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Please fix my matlab code so that Improved Euler Method is correct % Define the parameters y 0 = 1 ; h _ values =
Please fix my matlab code so that Improved Euler Method is correct Define the parameters
y;
hvalues ;
tend ;
Initialize time vector
t :maxhvalues:tend;
Preallocate arrays
n lengtht;
yimprovedeuler zeroslengthhvalues n;
yrk zeroslengthhvalues n;
exactsolution zeros n;
Initial conditions
yimprovedeuler: y;
yrk: y;
exactsolution y;
Define the differential equation function
dydt @t y y y cosexpt;
Improved Euler's method for each time step
for j :lengthhvalues
h hvaluesj;
for i :n
k h dydtti yimprovedeulerj i;
k h dydtti h yimprovedeulerj i k;
yimprovedeulerj i yimprovedeulerj ik k;
end
end
RK method for each time step
for j :lengthhvalues
h hvaluesj;
for i :n
k h dydtti yrkj i;
k h dydtti h yrkj i k;
k h dydtti h yrkj i k;
k h dydtti h yrkj i k;
yrkj i yrkj ikkk k;
end
end
Exact solution using numerical integration
opts odesetRelTole 'AbsTol', e;
texact, yexact odedydt t y opts;
Plot Improved Euler's method approximations
figure;
hold on;
for j :lengthhvalues
plott yimprovedeulerj: 'LineWidth', ;
end
plottexact, yexact, k 'LineWidth', ;
titleImproved Euler's Method Approximation";
xlabelt;
ylabely;
legendh h h h 'Exact Solution';
grid on;
Plot RK method approximations
figure;
hold on;
for j :lengthhvalues
plott yrkj: 'LineWidth', ;
end
plottexact, yexact, k 'LineWidth', ;
titleRK Method Approximation";
xlabelt;
ylabely;
legendh h h h 'Exact Solution';
grid on;
Discussion of results
dispResults Discussion:";
dispImproved Euler's method generally improves accuracy with smaller h values due to its higherorder approximation.";
dispRK method consistently shows higher accuracy compared to Improved Euler's method across all tested h values.";
dispExact solution closely matches RK method, validating its accuracy.";
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