06_initial_value_ode

#Problem 2 a) Analytically b) Eueler's Method

y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)-y(i)*h;
end

c)Heun's Approach

y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y_o=y(i)-y(i)*h;
    y(i+1)=y(i)+((-y(i)+y_o)/2)*h;
end

Time(s)	yTrue	yEuler	yHeuns
0	1	1	1
1	0.3679	0	0.5
2	0.1353	0	0.25
3	0.0498	0	0.125

#Problem 3 a) b)

y_i=1;%initial conditions
dy_i=0;
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)-3*(sin(3*(i-1)))*h;
end

c)

y_i=1;%initial conditions
n=3;%number of steps
h=1;%step size
y=zeros(1,n+1);
y(1)=y_i;
for i=1:n;
    y(i+1)=y(i)+((-3*sin(3*(i-1))+-3*sin(3*(i))))/2*h;
end

Time(s)	yTrue	yEuler	yHeuns
0	1	1	1
1	-0.99	1	0.7883
2	0.96	0.5766	0.9958
3	-0.91	1.4149	0.7967

#Problem 4 a)

g=9.81;
cd=.25;
m=60;
for t=1:13
    v(t)=sqrt((g*m)/cd)*tanh(sqrt((g*cd)/m)*(t-1));
    x(t)=1000-(m/cd)*log(cosh(sqrt((g*cd)/m)*(t-1)));
end

b)

g=9.81;
cd=.25;
m=60;
x_i=1000;
v_i=0;
timesteps=12;
h=1;
v=zeros(1,timesteps+1);
x=zeros(1,timesteps+1);
v(1)=v_i;
x(1)=x_i;
for i=1:timesteps
    v(i+1)=v(i)+(g-(cd/m)*v(i)^2)*h;
    x(i+1)=x(i)-v(i)*h;
end

c)

g=9.81;
cd=.25;
m=60;
x_i=1000;
v_i=0;
timesteps=12;
h=1;
v=zeros(1,timesteps+1);
x=zeros(1,timesteps+1);
v(1)=v_i;
x(1)=x_i;
for i=1:timesteps
    v_o=v(i)+(g-(cd/m)*v(i)^2)*h;
    v(i+1)=v(i)+(((g-(cd/m)*v(i)^2)+v_o)/2)*h;
    x_o=-v(i)*h;
    x(i+1)=x(i)+(-v(i)+x_o)/2*h;
end

Time (s)	yTrue (m)	yEuler (m)	yHeuns (m)
0	1000	1000	1000
1	995.13	1000	1000
2	980.8924	990.19	990.190
3	958.323	970.97	966.066
4	928.8272	943.48	922.495
5	893.898	909.33	855.238
6	854.897	870.23	763.391
7	812.945	827.69	650.960
8	768.911	782.88	525.173
9	723.4515	736.62	392.609
10	671.962	689.47	257.175
11	629.82	641.78	120.640
12	582.22	593.75	-16.298

Time Step	vTrue	vEuler	vHuens
0	0	0	0
1	9.68	9.81	9.81
2	18.61	19.21902	24.12402
3	26.28	27.48997	43.57116
4	32.45	34.15123	67.25655
5	37.17	39.10162	91.84714
6	40.64	42.54105	112.43114
7	43.11	44.81046	125.78687
8	44.84	46.25389	132.56390
9	46.07	47.14963	135.43424
10	46.84	47.69676	136.53455
11	47.59	48.02768	136.93815
12	47.77	48.22660	137.08365

#Problem 5

g=9.8;
v_t=5.5;
Cd=1/40;
Cl=1;
v_i=10;
theta_i=0;
y_i=2;
x_i=0;
t=20;
h1=0.1;
h2=0.01;
dt1=t/h1;
dt2=t/h2;
v1=zeros(1,dt1+1);
v1(1)=v_i;
theta1=zeros(1,dt1+1);
theta1(1)=theta_i;
y1=zeros(1,dt1+1);
y1(1)=y_i;
x1=zeros(1,dt1+1);
x1(1)=x_i;
v2=zeros(1,dt2+1);
v2(1)=v_i;
theta2=zeros(1,dt2+1);
theta2(1)=theta_i;
y2=zeros(1,dt2+1);
y2(1)=y_i;
x2=zeros(1,dt2+1);
x2(1)=x_i;
for i=1:dt1
    v1(i+1)=v1(i)+h1*(-g*sin(theta1(i))-(Cd/Cl)*(g/v_t^2)*(v1(i)^2));
    theta1(i+1)=theta1(i)+h1*(-(g/v1(i))*cos(theta1(i))+(g/v_t^2)*v1(i));
    x1(i+1)=x1(i)+h1*v1(i)*cos(theta1(i));
    y1(i+1)=y1(i)+h1*v1(i)*sin(theta1(i));
end
for i=1:dt2
    v2(i+1)=v2(i)+h2*(-g*sin(theta2(i))-(Cd/Cl)*(g/v_t^2)*(v2(i)^2));
    theta2(i+1)=theta2(i)+h2*(-(g/v2(i))*cos(theta2(i))+(g/v_t^2)*v2(i));
    x2(i+1)=x2(i)+h2*v2(i)*cos(theta2(i));
    y2(i+1)=y2(i)+h2*v2(i)*sin(theta2(i));
end
plot(x1,y1,'l',x2,y2)
xlabel('x (m)')
ylabel('y (m)')