euler.m

chapters/differential_equations/euler/code/euler.m

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+clc;clear;close all
+
+%==========================================================================
+% Define Function
+f =@(x) -3*x;
+
+% Define Initial and Final time
+tInit=0;
+tLast=5;
+
+% Define number of points
+N=1e2;
+
+% Set Initial Conditions
+yInit=1;
+%==========================================================================
+
+dt=(tLast-tInit)/(N-1); % Calculate dt
+t=[tInit:dt:tLast];     % Preallocate time array
+y=zeros(1,length(t));   % Preallocate solution array
+y(1)=yInit;             % Impose Initial Conditions
+
+% Loop over time
+for i=1:length(t)-1
+
+    t(i+1) = t(i) + dt;             % Calculate next time 
+    y(i+1) = y(i) + f( y(i) )*dt;   % Update solution
+
+end
+
+% Plot numerical solution
+plot(t,y)
+
+% Create analytical solution
+g=@(x) exp(-3*x);
+z=g(t);
+
+% Plot analytical solution on the same graph
+hold on
+plot(t,z,'--')
+
+% Set axis, title and legend
+xlabel('t');ylabel('y(t)');
+title('Analytical VS Numerical Solution')
+grid
+legend('Numerical','Analytical')