Question

Use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval. This question is from the differential equation.

y'-4y = x/y^2(y+1) , y(0) = 1; h=0.1, 0.05 , 0.025, on [0, 1]

Answer 1

clear all
close all

%function for which RK4 method have to calculate
f=@(x,y) 4.*y+(x./((y.^2).*(y+1)));
fprintf('function for which RK4 method have to calculate f(x,y)= ')
disp(f)
%all h
h=[0.1 0.05 0.025];
    %step size h
    %initial value
    y0=1; x0=0;
  
    xx=0:0.1:1;
    fprintf('\tx\ty_h=0.1 \ty_h=0.05 \ty_h=0.025\n')
    for it=1:length(xx)
        x_end=xx(it);
        [t_result1,y1_result1] = RK4_method(f,y0,x0,x_end,h(1));
        [t_result2,y1_result2] = RK4_method(f,y0,x0,x_end,h(2));
        [t_result3,y1_result3] = RK4_method(f,y0,x0,x_end,h(3));
        fprintf('\t%2.2f \t%f\t%f\t%f\n',t_result1(end),y1_result1(end),y1_result2(end),y1_result3(end))
    end

%%Matlab function for Runge Kutta Method
function [t_rk,y_rk]=RK4_method(f,yinit,tinit,tend,h)
    % RK4 method
    % h amount of intervals
    t=tinit;         % initial t
    y=yinit;         % initial y
    t_eval=tend;     % at what point we have to evaluate
    n=(t_eval-t)/h; % Number of steps
    t_rk(1)=t;
    y_rk(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)));
       dy=(1/6)*(k1+2*k2+2*k3+k4);
       t=t+h;
       y=y+dy;
       t_rk(i+1)=t;
       y_rk(i+1)=y;
    end
end

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