Asked by Christopher Salerno
on 8 Dec 2018

1. Write your own 4th order Runge-Kutta integration routine based on the general equations. Do not use Matlab functions, element-by-element operations, or matrix operations.

Answer by Christopher Salerno
on 8 Dec 2018

Accepted Answer

clear all

close all

clc

h = ___; % set the step size

x = ________; % set the interval of x

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

y(__) = ___; % set the intial value for y

n = length(x)-1;

y_dot =@(x,y)(___________); %insert function to be solved

for i = 1:n

k1 = y_dot(x(i),y(i));

k2 = y_dot(x(i)+.5*h,y(i)+.5*k1*h);

k3 = y_dot(x(i)+.5*h,y(i)+.5*k2*h);

k4 = y_dot(x(i)+h,y(i)+k3*h);

y(i+1) = y(i)+((k1+2*k2+2*k3+k4)/6)*h;

end

[t,y_check] = ode45(y_dot,x,2);

plot(x,y)

title('Eulers Method')

figure

plot(x,y_check)

title('ode45 Check')

Anthony Fitch
on 9 Dec 2018

Nice Job, my Code came out very similar to yours :)

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Answer by Anthony Fitch
on 9 Dec 2018

%% Problem 1

%4th order Runge-Kutta integration routine

clear,clc

%Input custom values and custom first-order differential equation

Step_Value = input('Enter Step Value: ')

x_beg = input('Enter intial value of "x" at the beginning of the interval: ')

x_end = input('Enter final value of "x" at the end of the interval: ')

x_intial=input('Enter intial "x" value: ')

y_intial=input('Enter intial "y" value: ')

% Routine starts here

F_xy=@(x,y) 'Enter equation here: ';

h=Step_Value;

x=x_beg:h:x_end;

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

y(x_intial)=y_intial;

for i=1:(length(x)-1)

k1 = F_xy(x(i),y(i));

k2 = F_xy(x(i)+0.5*h,y(i)+0.5*h*k1);

k3 = F_xy((x(i)+0.5*h),(y(i)+0.5*h*k2));

k4 = F_xy((x(i)+h),(y(i)+k3*h));

y(i+1) = y(i) + (1/6)*(k1+2*k2+2*k3+k4)*h;

end

%Checking custom routine and MATLAB function ode45

[x_c,y_c] = ode45(F_xy,x,y_intial);

%Plot in one figure

figure(1)

plot(x,y)

hold on

plot(x_c,y_c)

hold off

y_c2=y_c';

%Calculates error between custom runge-kutta routine and MATLAB function

%ode45

disp('Error between Runge-Kutta and ode45')

err = immse(y,y_c2)

Christopher Salerno
on 9 Dec 2018

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