Question

///MATLAB/// Consider the differential equation over the interval [0,4] with initial condition y(0)=0.

3. Consider the differential equation n y' = (t3 - t2 -7t - 5)e over the interval [0,4 with initial condition y(0) = 0. (a) Plot the approximate solutions obtained using the methods of Euler, midpoint and the classic fourth order Runge Kutta with n 40 superimposed over the exact solution in the same figure. To plot multiple curves in the same figure, make use of the following command: plot(t,y,.',t1,y1,'. . .',t2,y2,..'.) (b) Plot the error curves corresponding to the 3 methods over the interval [0, T]. Use the Matlab subplot command for this plot.

Answer 1

Matlab code for Euler, Improved Euler and RK4 method clear all close all %Function for which solution have to do f-e (t,y) (t.^3-t.^2-7. "t-5).*exp(t); Euler method number of intervals S initial t initial y y0 t eval-4 h- (t_eval-t) /n t2 (1)-t y2 (1)-y for i-1:n at what point we have to evaluate step size %Eular Steps m double (f (t ,y) ); t-t+h; y yh*m t2 (i+1t y2 (i+1 y end fprintf\n\ tThe solution using Euler Method for h-.2f at x(%.lf) is %f\n',h, t2 (end), y2 (end)) %Midpoint method t 0; y 0 initial t initial x t3 (1 t y3 (1 y for i-1:n %midpoint steps m1-double(f (t,y) ); m2 double (f ( (t+h), (y+h*ml))); y=y+double(h* ( (m1+m2) /2)); t-t+h; y3(i+1)-y t3 (i+1) t end fprintf(\n\tThe solution using improved Euler Method for h= .2f at x(.1f) is tf\n',h,t3(end),y3(end)) &RK4 method t 0 y 0 S initial t initial x t4 (1)-t y4 (1)-y for i-1:n RK4 Steps kl-h*double (f (t,y)); k2-h*double ( f ( (t+h/2), (y+kl/2)) ); 1

k3=h* double (f ( (t+h/2), (y+k2/2))); k4-h* double (f ( (t+h) , (y+k3))) dx(1/6) (k1+2*k2 +2 *k3+k4); t-t+h y=y+dx; t4(i+1)-t y4 (i+1y end ntE\n\tThe solution using Runge Kutta 4 for h=8.2f at f\n',h,t4 (end),y4 (end)) x(.lf) SExact solution syms y (t) eqndiff (y,t) cond y(0) 0; ySol (t)dsolve(eqn,cond) (t. 3-t.^2-7.*t-5).*exp (t); y_ext-ySol (t4); fprintfExact solution is y(t)=') disp (ySol) Plotting solution using Euler method figure(1) plot(t2,y2, 'r',t3,у3,'Ь",t4,y4, 'g', t4,y_ext, 'm') fprintf\n\n\tError norm using Euler method is %f. \n',norm(y_ext- y2)) fprintf\tError norm using Modified Euler method is f , norm (y ext-y3)) fprintf \tError norm using Runge Kutta method is %f.\n', norm (y ext- y4)) \n xlabel't' ylabely(t)') titleSolution plot y(t) vs. t' legend Euler Method', Midpoint', 'RK4 Method', 'Exact solution', Location 'northeast') grid on errl-abs (y_ext-y2); err2-abs (y_ext-y3) err3-abs (y_ext-y4) figure(2) subplot (3,1,1) plot(t2,errl) title Error plot for Euler method') xlabel't') ylabelerror') subplot (3,1,2) plot (t2,err2) title Error plot for Midpoint method') 2

xlabel't) ylabelerror') subplot (3,1,3) plot(t2,err3) title Error plot for RK4 method') xlabel't ylabelerror') 응동용을용동동을을동 을용동동을을 End of Code 응용용용용용용용응용용용용용용 The solution using Euler Method for h-0.10 at x(4.0) is -142.2024 00 The solution using improved Euler Method for h=0.10 at x(4. 0) is -101.003787 The solution using Runge Kutta 4 for h-0.10 at x(4.0) is -103.19594 7 Exact solution is y(t) -exp (t)* (t ^3 - 4*t^2 + t - 6) + 6 symbolic function inputs: t Error norm using Euler method is 59.947145 Error norm using Modified Euler method is 3.593554 . Error norm using Runge Kutta method is 0.000660 Solution plot yt) vs. t hod RK4 Method Exact solution -100 -150 -200 -250 nor 0.5 15 1 2 25 3 5

