Question

help, pls tq.

4. Consider the first order autonomous system d13-1 0)-1. (a) Estimate the solution of the system (1) at t0.2 using two steps of Euler's method with 2v, u(0)0 step-size h 0.1 T1+C2+A1-4 (b) An autonomous system of two first order differential equations can be written as: du dt=f(mu), u(to) = uo, dv dt=g(u, u), u(to) to. The Improved Euler's scheme for the system of two first order equations is tn+1 = tn + h, Use the Improved Euler's scheme to find an approximate solution of the system (1) at t = 12, if the step size h = 0.1. T2+C1+A2-5

Answer 1

Matlab code for Euler and Heun's method for 2d clear all close all %functions for Euler and Huen method ⅝step size %Initial guess u-0-0; v-0-1; u_cu 1er ( 1 ) =u_0 ; veu 1er ( 1 )-v-0 ; t-euler ( 1 ) #0 ; t-end-0. 2 ; n (t _end-t_euler (1))/h; %iteration for y and z using Euler method fprintf( Iterations for u and v using Euler methodInin fprintf( "Initial condition for u(0)=%f and v(0)=%f .\n',u 0,V 0) for i-1: n teuler(i+1)-t_euler (i)+hi u-euler ( i+1 ) ฃ-euler ( í ) +h* f ( t-euler ( í), u_culer (i) , v-euler ( í)) ; v euler(i+1)-v_euler (i)+h*g(t_euler(i),u_euler (i),v_euler(i)) fprint f ("After %d iteration\n',1) fprint f ("\t t=%f\t, u(82.2f)- %f\t v(82.2f)= %f\n', At t_euler(i+1) ,t_euler(i+1), u_euler(i+1), t_euler (i+1),v_euler(i end titeration for y and z using Huen method fprintf (nnIterations for u and v using Improved Euler method\n\n') fprint f ( "Initial condition for u(0)-%f and v(0)-%f .\n',u-0 , v-0 ) u huen (1)-u 0 v hu t_huen (1) 0; t_end-1.2; n-(t end-t huen(1))/h; en(1)-v_o; for i=1:n+ 1 11-h*g (t_huen(), u huen(i),v_huen(i)); k2-h * f(t-hue n (1+1) ,u-huen( il+k 1, v-huen (i )+11) ; 12-h*g(t_huen(i+1),u_huen(i) +k1,v_huen (i)+11)

u huen (i+1)-u_huen (i)+ (1/2) (k1+k2); v huen (i+1)-v_huen (i)+(1/2)*(11+12) tprint f ("After %d iteration\n',1) fprint f ("\t At t-%f\t, u(82.2f)- %f\t v(82.2f,- %f\n', t huen (i+1), t huen (i+1), u huen (i+1), t huen (i+1), v huen(i+1)) end 88888888888888888888 End of Code 움888888888888888888 Iterations for u and v using Euler method Initial condition for u(0)=0.000000 and v(0)·1.000000 . After 1 iteration At t=0 . 100000 u(0.10). 0.200000 v(0.10)=-0.400000 , After 2 iteration At t-0.200000u(0.20)- 0.120000 v(0.20)-0-364000 Iterations for u and v using Improved Euler method Initial condition for u(0)-0.000000 and v(0)-1.000000 . After 1 iteration At t-0.100000 ,u(0.10)0.060000 v(0.10)- 0.318000 At t-0.200000 , u(0.20)= 0.117243 v(0.20)= 0.321240 At t-0.300000 u(0.30)0.182285 v(0.30)0. 367328 At t-0.400000,u(0.40)- 0.260557v(0.40)- 0.436812 At t=0 . 500000 , u(0.50,-0.355079 v(0.50)= 0.517822 At t-0.600000 ,u(0.60)- 0.467265 v(0.60)- 0.603648 At t-o. 700000u(0. 70)- 0.597724 v(0.70)- 0.690566 At t-0.800000 ,u(0.80)- 0.746778 v(0.80)- 0.775594 At t 0.900000u(0.90)- 0.914761 v(0.90)- 0.853794 At t-1.000000u(1.00)- 1.102384 v(1.00)- 0.911943 At t-l.100000u(1.10)- 1.311653 v(1.10)- 0.905534 At t-1.200000u(1.20)- 1.548476 v(1.20)- 0.656776 After 2 İter at 10 After 3 iteration After 4 iteration After 5 iteration After 6 iteration After 7 iteratiorn After 8 iterationn After 9 iterationn After 10 iteration After 11 iteration After 12 iteration Published with MATLAB® R2018a

%Matlab code for Euler and Heun's method for 2d
clear all
close all
%functions for Euler and Huen method

f=@(t,u,v) 2*v;
g=@(t,u,v) 13.*u-14.*v.^2;

%step size
h=0.1;

%Initial guess
u_0=0; v_0=1;

u_euler(1)=u_0; v_euler(1)=v_0;
t_euler(1)=0; t_end=0.2;
n=(t_end-t_euler(1))/h;

%iteration for y and z using Euler method
fprintf('Iterations for u and v using Euler method\n\n')
fprintf('Initial condition for u(0)=%f and v(0)=%f .\n',u_0,v_0)
for i=1:n
  
    t_euler(i+1)=t_euler(i)+h;
    u_euler(i+1)=u_euler(i)+h*f(t_euler(i),u_euler(i),v_euler(i));
    v_euler(i+1)=v_euler(i)+h*g(t_euler(i),u_euler(i),v_euler(i));
  
    fprintf('After %d iteration\n',i)
  
    fprintf('\t At t=%f\t, u(%2.2f)= %f\t v(%2.2f)= %f\n',...
        t_euler(i+1),t_euler(i+1),u_euler(i+1),t_euler(i+1),v_euler(i+1))
  
  
end

%iteration for y and z using Huen method
fprintf('\n\nIterations for u and v using Improved Euler method\n\n')
fprintf('Initial condition for u(0)=%f and v(0)=%f .\n',u_0,v_0)
u_huen(1)=u_0; v_huen(1)=v_0;
t_huen(1)=0; t_end=1.2;
n=(t_end-t_huen(1))/h;

for i=1:n+1
  
    t_huen(i+1)=t_huen(i)+h;
    k1=h*f(t_huen(i),u_huen(i),v_huen(i));
    l1=h*g(t_huen(i),u_huen(i),v_huen(i));
  
    k2=h*f(t_huen(i+1),u_huen(i)+k1,v_huen(i)+l1);
    l2=h*g(t_huen(i+1),u_huen(i)+k1,v_huen(i)+l1);
  
    u_huen(i+1)=u_huen(i)+(1/2)*(k1+k2);
    v_huen(i+1)=v_huen(i)+(1/2)*(l1+l2);
  
    fprintf('After %d iteration\n',i)
  
    fprintf('\t At t=%f\t, u(%2.2f)= %f\t v(%2.2f)= %f\n',...
        t_huen(i+1),t_huen(i+1),u_huen(i+1),t_huen(i+1),v_huen(i+1))
  
end

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