Forward Euler for ODE

We solve the ordinary differential equation

$\frac{\mathrm{d}y}{\mathrm{d}t} = -ky(t)$

with initial conditions $y(0)=C$ , $C=10$ , on the interval $(0, T]$ , $T=5$ , and concentration $k=1$ using forward Euler with timestep size $\frac{T}{N}$ , $N=10$ .

We note, that this equation has the known analytical solution

$y(t) = C \mathrm{e}^{-kt},$

so we can compare to the numerical result.

N = 10;  % interval [0,T] discretized with N+1 points
C = 10;  % value C (0)
T = 5;   % maximum time
k = 1;   % concentration
h = T/N; % the step size h, we consider equi-distance division

nC = C*ones(N+1,1);  % numerical solution
aC = C*ones(N+1,1);  % analytical solution

for i=1:N
    nC(i+1) = (1-k*h)*nC(i);
    aC(i+1) = C * exp(-k*h*i);
end

x = 0:h:T;          % x-axis values

plot(x,aC,'r-o','LineWidth',2); hold on;
plot(x,nC,'b-o','LineWidth',2); hold on;
legend('analytical solution', 'numerical solution');