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README.md

@@ -16,9 +16,10 @@ for i=2:length(x)
     dy=prob2_ode(x,y_n(i-1));
     y_n(i)=y_n(i-1)+ dt(2)*dy;
 end  
-plot(x,exp(-x),'k-',x,y_n,'o');
+plot(x,exp(-x),x,y_n)
+legend('Analytical','Euler','Location','Northeast')
 ```
-![](assets/README-66741291.png)
+![](assets/README-be88b588.png)
 
 ## Part C
 
@@ -46,7 +47,7 @@ y(1)=y0;
     end
 end
 ```
-![](assets/README-23f54e97.png)
+![](assets/README-8b7eb1c9.png)
 
 # Problem 3
 ## Part A
@@ -64,6 +65,31 @@ for i=2:length(x)
         dy=prob3_ode(x,y_n(i-1,:));
         y_n(i,:)=y_n(i-1,:)+ dt(2)*dy;
 end
-plot(x,cos(3*x),'k-',x,y_n(:,1),'o');
+plot(x,cos(3*x),x,y_n(:,1));
 ```
-![](assets/README-4e4bb548.png)
+![](assets/README-bec1ab77.png)
+
+## Part C
+```
+function y = prob3ptC(prob3_ode, dt, y0,tlimits)
+    t = [tlimits(1):dt:tlimits(2)];
+    y = zeros(length(t),length(y0));
+    y(1,:) = y0;
+    for i = 2:length(t)
+        dy = prob3_ode(t(i-1),y(i-1,:));
+        y(i,:) = y(i-1,:)+dt*dy;
+        dyc = (dy + prob3_ode(t(i),y(i,:)))/2;
+        y(i,:) = y(i-1,:)+dt*dyc;
+        n = 1;
+        while (1)
+            n = n+1;
+            yold = y(i,:);
+            dyc = (dy+prob3_ode(t(i),y(i,:)))/2;
+            y(i,:) = y(i-1,:)+dt*dyc;
+            ea = abs(y(i,:)- yold)./y(i,:)*100;
+            if max(ea) < 0.0001 || n>100; break, end
+        end
+    end
+end
+```
+![](assets/README-fcbec9a1.png)

prob2plot.m

@@ -11,7 +11,7 @@ for i = 2:length(t)
 end
 a = figure(1);
 y_heun = Prob2ptC (@(t,y) prob2_ode(t,y), 0.1, 1, [0 3]);
-plot(t,exp(-t),'k-',t,y,'Xg',t,y_heun,'Or'); 
+plot(t,exp(-t),t,y,t,y_heun); 
 legend('Analytical','Euler','Heun','Location','Northeast')
 xlabel('Time')
 ylabel('Position')

prob2ptB.asv

@@ -0,0 +1,12 @@
+dt=[0.5 .1 .001];
+figure();
+hold on
+     x=[0:dt(2):3]';
+     y_n=zeros(size(x));
+    y_n(1)=1;
+for i=2:length(x)
+    dy=prob2_ode(x,y_n(i-1));
+    y_n(i)=y_n(i-1)+ dt(2)*dy;
+end  
+plot(x,exp(-x),x,y_n)
+legend('Analytical','Euler','Location','Northeast')

prob2ptB.m

@@ -8,4 +8,5 @@ for i=2:length(x)
     dy=prob2_ode(x,y_n(i-1));
     y_n(i)=y_n(i-1)+ dt(2)*dy;
 end  
-plot(x,exp(-x),'k-',x,y_n,'o');
+plot(x,exp(-x),x,y_n)
+legend('Analytical','Euler','Location','Northeast')

prob3plot.asv

@@ -0,0 +1,17 @@
+set(0, 'defaultAxesFontSize', 16)
+set(0,'defaultTextFontSize',14)
+set(0,'defaultLineLineWidth',1)
+dt = 0.01;
+t = [0:dt:3];
+y = zeros(length(t),2);
+y(1,:) = [1,0];
+for i = 2:length(t)
+    dy = prob3_ode(t,y(i-1,:));
+    y(i,:) = y(i-1,:)+dt*dy;
+end
+b = figure(2);
+y_heun = prob3ptC (@(t,y) prob3_ode(t,y),0.01,[1,0],[0 3]);
+plot(t,cos(3*t),t, y(:,1),t,y_heun(:,1))
+legend('Analytical','Euler','Heun','Location','Northeast')
+xlabel('Time')
+ylabel('Position')

prob3plot.m

@@ -0,0 +1,17 @@
+set(0, 'defaultAxesFontSize', 16)
+set(0,'defaultTextFontSize',14)
+set(0,'defaultLineLineWidth',1)
+dt = 0.01;
+t = [0:dt:3];
+y = zeros(length(t),2);
+y(1,:) = [1,0];
+for i = 2:length(t)
+    dy = prob3_ode(t,y(i-1,:));
+    y(i,:) = y(i-1,:)+dt*dy;
+end
+b = figure(2);
+y_heun = prob3ptC (@(t,y) prob3_ode(t,y),0.01,[1,0],[0 3]);
+plot(t,cos(3*t),t, y(:,1),t,y_heun(:,1))
+legend('Analytical','Euler','Heun','Location','Northeast')
+xlabel('Time')
+ylabel('Position')

prob3ptB.m

@@ -8,4 +8,5 @@ for i=2:length(x)
         dy=prob3_ode(x,y_n(i-1,:));
         y_n(i,:)=y_n(i-1,:)+ dt(2)*dy;
 end
-plot(x,cos(3*x),'k-',x,y_n(:,1),'o');
+plot(x,cos(3*x),x,y_n(:,1))
+legend('Analytical','Euler','Location','Northeast')

prob3ptC.m

@@ -0,0 +1,20 @@
+function y = prob3ptC(prob3_ode, dt, y0,tlimits)
+    t = [tlimits(1):dt:tlimits(2)];
+    y = zeros(length(t),length(y0));
+    y(1,:) = y0;
+    for i = 2:length(t)
+        dy = prob3_ode(t(i-1),y(i-1,:));
+        y(i,:) = y(i-1,:)+dt*dy;
+        dyc = (dy + prob3_ode(t(i),y(i,:)))/2;
+        y(i,:) = y(i-1,:)+dt*dyc;
+        n = 1;
+        while (1) 
+            n = n+1;
+            yold = y(i,:);
+            dyc = (dy+prob3_ode(t(i),y(i,:)))/2;
+            y(i,:) = y(i-1,:)+dt*dyc;
+            ea = abs(y(i,:)- yold)./y(i,:)*100; 
+            if max(ea) < 0.0001 || n>100; break, end
+        end
+    end
+end