to find out how this is relevant for the behaviour of the solution. [1 mark dt 0.01. %dt is the numericaL separation in time tmax- 20; syou can play around with this end time t -10 : dt: tmax]; %t is a vector containing the t values at which we wiu find y(t) N- length(t); %the number of specific times we want to find the solution at %We first have to recast the problem into set of three first order ODEs To do %this wel set v = [x1,x2,x3] where x1 is y, x2 is dy/dt and x3 is d^2y/dx^2. You did th v = zero s (N,3); %this line makes a vector for all the v vectors we want to tind at eve v(1..) -% ??? Here we store x1(0) and x2(0) and x3(0) as the initial vector-that A = % ??? You might want to store the matrix A from the question to make the proceedi | for i = 2:N %we start our calculation, not at the first point which we just prescribe %you may need to consult your module 2 reading material to remind yourself of how to k1s ??? Note that this will be a vector V1-44% ??? (this is the Euler approximation to the value of V(1)1 k2 = 48% ??? This will also be a vector end plot(t,y)

Answer 1

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SMatlab code for Heun's method clear all close all dt-0.01i tmax-20; t-[0:dt:tmax ] N-length(t) v zeros (N, 3) y( 1.. )一0 5-39] ; A [0 1 0:0 0 1-124-31; for i-2:N K1-dt*A* (v(i-1,:))'; end yay ( : , 1) ; plot (t,y, 'Linewidth',2) Exact solution yy-l(tt) -3*exp(-3*tt)+3*cos (2*tt)-2*sin (2*tt) hold on plot(t,yy(t), '-') legend(Heun Method", 'Exact solution') title( 'Solution of y(x) using Heun method) xlabel ( ·x' ) ylabel (y(x) grid on 868888888888888888 End of Code 웅8888888888888888

Solution of y(x) using Heun method Heun Method 一一Exact solution -3 0 24 6 8 10 1 14 16 820 Published with MATLAB R2018a

%%Matlab code for Heun's method
clear all
close all

dt=0.01;
tmax=20;
t=[0:dt:tmax];
N=length(t);

v=zeros(N,3);
v(1,:)=[0 5 -39];

A=[0 1 0;0 0 1;-12 -4 -3];

for i=2:N
  
    K1=dt*A*(v(i-1,:))';
    K2=dt*A*((v(i-1,:)+K1'))';
  
    v(i,:)=v(i-1,:)+(K1'+K2')/2;
  
end

y=v(:,1);

plot(t,y,'Linewidth',2)

%Exact solution
yy=@(tt) -3*exp(-3*tt)+3*cos(2*tt)-2*sin(2*tt);

hold on
plot(t,yy(t),'--')
legend('Heun Method','Exact solution')
title('Solution of y(x) using Heun method')
xlabel('x')
ylabel('y(x)')

grid on

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