Use the 4th order RungeKutta method for systems to approxima

Use the 4th order Runge–Kutta method for systems to approximate the solution of the following system of first–order differential equations for 0 t 1, and h = 0.2, and compare the results to the actual solution. u1 = 3u1 +2u2 (2t^2+1)e^2t u2 = 4u1 +u2 +(t^2+2t-4)e^2t The initial conditions are u1(0) = 1 and u2(0) = 1. For comparison, the exact solutions are u1(t)=1/3e^5t1/3e^t+e^2t and u2(t)=1/3e^5t+2/3e^t+t^2e^2t.

Solution

Fourth-order Runge-Kutta method (h = 0.2)

u1p=\'3*u1+2*u2-(2*t*t+1)*exp(2*t)\';

u2p=\'4*u1+u2+(t*t+2*t-4)*exp(2*t)\';

h=0.2;t=0;u1=1;u2=1;hs=h/2;

for i=1:5

u1s=u1;u2s=u2;

k11=eval(u1p);k12=eval(u2p);

t=t+hs;u1=u1s+hs*k11;u2=u2s+hs*k12;

k21=eval(u1p);k22=eval(u2p);

u1=u1s+hs*k21;u2=u2s+hs*k22;

k31=eval(u1p);k32=eval(u2p);

t=t+hs;u1=u1s+h*k31;u2=u2s+h*k32;

k41=eval(u1p);k42=eval(u2p);

u1=u1s+h*(k11+2*k21+2*k31+k41)/6;

u2=u2s+h*(k12+2*k22+2*k32+k42)/6;

end

fprintf(\'Calculated u1(1) = %8.6f , u2(1) = %8.6f\ \',u1,u2)

u1c=exp(5)/3-exp(-1)/3+exp(2);

u2c=exp(5)/3+2*exp(-1)+exp(2);

fprintf(\'Actual     u1(1) = %8.6f , u2(1) = %8.6f\ \',u1c,u2c)

>> s5p3

Calculated u1(1) = 55.661181 , u2(1) = 56.030503

Actual     u1(1) = 56.737483 , u2(1) = 57.595868