Question

SOLVE USING MATLAB

Problem 22.1A. Solve the following initial value problem over the interval fromt 0 to 5 where y(0) 8. Display all your results on the same graph. dt The analytical solution is given by: y(0) - 4e-0.5t (a) Using the analytical solution. (b) Using Eulers method with h 0.5 and 0.25 (c) Using the midpoint method with h 0.5. (d) Using the fourth-order RK method with h 0.5.

Answer 1

clear all;
y(1)=8;
% analytical y= 4+(y(0)-4)*e-0.5t
t=0.01:0.01:5;
y1=4+(y(1)-4)*exp(-0.5*t);
% Eulers method y_n+1=y_n+h*f(t,y) with h=.5
h1=0.5;
t2=0:h1:5;
for n=2:length(t2)
y2(1)=y(1);
y2(n)=y2(n-1)+h1*(2-((y2(n-1))/2));
end
% Eulers method y_n+1=y_n+h*f(t,y) with h=.25
h2=0.25;
t3=0:h2:5;
for n=2:length(t3)
y3(1)=y(1);
y3(n)=y3(n-1)+h2*(2-((y3(n-1))/2));
end
%mid point method
h3=.5;
t4=0:.5*h3:5;
for n=2:length(t4)
y4(1)=y(1);
y4(n)=y4(n-1)+.5*h3*(2-((y3(n-1))/2));
end
% fourth-order RK method with h=.5
h5=0.5;

t5=0:h5:5;
for n=2:length(t5)
y5(1)=y(1);
K1=2-((y5(n-1))/2);
K2=2-((y5(n-1)+(K1*h5/2))/2);
K3=2-((y5(n-1)+(K2*h5/2))/2);
K4=2-((y5(n-1)+(K3*h5))/2);
m=(K1/6)+(K2/3)+(K3/3)+(K4/6);
y5(n)=y5(n-1)+m*h5;
end
plot(t,y1,t2,y2,t3,y3,'*y',t4,y4,'r',t5,y5,'*');grid on;
legend('analytical','Euler,h=0.5','Euler,h=0.25','midpoint,h=0.5','4th order RK h=0.5')

_analytical Euler,h=0.5 Euler,h-0.25 midpoint, h:0.5 7.5 4th order RK h 0.5 6.5 5.5 5 0.5 1.5 2.5 3.5 4.5