Question

Solve using Matlab

Use the forward Euler method, Vi+,-Vi+(4+1-tinti ,Vi) for i= 0,1,2, , taking yo y(to) to be the initial condition, to approximate the solution at t-2 of the IVP y'=y-t2 + 1, 0-t-2, y(0) = 0.5. Use N = 2k, k = 1, 2, , 20 equispaced time steps (so to = 0 and tN-1 = 2). Make a convergence plot, computing the error by comparing with the exact solution, y: t1)2 -exp(t)/2, and plotting the error as a function of the step-size.

Answer 1

clc;clear all;
format long
t0=0;
tf=2;
f=@(t,y) y-t^2+1;
fact=@(t) (t+1)^2-exp(t)/2;
yact=fact(2);
k=1:1:20;
N=2.^k;
h=(tf-t0)./(N-1);
m=length(h);

for j=1:m
    y0=0.5;
    t=t0:h(j):tf;
    mm=length(t);
    for i=1:mm
         y(i)=y0+h(j)*f(t(i),y0);
         y0=y(i);
    end
   err(j)=abs(y0-yact);

end
plot(h,err)
set(gca, 'XDir','reverse')

xlabel('h')
ylabel('error')
axis([0 2 0 5])

Figure 1 File Edit View Insert Tools Desktop Window Help 4.5 3.5 2.5 0.5 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0