Question

use matlab

Assignment: 1) Write a function program that implements the 4th Order Runge Kutta Method. The program must plot each of the k values for each iteration (one plot per k value), and the approximated solution (approximated solution curve). Use the subplot command. There should be a total of five plots. If a function program found on the internet was used, then please cite the source. Show the original program and then show the program after any modifications. Submission Function Program(s) K1 K2 K3 K4 2) Solve the following Initial Value Problem across the interval t = 2 to 5, by writing a script that will utilize your function program written for Part 1. dt = -0.5y +et y(2.0) = 4.143883 h = 0,5 Submission Script Program Output

Answer 1

88 Matlab code using RK4 for 2nd order differential equation clear all close all tinitial condition t_in=2; t_end=5; function f=@(t,y) -0.5.*y+exp(-t); $initial guess yl in=4.143883; h=0.5; (y1, K1, K2,K3, K4, tt ]=RK4(f, h, yl_in,t_in, t_end); function for RK4 method function (y1, K1, K2,K3, K4, t]=RK4(f,h, yi_in, t_in, t_end) $initial conditions t(1)=t_in;yl(1)=yl_in; t_in=t(1); Initial t t_max=t_end; Final t n=(t_max-t_in) /h; $number of steps Runge Kutta 4 iterations for i=1:n+1 k0=h*f(t(i),yl(i)); Kl(i)=k0; kl=h*£(t(i)+(1/2)*h, yl(i)+(1/2)*k0); K2 (i)=kl; k2=h*f(t(i)+(1/2)*h, yl(i)+(1/2)*kl); K3 (i)=k2; k3=h*f(t(i)+h, yl(i)+k2); K4 (i)=k3; if icon t(i+1)=t_in+i+h; yl(i+1)=double (y1(i)+(1/6)*(k0+2*k1+2*k2+3)); end end figure(1) subplot(5,1,1) plot(t,K1) xlabel('t') ylabel('ki') subplot (5,1,2) plot(t,K2) xlabel('t') ylabel('K2') subplot(5,1,3) plot(t,K3) xlabel('t') ylabel('K3') subplot(5,1,4)

plot(t, K4) xlabel('t') ylabel('64') subplot(5,1,5) plot(t,y1) xlabel('t') ylabel('y(t)') end $$$$$$$$$$$$$$$$$$ End of Code %%%%%%%%%%%%%%%% KI F H k2 F - k3 FFF F yoooo GAN F F - FH N Published with MATLABE R2018

%%Matlab code using RK4 for 2nd order differential equation
clear all
close all
%initial condition
t_in=2; t_end=5;
%function
f=@(t,y) -0.5.*y+exp(-t);

%initial guess
y1_in=4.143883;
h=0.5;
[y1,K1,K2,K3,K4,tt]=RK4(f,h,y1_in,t_in,t_end);

%function for RK4 method
function [y1,K1,K2,K3,K4,t]=RK4(f,h,y1_in,t_in,t_end)
  
    %initial conditions
    t(1)=t_in;y1(1)=y1_in;
         
    t_in=t(1);            %Initial t
    t_max=t_end;          %Final t
    n=(t_max-t_in)/h; %number of steps
    %Runge Kutta 4 iterations
    for i=1:n+1

        k0=h*f(t(i),y1(i));
        K1(i)=k0;
        k1=h*f(t(i)+(1/2)*h,y1(i)+(1/2)*k0);
        K2(i)=k1;
        k2=h*f(t(i)+(1/2)*h,y1(i)+(1/2)*k1);
        K3(i)=k2;
        k3=h*f(t(i)+h,y1(i)+k2);
        K4(i)=k3;
        if i<=n
            t(i+1)=t_in+i*h;
            y1(i+1)=double(y1(i)+(1/6)*(k0+2*k1+2*k2+k3));
        end
    end
    figure(1)
     subplot(5,1,1)
    plot(t,K1)
    xlabel('t')
    ylabel('K1')
     subplot(5,1,2)
    plot(t,K2)
    xlabel('t')
    ylabel('K2')
    subplot(5,1,3)
    plot(t,K3)
    xlabel('t')
    ylabel('K3')
    subplot(5,1,4)
    plot(t,K4)
    xlabel('t')
    ylabel('K4')
    subplot(5,1,5)
    plot(t,y1)
    xlabel('t')
    ylabel('y(t)')
  
end

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%