CourseWork Questions

This document contains questions and solutions for CourseWork Questions. It includes solutions for Question 1, Question 2, Question 12, and Question 14. The solutions cover topics such as plotting functions, gas laws, solving differential equations, and applying the shooting method.

CourseWork Questions............................................................................................................................1
Question 1...............................................................................................................................................1
Question 2...............................................................................................................................................3
Question 12.............................................................................................................................................4
Question 14.............................................................................................................................................6
CourseWork Questions
clear all
close all
clc
Question 1
% Question 1: Part A:1
x=0:0.001:3;
fx=(14*exp(x-2))-(12*exp(x-2))-7*x.^3+20*x.^2-26*x+12;
fx1=diff(fx);
plot(x,fx,'r-.','LineWidth',1.2)
grid on
xlabel('x')
ylabel('f(x)')
% Question 1: Part A:2
fc=@(x) ((14*exp(x-2))-(12*exp(x-2))-7*x.^3+20*x.^2-26*x+12);
xb = fzero(fc,15)
init=20;
tol=1e-2;
maxiter=1e4;
%[root, error_estimate]=newton('fx','fx1',200,tol,maxiter)
%Question 1: Part B
fb=(1-3./(4*x)).^(1/3);
y=abs(fb);
figure(2)
plot(x,y)
xlabel('x')
ylabel('f(x)')
grid on
y1=@(x) (1-3/(4*x))^(1/3);
%xc = fzero(y1,15)

xb =
9.8090

Question 2
clear
close all
% v - molar volumes vector
% fg - value of functions at v
%For oxygen
R=0.0820578; %molar gas constant
a=1.36; %L^2 atm/mole^2
b=0.003183; %L/mole
P=15; %pressure (atm)
T=320; %one mole of oxygen
v=linspace(0,50);
y=myVanderWaals(v,R,a,b,P,T);
figure(1)
plot(v,y,'r-.')
grid on
xlabel('Samples')
ylabel('Van der Waals')
title('Gas Laws')

legend('y')
clear
% For Benzene vapor
R=0.0820578; %molar gas constant
a=18.0; %L^2 atm/mole^2
b=0.1154; %L/mole
P=20; %pressure (atm)
T=700; %one mole of oxygen
v=linspace(0,50);
y=myVanderWaals(v,R,a,b,P,T);
figure(1)
plot(v,y,'b-.')
grid on
% v0=ginput(1);
xlabel('Samples')
ylabel('Van der Waals')
title('Gas Laws')
legend('y')

Question 12
clear
tspan = [0 0.15];
c = 83;
val_b = 0.14;
omega = 120*pi;
E=165;
N=600;
A=omega*E/N;
ode_fun = @(t,u,c,val_b,A,omega)[u(2);-u(1)-c*u(2)-val_b*u(1).^3+A*cos(omega*t)];
[t,u] = ode15s(@(t,u)ode_fun(t,u,c,val_b,A,omega),tspan,[0 0]);
figure(1);
plot(t,u);
xlabel('Time');
ylabel('State');
grid on
acc = -u(:,1)-c*u(:,2)-val_b*u(:,1).^3+A*sin(omega*t);
figure(2);
plot(t,acc);
xlabel('Time(sec)');
ylabel('Acceleration');
grid on

Question 14
clear
clc
close all
% applying the shooting method
%[y1, y2]=shooting_method(fun,0.001,1e-5,1,2,[0 0],'fd')
clf
alpha=1/2;
beta=log(2);
a=1;
b=2;
figure(1)
[xv,yv]=ode45('funsysa',[a b],[alpha;0;0;1]);
plot(xv,yv(:,1),'r-.')
hold on
n=length(yv(:,1));
y1n=yv(n,1);
y2n=yv(n,3);
yvsol=yv(:,1)+(beta-y1n)/y2n*yv(:,3);
truesol=-1./(xv.^2).*(2-4*xv+3/2*xv.^2-xv.^2.*log(xv));
plot(xv,yvsol,'b-')
title('Shooting Method: y"=-4/x*y-2/x^2*y+2*ln(x)/x^2, y(1)=1/2,y(2)=ln2')
text(1.2,0.4,'IVP: y_1"=-4/x*y_1-2/x^2*y_1+2*ln(x)/x^2, y_1(1)=1/2,y_1(1)=0')
text(1.2,0.6,'- y=y_1+(beta-y_1(2))/y_2(2)*y_2')
axis([1 2 0 0.8])
grid on
hold off
figure(2)
semilogy(xv,abs(yvsol-truesol),'b-.')
title('Shooting Method: y"=-4/x*y-2/x^2*y+2*ln(x)/x^2, y(1)=1/2,y(2)+ln2')
ylabel('|y(x_i)-y_i|')
xlabel('x')
grid on