Solving ODE in MATLAB using Runge-Kutta method of order 4

Write a computer program that will solve the initial value problem dy/dx = f(x, y) subjected to y(x0) = y0 by classical fourth order Runge-Kutta method.

Inputs: x0, y0, f(x, y), h = step-size, and n = number of steps. Using the code obtain the solutions of the ODE

Compare each numerically obtained solution with corresponding analytical solution by plotting them in a single figure.

[Torsten]

If you are unable or too lazy to write your own code, search the internet for

matlab & classical runge kutta

[Shankhadeep Saha]

NOT LAZY! Just can't figure out the approach to the question. Can you give a little idea on what will be the function i'll be operating on?

[Torsten]

So what is your specific question ?

[James Tursa]

What have you done so far? What specific problems are you having with your code?

[Shankhadeep Saha]

clc;

clear all;

g = input("Enter your function after substitution: ");

x(1) = input("Enter the initial value of independent variable: ");

y(1) = input("Enter the initial value of dependent varible: ");

z(1) = input("Enter the initial value of dy/dx: ");

h = input("Enter step-size: ");

n = input("Enter no. of steps: ");

f = @(x,y,z) z;

for i=1:n

k1 = h*f(x(i),y(i),z(i));

l1 = h*g(x(i),y(i),z(i));

k2 = h*f(x(i)+(h/2),y(i)+(k1/2),z(i)+(l1/2));

l2 = h*g(x(i)+(h/2),y(i)+(k1/2),z(i)+(l1/2));

k3 = h*f(x(i)+(h/2),y(i)+(k2/2),z(i)+(l2/2));

l3 = h*g(x(i)+(h/2),y(i)+(k2/2),z(i)+(l2/2));

k4 = h*f(x(i)+(h),y(i)+(k3),z(i)+(l3));

l4 = h*g(x(i)+(h),y(i)+(k3),z(i)+(l3));

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

x(i+1) = x(i) + h;

z(i+1) = y(i+1)/x(i+1);

end

This is the code I have written for 2nd degree ODE. Now what I change the value of z(i+1) for the next iteration?

[James Tursa]

It is unclear what your z variable is and how it is related to your differential equation. It is also unclear how the f and g functions are related. Can you provide more detail about this? Is the user supposed to be able to enter an arbitrary g function and your program should handle it? Is z supposed to be ydot?

[Shankhadeep Saha]

The variable z is actually dy/dx. Function f(x,y,z) = dy/dx = z (also given in the code). Then I have substituted z in the original 2nd degree ODE in order to cconvert it into 1st degree ODE. Now g(x,y,z) = the ode after substitution.

[James Tursa]

Still not clear. Is the user supposed to know that z is dy/dx? How would they know this? Is the user supposed to enter a function handle directly using x, y, and z with the assumption that z is dy/dx?

[Shankhadeep Saha]

Yes, the user knows that. He will input the function interms of x, y and z. Like in this particular case user will input g(x,y,z) = sin 2x -0.2z-y = dz/dx

Answer 1

[James Tursa]

Why aren't you using this to update z:

z(i+1) = z(i) + (1/6)*(l1+(2*l2)+(2*l3)+l4);