%% Project C - solving an initial problem
% please write your "candidate number" here:
% please do NOT write your name or college in this file
clc
clear all
close all
%% 5.1 Matlab's symbolic toolbox
syms t y
f(t,y)  = sin((t^2)*y);
ob = matlabFunction(f,'vars',[t,y]);
t0 = [0:0.1:8];
Sol = ob(t0,1);
int = sum(Sol);
figure
plot(t0,Sol,'-bo')
xlabel('t','fontsize',15)
ylabel('y','fontsize',15)
title('Solving Function sin(t^2)*y Using symbolic toolbox','fontsize',15)
%% 5.2 trapezium rule
n = 81;
lowerLimit = 0;
upperLimit = 8;
% Discretization into equally spaced segments
x = linspace(lowerLimit,upperLimit,n+1);
% Calculate the function y
y = sin(x.^2);
% Trapezoidal formula
A = 0;
for i=2:n
A = A + y(i);
end
M = 2*A;
I = (upperLimit-lowerLimit)*(y(1)+M+y(end))/(2*n);
figure
plot(x,y,'-bo')
xlabel('t','fontsize',15)
ylabel('y','fontsize',15)
title('Solving Function sin(t^2)*y Using trapezium rule','fontsize',15)

%% 5.3
% Excercise_5.3 Solution
% ======================
tspan = [0 8];
y0 = 1;
[t,y] = ode23s(@(t,y) sin(t^2)*y, tspan, y0);
figure
plot(t,y,'-bo')
title('Solving Function sin(t^2)*y Using ODE23s','fontsize',15)
grid on
xlabel('t','fontsize',15)
ylabel('y','fontsize',15)
%% 5.4
n = 81;
t0 = 0;
T = 8;
y0 = 1;
% Calculate the function y
h = (T - t0) / n;
% [t, y] = Heun(f, t0, T, h, y0);
[t,y] = Heun(inline('sin(y^2*t)','t','y'),t0,T,h,y0);
figure
plot(t,y,'-bo')
title('Solving Function sin(t^2)*y Using Heun method','fontsize',15)
grid on
xlabel('t','fontsize',15)
ylabel('y','fontsize',15)