Question

(16 marks) Consider the initial value problem (a) Without using pre-built commands write an m-file function that uses the fourth-order Runge-Kutta method to estimate the value of y(n) for a given value n and a given step size h (b) Use the m-file function built in part (a) to compute an estimate of y(2) using step size h = 0.5 and h = 0.25. Fron these two estimates, approximate the step size needed to estimate y(2) correct to 4 decimal places, explaining your answer. Compute y(2) using the approximate step size and then with half the approximate step size. Comment on your results. (30 marks)

Answer 1

Answer for (a):

clc;                                               % Clears the screen

clear all;

h=1.5;                                             % step size

x = 0:h:3;                                         % Calculates up to y(3)

y = zeros(1,length(x));

y(1) = 5;                                          % initial condition

F_xy = @(t,r) t*t- sqrt(t*r);                    % change the function as you desire

for i=1:(length(x)-1)                              % calculation loop

    k_1 = F_xy(x(i),y(i));

    k_2 = F_xy(x(i)+0.5*h,y(i)+0.5*h*k_1);

    k_3 = F_xy((x(i)+0.5*h),(y(i)+0.5*h*k_2));

    k_4 = F_xy((x(i)+h),(y(i)+k_3*h));

    y(i+1) = y(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; % main equation