Question

Numerical Methods

Consider the following IVP dy=0.01(70-y)(50-y), with y(0)-0 (a) [10 marks Use the Runge-Kutta method of order four to obtain an approximate solution to the ODE at the points t-0.5 and t1 with a step sizeh 0.5. b) [8 marks Find the exact solution analytically. (c) 7 marks] Use MATLAB to plot the graph of the true and approximate solutions in one figure over the interval [.201. Display graphically the true errors after each steps of calculations.

Answer 1

ка 이 = clt C40-j)(so-נג o.cs i dly - cl doo dd = 014

So-J 5 (So-3) 5 Is e-t 子0- fes 120-29 - 2 0 tS t(s Is

(c)

%%%% Matlab code %%

clc;
close all;
clear all;
format long

f=@(t,y) 0.01*(70-y)*(50-y);
y(1)=0;
h=0.5;
t=0:h:20;
y_e(1)=0;

for k=1:length(t)-1
k1=h*feval(f,t(k),y(k));
k2=h*feval(f,t(k)+h/2,y(k)+k1/2);
k3=h*feval(f,t(k)+h/2,y(k)+k2/2);
k4=h*feval(f,t(k)+h,y(k)+k3);
y(k+1)=y(k)+(k1+2*k2+2*k3+k4)/6;
  
%%% Exact solution
y_e(k+1)=350*(exp(t(k)/5)-1)/(7*exp(t(k)/5)-5);
  
  
  

  
end

%%%Error
err=abs(y_e-y);

plot(t,y_e);
hold on
plot(t,y,'*');
legend('Analytical solution','Solution by R-K method');
xlabel('t');
ylabel('y')

50 Analytical solution Solution by R-K method 45 40 35 30 25 20 15 10 0 268 10 12 14 16 18 20