Formulas for Numerical Methods

In what follows, \( f_m = f(x_m) \) and \( y_m = y(x_m) \).
Examples:
\( f_0 = f(x_0) \) , \( f_1 = f(x_1) \) , \( f_2 = f(x_2) \) ...
\( y_0 = y(x_0) \) , \( y_1 = y(x_1) \) , \( y_2 = y(x_2) \) ...

Trapezoidal Rule for Integration

\( \displaystyle \int_{x_0}^{x_m} f(x) dx = h \left( \dfrac{f_0}{2} + f_1 + f_2 + f_3 + ... + f_{m-1}+\dfrac{f_m}{2}\right) \)
\( h = \dfrac{x_m - x_0}{m} \)

Simpson's Rule for Integration

The number of intervals must be even and is taken as \( 2 m \)
\( \displaystyle \int_{x_0}^{x_{2m}} f(x) dx = \dfrac{h}{3} \left[ f_0 + 4( f_1 + f_3 + f_5 + ...) + 2 ( f_2 + f_4 +f_6 + .... )+ f_{2m} \right] \)
\( h = \dfrac{x_{2m} - x_0}{2 m} \)

Euler's Method for Differential Equations

\( y' = f(x,y) \)
\( y_{n+1} = y_n + h f(x_n , y_n) + O(h^3) \)

Second-Order Runge-Kutta Method for Differential Equations

\( y' = f(x,y) \)
\( y_{n+1} = y_n + \dfrac{1}{2} (k_1 + k_2) + O(h^3) \)
where
\( k_1 = h f(x_n , y_n) \)
\( k_2 = h f(x_n+h , y_n+k_1) \)

Fourth-Order Runge-Kutta Method for Differential Equations

\( y' = f(x,y) \)
\( y_{n+1} = y_n + \dfrac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) + O(h^5) \)
where
\( k_1 = h f(x_n , y_n) \)
\( k_2 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} ) \)
\( k_3 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} ) \)
\( k_4 = h f(x_n+ h , y_n + k_3) \)

Second-Order Runge-Kutta Method for Systems of Differential Equations

\( y' = f(x,y,z) \) , \( z' = g(x,y,z) \)
\( y_{n+1} = y_n + \dfrac{1}{2} (l_1 + l_2) \) *(Note: symmetric formulations apply)*
\( y_{n+1} = y_n + \dfrac{1}{2} (k_1 + k_2) + O(h^3) \)
\( z_{n+1} = z_n + \dfrac{1}{2} (l_1 + l_2) + O(h^3) \)
where
\( k_1 = h f(x_n , y_n , z_n) \) , \( l_1 = h g(x_n , y_n , z_n) \)
\( k_2 = h f(x_n+h , y_n + k_1 , z_n + l_1) \) , \( l_2 = h g(x_n+h , y_n + k_1 , z_n + l_1) \)

Fourth-Order Runge-Kutta Method for Systems of Differential Equations

\( y' = f(x,y,z) \) , \( z' = g(x,y,z) \)
\( y_{n+1} = y_n + \dfrac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) + O(h^5) \)
\( z_{n+1} = z_n + \dfrac{1}{6} (l_1 + 2 l_2 + 2 l_3 + l_4) + O(h^5) \)
where
\( k_1 = h f(x_n , y_n , z_n) \) , \( l_1 = h g(x_n , y_n , z_n) \)
\( k_2 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} , z_n + \dfrac{l_1}{2}) \) , \( l_2 = h g(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} , z_n + \dfrac{l_1}{2} ) \)
\( k_3 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} , z_n + \dfrac{l_2}{2} ) \) , \( l_3 = h g(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} , z_n + \dfrac{l_2}{2}) \)
\( k_4 = h f(x_n + h , y_n + k_3 , z_n + l_3) \) , \( l_4 = h g(x_n + h , y_n + k_3 , z_n + l_3) \)

Note: The fourth-order Runge-Kutta method for systems of differential equations is more stable. I used it in my doctoral thesis on "Evanescent Field Effects on Pulse Propagation in Transversely Inhomogeneous Waveguides", page 43, and it yielded excellent results.