Error plot for Euler method 20 0 05 1 15 25 35 4 4 5 t Error plot for Midpoint method 05 15 25 3.5 t. Error plot for RK4 method x104 0.5 1 A 2 25 4 5 Published with MATLAB R2018a 4

%%Matlab code for Euler, Improved Euler and RK4 method
clear all
close all
%Function for which solution have to do
f=@(t,y) (t.^3-t.^2-7.*t-5).*exp(t);
    %Euler method
    n=40;            % number of intervals
    t=0;             % initial t
    y=0;             % initial y
    t_eval=4;        % at what point we have to evaluate
    h=(t_eval-t)/n; % step size
    t2(1)=t;
    y2(1)=y;
    for i=1:n
        %Eular Steps
        m=double(f(t,y));
        t=t+h;
        y=y+h*m;
        t2(i+1)=t;
        y2(i+1)=y;
    end
    fprintf('\n\tThe solution using Euler Method for h=%.2f at x(%.1f) is %f\n',h,t2(end),y2(end))
    %Midpoint method
    t=0;             % initial t
    y=0;             % initial x

    t3(1)=t;
    y3(1)=y;
    for i=1:n
    %midpoint steps
       m1=double(f(t,y));
       m2=double(f((t+h),(y+h*m1)));
       y=y+double(h*((m1+m2)/2));
       t=t+h;
       y3(i+1)=y;
       t3(i+1)=t;
    end
  
    fprintf('\n\tThe solution using improved Euler Method for h=%.2f at x(%.1f) is %f\n',h,t3(end),y3(end))
  
    %RK4 method
    t=0;             % initial t
    y=0;             % initial x

    t4(1)=t;
    y4(1)=y;
    for i=1:n
    %RK4 Steps
       k1=h*double(f(t,y));
       k2=h*double(f((t+h/2),(y+k1/2)));
       k3=h*double(f((t+h/2),(y+k2/2)));
       k4=h*double(f((t+h),(y+k3)));
       dx=(1/6)*(k1+2*k2+2*k3+k4);
       t=t+h;
       y=y+dx;
       t4(i+1)=t;
       y4(i+1)=y;
    end
    fprintf('\n\tThe solution using Runge Kutta 4 for h=%.2f at x(%.1f) is %f\n',h,t4(end),y4(end))

%%Exact solution
syms y(t)
eqn = diff(y,t) == (t.^3-t.^2-7.*t-5).*exp(t);
cond = y(0) == 0;
ySol(t) = dsolve(eqn,cond);

y_ext=ySol(t4);
fprintf('Exact solution is y(t)=')
disp(ySol)

%%Plotting solution using Euler method
figure(1)
plot(t2,y2,'r',t3,y3,'b',t4,y4,'g',t4,y_ext,'m')

fprintf('\n\n\tError norm using Euler method is %f.\n',norm(y_ext-y2))
fprintf('\tError norm using Modified Euler method is %f.\n',norm(y_ext-y3))
fprintf('\tError norm using Runge Kutta method is %f.\n',norm(y_ext-y4))

xlabel('t')
ylabel('y(t)')
title('Solution plot y(t) vs. t')
legend('Euler Method','Midpoint','RK4 Method','Exact solution','Location','northeast')
grid on

err1=abs(y_ext-y2);
err2=abs(y_ext-y3);
err3=abs(y_ext-y4);

figure(2)
subplot(3,1,1)
plot(t2,err1)
title('Error plot for Euler method')
xlabel('t')
ylabel('error')

subplot(3,1,2)
plot(t2,err2)
title('Error plot for Midpoint method')
xlabel('t')
ylabel('error')

subplot(3,1,3)
plot(t2,err3)
title('Error plot for RK4 method')
xlabel('t')
ylabel('error')

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