Runge-Kutta Method Calculator

Runge-Kutta Method Calculator

Runge-Kutta Method Calculator is used to calculate ordinary differential equations (ODEs) numerically.

What is Runge-kutta method?

Runge-kutta method is a numerical technique to find the solution of the ordinary differential equation. The method provides us with the value of y as the corresponding value of x. It is also known as the RK method. Here we use the most common RK-4 which is known as runge-kutta of order fourth.

Formula of Runge-kutta method

The formula of the Runge-Kutta method of order fourth is given as:

y_i+1=y_i+h/6(k₁+2k₂+2k₃+k₄) where

x_i+1 = x_i+h

where i is the number of iterations.

For the value of k’s, we have

• k₁= f(x_i,y_i)
• k₂= f(x_i+h/2, y_i+hk₁/2)
• k₃=f(x_i+h/2,y_i+hk₂/2)
• k₄= f(x_i+h,y_i+hk₃)

How to calculate Runge-kutta method problems?

Here is a numerical example to understand this iterative method.

Example:

If we have differential function x²+1 with the "2" number of iterations. The initial value of the x is 1 to 3 and the initial value of y is 2.

Solution:

Step 1: The step size is

h= (3-1)/2 = 1, x₀= 1, y₀ = 2 and n= 2

f(x, y) = x^2+1

Step 2: For 1^st iteration

x₁= x₀+h = 1+1 = 2

k₁= f(x₀,y₀)= f(1,2) = 2

k₂= f(x₀+h/2, y₀+hk₁/2) = f(1+1/2,2+(1)(2)/2) = f(3/2,3) = 3.25

k₃= f(x₀+h/2, y₀+hk₂/2) = f(1+1/2,2+(1)(3.25)/2) = f(2/2,3.625) = 3.25

k₄ = f(x₀+h,y₀+hk₃) = f(1+1,2+(1)(3.25)) = f(2,5.25) = 5

y₁ = y(x₁) = y(2) = y₀+ h/6(k₁+2k₂+2k₃+k₄) = 2+1/6(2+2(3.25)+2(3.25)+5) = 5.333

Step 3: For the 2^nd iteration we have x₁ and y₁ to calculate x₂ and y₂.

x₂= x₁+h = 2+1 = 3

k₁= f(x₁,y₁) = f(2,5.333) = 5

k₂= f(x₁+h/2, y₁+hk₁/2) = f(2+1/2,5.333+(1)(5)/2) = f(5/2,7.83) = 7.25

k₃= f(x₁+h/2, y₁+hk₂/2) = f(2+1/2,5.333+(1)(7.25)/2= f(5/2,8.958) = 7.25

k₄ = f(x₁+h,y₁+hk₃) = f(2+1,5.333+(1)(7.25)) = f(3,12.583) = 10

y₂ = y(x₂) = y (3) = y₁+ h/6(k₁+2k₂+2k₃+k₄) = 5.333+1/6(5+2(7.25) +2(7.25) +10) = 12.666

Hence the approximate value of y₂ = 12.666.