The document discusses Runge-Kutta methods for solving differential equations. It describes the second and third-order Runge-Kutta methods, which evaluate the integrand multiple times per step to improve accuracy over Euler's method. Higher-order Runge-Kutta methods can also be derived, though above fourth order the derivation becomes complicated. The fourth-order Runge-Kutta method evaluates the integrand four times per step and is commonly used. An example problem applies the fourth-order method to solve a differential equation.

1. Runge- Kutta Methods

2. Runge- Kutta Methods
• Runge–Kutta method is an effective and widely used
method for solving the initial-value problems of differential
equations. Runge–Kutta method can be used to construct
high order accurate numerical method by functions' self
without needing the high order derivatives of functions.
• The Runge-Kutta method attempts to overcome the
problem of the Euler's method, as far as the choice of a
sufficiently small step size is concerned, to reach a
reasonable accuracy in the problem resolution.

3. Second-order Runge-Kutta Method
(Improved Euler’s Method)
• The improved Euler method is a Runge-Kutta
method for approximating the solution of the
initial value problem y'(x) = f(x,y); y(x0) =
y0 which evaluates the integrand,f(x,y), twice
for each step.

4. The essential formulas to compute the value of
y(n+1):
yn+1 = yn + k2 ,
k1 = h * f(xn, yn),
k2 = h * f(xn + h / 2, yn + k1 / 2 ),
xn = x0 + n h.

5. Third-order Runge-Kutta Method
• This method is a third order Runge-Kutta
method for approximating the solution of the
initial value problem y'(x) = f(x,y); y(x0) = y0
which evaluates the integrand,f(x,y), three
times per step.

6. The essential formulas to compute the value of
y(n+1):
yn+1 = yn + 1/6 ( k1 + 4 k2 + k3 ),
k1 = h f(xn, yn),
k2 = h f(xn + h / 2, yn + k1 / 2 ),
k3 = h f(xn + h, yn - k1 + 2 k2 ),
xn = x0 + n h.

7. Higher Order Runge-Kutta Methods
• Runge-Kutta methods of any order can be
derived, although the derivation of an order
higher than four can become extremely
complicated. The table below gives the order
of the method and the minimum number of
stages required.

8. Because the fifth-order method requires two
more stages than the fourth-order method,
the fourth order method has found some
popularity.

9. Classical Runge-Kutta Fourth Order Method
• This method is the classical fourth order
Runge-Kutta method for approximating the
solution of the initial value problem y'(x) =
f(x,y); y(x0) = y0 which evaluates the
integrand,f(x,y), four times per step.

10. The essential formulas to compute the value of
y(n+1):
yn+1 = yn + K
yn+1 = yn + 1/6 ( k1 + 2 k2 + 2 k3 + k4 ),
k1 = h f(xn, yn),
k2 = h f(xn + h / 2, yn + k1 / 2 ),
k3 = h f(xn + h / 2, yn + k2 / 2 ),
k4 = h f(xn + h, yn + k3 ),
xi = x0 + n h.

11. Sample Problem:
• Apply the Runge-Kutta method of 4th order to
find approximate value of y at x=0.2 if given
that y=1 when x=0 in steps of h=0.1.

12. Solution: