Question

3. (25 points) Given a series of ODES: dy = 6e– y2 +224/7 = 62 + 3y Given initial conditions y(0)=0, 2(0)=1, and I = 1; dra dx dx x=0 solve the system using 2nd-order Runge:Kutta method (Heun's method) with step size of h = 1. dy (Hint: Treat- dy dz as separate ODES) dx2

Answer 1

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%%Matlab code for solving ode clear all close all %Initial conditions for ode u0=[0;1;1]; h=1; % Solution for equation using ode 45 minimum and maximum time span tspan=[05]; % Solution of ODEs using ode45 matlab function [x, s )= RK2 System (@(t,y) odel_equation(t,y), tspan, 0,h); fprintf('Solution using RK2 method for h=1\n') for i=1:length(s) fprintf('At x=%2.2f y(%2.2f) =%2.2e, \n\t dy (2.2f)/dx=82.2e \n\t z(%2.2f )=%2.2e\n',... x(i),x(i),s(1,i),x(i),s(2,i),x(i),s(3,1)) end figure(1) plot 3(s(1,:),s(2,:),s(3,:),'b', 'linewidth', 2) xlabel('y(x)') ylabel('dy (x)/dt') zlabel('z(x)') title('Solution plot using RK2') box on grid on view ( [00]); figure (2) plot(x,s(1,:), 'b', 'linewidth', 2) ylabel('y(x)') xlabel('x') title('Solution plot using RK2 x vs. y(x)') box on grid on figure (3) plot(x,s(2,:), 'b', 'linewidth',2) ylabel('dy(x)/dx'). xlabel('x') title('Solution plot using RK2 x vs. dy(x)/dx') box on grid on figure (4) plot (x,S(3,:), 'b', 'linewidth', 2) ylabel('z(x)') xlabel('x') title('Solution plot using RK2 X vs. z(x)') box on grid on 응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응 %Function for evaluating the ODE

function dydt = odel_equation(x,y) eql = y(2); eq2 = 6. *exp(3.*x)-(y(1)).^2+2.*y(3); eq3 = 6.*(y(3)).^2+3.*y(1); %Evaluate the ODE for our present problem dydt = [eql; eq2; eq3]; end 응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응 %%RK4 system for multidimentional problem function [t,y)=RK2 System (Func, Tspan, Yo,h) finitialization t0= Tspan(1); tf= Tspan(2); N=(tf-to)/h ; t=t0:h:tf; y=zeros(length(YO),N+1); y(:,1) = YO; for i=1:N kl=h*Func(t(i),y(:,i)); k2=h*Func(t(i) +h/2,y(:,i)+k1/2); y(:,i+1)=y(:,i)+k2; end end 응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응응 Solution using RK2 method for h=1 At x=0.00 y(0.00)=0.00e+00, dy (0.00)/dx=1.00e+00 2(0.00)=1.00e+00 At x=1.00 y(1.00)=5.00e+00, dy (1.00)/dx=3.56e+01 2(1.00)=9.85e+01 At x=2.00 y(2.00)=1.87e+02, dy (2.00)/dx=5.85e+04 2(2.00)=5.12e+09 At x=3.00 y(3.00)=5.12e+09, dy (3.00)/dx=1.57e+20 2(3.00)=3.71e+40 At x=4.00 y(4.00)=3.71e+40, dy (4.00)/dx=8.27e+81 214.00)=1.02e+164 At x=5.00 y(5.00)=1.02e+164, dy (5.00)/dx=Inf 2(5.00)=Inf

10763 Solution plot using RK2 12 (x)Z O 0.5 1 1.5 2.5 3 3.5 4 2 y(x) *100 10163 Solution plot using RK2 x vs. y(x) 12 y(x) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Solution plot using RK2 x vs. dy(x)/dx 9x1087 хрохир о 0.5 1 15 2 25 3 3.5 4 10 103 Solution plot using RK2 x vs. z(x) 12 2(x) о 0.5 1 15 2 25 3 3.5 4

%%Matlab code for solving ode
clear all
close all

%Initial conditions for ode
    u0=[0;1;1];
    h=1;
%Solution for equation using ode45
        %minimum and maximum time span
        tspan=[0 5];
        %Solution of ODEs using ode45 matlab function
        [x,s]= RK2System(@(t,y) odel_equation(t,y), tspan, u0,h);
        fprintf('Solution using RK2 method for h=1\n')
        for i=1:length(s)
            fprintf('At x=%2.2f y(%2.2f)=%2.2e,\n\t dy(%2.2f)/dx=%2.2e\n\t z(%2.2f)=%2.2e\n',...
                x(i),x(i),s(1,i),x(i),s(2,i),x(i),s(3,i))
        end
        figure(1)
        plot3(s(1,:),s(2,:),s(3,:),'b','linewidth',2)
        xlabel('y(x)')
        ylabel('dy(x)/dt')
        zlabel('z(x)')
        title('Solution plot using RK2')
        box on
        grid on
        view([0 0]);
        figure(2)
        plot(x,s(1,:),'b','linewidth',2)
        ylabel('y(x)')
        xlabel('x')
        title('Solution plot using RK2 x vs. y(x)')
        box on
        grid on
        figure(3)
        plot(x,s(2,:),'b','linewidth',2)
        ylabel('dy(x)/dx')
        xlabel('x')
        title('Solution plot using RK2 x vs. dy(x)/dx')
        box on
        grid on
        figure(4)
        plot(x,s(3,:),'b','linewidth',2)
        ylabel('z(x)')
        xlabel('x')
        title('Solution plot using RK2 x vs. z(x)')
        box on
        grid on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Function for evaluating the ODE
function dydt = odel_equation(x,y)

    eq1 = y(2);
    eq2 = 6.*exp(3.*x)-(y(1)).^2+2.*y(3);
    eq3 = 6.*(y(3)).^2+3.*y(1);

    %Evaluate the ODE for our present problem
    dydt = [eq1;eq2;eq3];
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%RK4 system for multidimentional problem
function [t,y]=RK2System(Func,Tspan,Y0,h)

    %initialization
    t0= Tspan(1);
    tf= Tspan(2);
    N=(tf-t0)/h ;
    t=t0:h:tf;
    y=zeros(length(Y0),N+1);
    y(:,1) = Y0;
  
    for i=1:N
        k1=h*Func(t(i),y(:,i));
        k2=h*Func(t(i)+h/2,y(:,i)+k1/2);
        y(:,i+1)=y(:,i)+k2;
    end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%