function Name_XXXXXXX_hwN %insert the exact file name (as explained in the submission
                           % instructions) after "function"



%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS III - MATH 2Z03
%%%                             ASSIGNMENT #3
%%%             EULER's AND HEUN's METHODS
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 9.2.
% - "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright,
%    (Jones and Bartlett, 4th edition) Sections 1.2, 2.3, 6.1.
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Instructions:
% - Submit your assignment electronically (via Email) to the
%   address specific to your last name as indicated on the course
%   website; hardcopy submissions will not be accepted.
% - It is obligatory to use the current MATLAB template file available at
%   http://www.math.mcmaster.ca/gabardo/M2Z03/frames/template.m; 
%   submissions non compliant with this template will not be accepted.
% - Make sure to enter your name and student I.D. number in the
%   appropriate section of the template.
% - Late submissions and submissions which do not comply with
%   these guidelines will not be accepted.
% - All graphs should contain suitable titles and legends.
%------------------------------------------------------------------------
% Written by Vladislav Bukshtynov, 2009

clc; close all; clear all;

% Student information
  
disp('  ------------   Student Information  ------------  ')  

% Please enter your information here
Name = 'Johny';
Surname = 'Good';
ID = 123456789;

fprintf('  Student:  %s %s (ID: %d) \n', Name, Surname, ID);



disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #1  ----------  ')  
disp('  ')
% Insert here your solution to Question #1
% (if you define your own functions, they may appear 
%  at the end of the file)
a=6;b=7;c=8;d=9;
A=a+1;B=1+b/10;
h = 0.1; % time step
y0 = 1+c/10; % initial value of y(0)
n = 1/h; % number of steps
t1(1)=1;
yHeun1(1)=y0;
for k = 1 : n
  t1(k+1) = t1(k) + h;
p=yHeun1(k)+h*(A*t1(k)+B*yHeun1(k)-5);
yHeun1(k+1)=yHeun1(k)+h/2*[(A*t1(k)+B*yHeun1(k)-5)+(A*t1(k+1)+B*p-5)];
end
Answer1=yHeun1(6)
h = 0.05; % time step
y0 = 1+c/10; % initial value of y(0)
n = 1/h; % number of steps
t2(1)=1;
yHeun2(1)=y0;
ytrue(1)=y0;
for k = 1 : n
  t2(k+1) = t2(k) + h;
p=yHeun2(k)+h*(A*t2(k)+B*yHeun2(k)-5);
yHeun2(k+1)=yHeun2(k)+h/2*[(A*t2(k)+B*yHeun2(k)-5)+(A*t2(k+1)+B*p-5)];
ytrue(k+1)=(5-A*t2(k+1))/B-(A/B^2)+exp(B*(t2(k+1)-1))*(y0-(5-A)/B+A/B^2);
end
Answer2=yHeun2(11)
figure;
plot(t1, yHeun1, '*',t2, yHeun2, '*',t2,  ytrue)
legend('yHeun1', 'yHeun2','ytrue');
xlabel('x'); ylabel('y');
grid on; 
disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')

disp('  ----------   Solution to Question #2  ----------  ')  
disp('  ')
% Insert here your solution to Question #2
% (if you define your own functions, they may appear 
%  at the end of the file)
r=c+1;s=d+1;
h = 0.05; % time step
y0 = 1; % initial value of y(0)
n = 2/h; % number of steps
t3(1)=0;
yMod1(1)=y0;
for k = 1 : n
  t3(k+1) = t3(k) + h;
p = yMod1(k) + 0.5*h*(4*sin(8*t3(k)))/((t3(k))^r+(yMod1(k))^s);
yMod1(k+1)= yMod1(k)+h*(4*sin(8*(t3(k)+h/2)))/((t3(k)+h/2)^r+(p)^s);
end
Answer3=yMod1(41)

h = 0.01; % time step
y0 = 1; % initial value of y(0)
n = 2/h; % number of steps
t4(1)=0;
yMod2(1)=y0;
for k = 1 : n
  t4(k+1) = t4(k) + h;
p = yMod2(k) + 0.5*h*(4*sin(8*t4(k)))/((t4(k))^r+(yMod2(k))^s);
yMod2(k+1)=yMod2(k)+h*(4*sin(8*(t4(k)+h/2)))/((t4(k)+h/2)^r+(p)^s);
end
Answer4=yMod2(201)
figure;
plot(t3, yMod1,'*', t4, yMod2,'*')
legend('yMod1', 'yMod2');
xlabel('x'); ylabel('y');
grid on; 





disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')

disp('  ----------            Answers         ----------  ')  
disp('  ')
% DO NOT TOUCH THIS PART (RESERVED FOR THE INSTRUCTOR'S USE)

Name 
Surname 
ID

Answer1
Answer2
Answer3
Answer4