Question

4. * Using your calculations from 3., plot the exact solution to dy = 1-y, dt y(0) = 1/2, for 0 <ts1, along with the numerical solution given by Euler's method and the trapezoid method, both with stepsize h = 0.1. Give the approximation of y(t = 1) for each numerical method. To distinguish your solutions: (i) Plot the Euler solution using crosses; do not join them with line segments. (ii) Plot the trapezoid solution using squares; again do not join with line segments. (iii) Plot the exact solution using a solid line with no markers. Enter help plot at the MATLAB prompt if you need help on how to plot with different markers.

Answer 1

Matlab code for (i)

clear t % Clears old time steps and
clear y % y values from previous runs
a=0;
b=1;
y0=0.5; % Initial value y(a)
h=0.1; % Time step
N=(b-a)/h;
t(1)=a;
y(1)=y0;
f=@(t,y)(1-y^2);
%while (norm(v,2))>10^-4 % For loop, sets next t,y values
for n=1:N
t(n+1)=t(n)+h;
y(n+1)=y(n)+h*f(t(n),y(n)); % Calls the function f(t,y)=dy/dt
end
y
plot(t(end),y(end),'*')

Output

y =

Columns 1 through 6

0.500000000000000 0.575000000000000 0.641937500000000 0.700729124609375 0.751626994001793 0.795132680190576

Columns 7 through 11

0.831909082279871 0.862701810161897 0.888276368836236 0.909372878092947 0.926676974951842

2 1.5 米 0.5 -0.5 0 0.2 0.4 0.6 0.8 11.214 1.6 1.8 2

Trapezoid rule

tspan =0:0.1:1;
y0 = 0.5;
[t,y] = ode23t(@(t,y) 1-y^2, tspan, y0)
plot(t(end),y(end),'square')

Output

t =

0
0.100000000000000
0.200000000000000
0.300000000000000
0.400000000000000
0.500000000000000
0.600000000000000
0.700000000000000
0.800000000000000
0.900000000000000
1.000000000000000

y =

0.500000000000000
0.570980566474321
0.634534816080768
0.690558579148676
0.739387374705229
0.781534279817847
0.817603838846551
0.848245447862247
0.874112554393215
0.895832147232664
0.913968745956985

2 1.5 0.5 -0.5 0 0.2 0.4 0.6 0.8 11.214 1.6 1.8 2

solation -Wj -) ..ヲ ヲヲヲラヲ ge

iii Matlab Code

t=0:0.1:1;
y=(3*exp(2*t)-1)./(3*exp(2*t)+1);
plot(t,y)

Output

0.95 n 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 0.9 1