How to create Runge-Kutta 4th order routine to solve first-order ODE's

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 1

[Christopher Salerno]

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]

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

[David After]

How can i solve this eqaution?

F''' + F*F"+ F'^2 = 0

with the boundary conditions

F(0)=F''(0)=0 and F'(infinity)=0 and eta is 0:0.1:6

i want to plot it and create a table for it (eta-f-f'-f'')

I am new to using the ode solver in matlab and am not sure how to make it solve a equation. Any suggestion would be appreciated.

please help me

thank you

my email : [e-mail]

[Jivansu Vyas]

In what form are we supposed to write the function?

[Yixuan Qi]

what is the "t" after the end

Answer 2

[Anthony Fitch]

%% 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]

I like the error checking and the user inputs. Very nice man!.. a bit better than my code.. thanks for posting your code so I can see how to be more user friendly. Love it!

[Andrew Mackintosh]

what if you have four functions to work out?

[David After]

How can i solve this eqaution?

F''' + F*F"+ F'^2 = 0

with the boundary conditions

F(0)=F''(0)=0 and F'(infinity)=0 and eta is 0:0.1:6

i want to plot it and create a table for it (eta-f-f'-f'')

I am new to using the ode solver in matlab and am not sure how to make it solve a equation. Any suggestion would be appreciated.

please help me

thank you

my email : [e-mail]