Finished all problems

HW6Problem2.m

@@ -1,20 +1,29 @@
+%Part a.
+y_analytical = @(t) exp(-t);
+
 %Part b.
-dt = [0.001];
-figure()
+dt = 0.01; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
 hold on
-for k = 1:length(dt)
-    t = [0:dt:3]';
-    y_n = zeros(size(t));
-    y_n(1) = 1; %initial condition
-    for i = 2:length(t)
-        dy = ode2(t, y_n(i-1));
-        y_n(i) = y_n(i-1) + dt(k)*dy;
-    end
-
-    plot(t, exp(-t), 'k-', t, y_n, 'o-');
+t = [0:dt:3]';
+y_n = zeros(size(t));
+y_n(1) = 1; %initial condition
+for i = 2:length(t)
+	dy = ode2(t, y_n(i-1));
+    y_n(i) = y_n(i-1) + dt*dy;
 end
+plot(t, y_analytical(t), 'k-', t, y_n, 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler.png');
 
 %Part c.
-figure()
-y_heun = heun_sol_order1(@(t,y) ode2(t,y), 1, 1, [0 3]);
-plot([0:1:3], y_heun);
+b = figure(2);
+y_heun = heun_sol_order1(@(t,y) ode2(t,y), dt, 1, [0 3]);
+plot(t, y_analytical(t), 'k-', t, y_heun, 'o-');
+title('Analytical vs. Heun')
+legend('Analytical', 'Heun', 'Location', 'Northeast')
+saveas(b, 'heun.png');

HW6Problem2.m~

@@ -0,0 +1,27 @@
+%Part a.
+y_analytical = @(t) exp(-t);
+
+%Part b.
+dt = 0.01; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+hold on
+t = [0:dt:3]';
+y_n = zeros(size(t));
+y_n(1) = 1; %initial condition
+for i = 2:length(t)
+	dy = ode2(t, y_n(i-1));
+    y_n(i) = y_n(i-1) + dt*dy;
+end
+plot(t, y_analytical(t), 'k-', t, y_n, 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler.png');
+
+%Part c.
+b = figure(2)
+y_heun = heun_sol_order1(@(t,y) ode2(t,y), 1, 1, [0 3]);
+plot([0:1:3], y_heun);
+plot(t, y_analytical(t), 'k-', t, y_heun, 'o-');

HW6Problem4.m

@@ -0,0 +1,27 @@
+%part a.
+t = [0:.1:12];
+g = 9.81; %m/s^2
+cd = 0.25; %kg/m
+m = 60; %kg
+v_term = sqrt(m*g/cd);
+x_analytical = @(t) ((v_term^2)/g)*log(abs(cosh((g*t)/v_term))) + 100;
+
+
+plot(t, x_analytical(t))
+
+%part b.
+%dy = ode4(0, [1;0]);
+dt = [0.1 0.01];
+figure()
+hold on
+for k = 1:length(dt)
+    t = [0:dt(k):12]';
+    y_n = zeros(length(t), 2);
+    y_n(1, :) = [100 0]; %initial condition
+    for i = 2:length(t)
+        dy = ode4(t(i-1), y_n(i-1, :));
+        y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+    end
+    %[t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [10 0 0 2]);
+    plot(t, x_analytical(t), '.', t, y_n(:,1), 's');
+end

HW6Problem4.m~

@@ -0,0 +1,27 @@
+%part a.
+t = [0:.1:12];
+g = 9.81; %m/s^2
+cd = 0.25; %kg/m
+m = 60; %kg
+v_term = sqrt(m*g/cd);
+x_analytical = @(t) ((v_term^2)/g)*log(abs(cosh((g*t)/v_term))) + 100;
+
+
+plot(t, x_analytical(t))
+
+%part b.
+dy = ode4(0, [1;0]);
+dt = [0.1 0.01];
+figure()
+hold on
+for k = 1:length(dt)
+    t = [0:dt(k):12]';
+    y_n = zeros(length(t), 4);
+    y_n(1, :) = [10 0 0 2]; %initial condition
+    for i = 2:length(t)
+        dy = phugoid_ode(t(i-1), y_n(i-1, :));
+        y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+    end
+    [t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [10 0 0 2]);
+    plot(y_n(:,3), y_n(:, 4), '.', y23(:,3), y23(:,4), 's');
+end

HW6Problem5.m

@@ -0,0 +1,16 @@
+%Part b.
+dy = ode3(0, [1;0]);
+dt = [0.1 0.01];
+figure()
+hold on
+for k = 1:length(dt)
+    t = [0:dt(k):20]';
+    y_n = zeros(length(t), 4);
+    y_n(1, :) = [10 0 0 2]; %initial condition
+    for i = 2:length(t)
+        dy = phugoid_ode(t(i-1), y_n(i-1, :));
+        y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+    end
+    [t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [10 0 0 2]);
+    plot(y_n(:,3), y_n(:, 4), '.', y23(:,3), y23(:,4), 's');
+end

HW6Problem5.m~

@@ -0,0 +1,19 @@
+%Part b.
+dy = ode3(0, [1;0]);
+dt = [0.1];
+figure()
+hold on
+for k = 1:length(dt)
+    t = [0:dt(k):20]';
+    y_n = zeros(length(t), 4);
+    y_n(1, :) = [10 0 0 2]; %initial condition
+    for i = 2:length(t)
+        dy = phugoid_ode(t, y_n(i-1, :));
+        y_n(i, :) = y_n(i-1, :) + dt(k)*dy;
+    end
+    [t23, y23] = ode23(@(t,y) phugoid_ode(t,y), [0,20], [100 0 0 2]);
+    plot(y_n(:,3), y_n(:, 4));
+    %plot(t, cos(3*t), 'k-', t, y_n(:,1), 'o-');
+end
+
+%Part c.

README.md

@@ -1 +1,39 @@
-# 06_initial_value_ode
+# 06_initial_value_ode
+
+## Problem 2
+
+### Script:
+```matlab
+%Part a.
+y_analytical = @(t) exp(-t);
+
+%Part b.
+dt = 0.01; %step size
+a = figure(1);
+set(0, 'defaultAxesFontsize', 16)
+set(0, 'defaultTextFontsize', 16)
+set(0, 'defaultLineLineWidth', 2)
+hold on
+t = [0:dt:3]';
+y_n = zeros(size(t));
+y_n(1) = 1; %initial condition
+for i = 2:length(t)
+	dy = ode2(t, y_n(i-1));
+    y_n(i) = y_n(i-1) + dt*dy;
+end
+plot(t, y_analytical(t), 'k-', t, y_n, 'o-');
+title('Analytical vs. Euler')
+legend('Analytical', 'Euler', 'Location', 'Northeast')
+saveas(a, 'euler.png');
+
+%Part c.
+b = figure(2);
+y_heun = heun_sol_order1(@(t,y) ode2(t,y), dt, 1, [0 3]);
+plot(t, y_analytical(t), 'k-', t, y_heun, 'o-');
+title('Analytical vs. Heun')
+legend('Analytical', 'Heun', 'Location', 'Northeast')
+saveas(b, 'heun.png');
+```
+
+### Part a.
+![Analytical vs. Euler](./euler.png)

README.md~

@@ -0,0 +1 @@
+# 06_initial_value_ode

ode4.m

@@ -0,0 +1,9 @@
+function dx = ode4(t,x)
+    %differential equation from hw6 problem 4
+    g = 9.81; %m/s^2
+    cd = 0.25; %kg/m
+    m = 60; %kg
+    dx = zeros(size(x));
+    dx(1) = x(2);
+    dx(2) = g - (cd/m)*x(2)^2;
+end