Friday, January 21, 2011

runge kutta method

Runge-Kutta Method of Order $ 4$

We are now ready to state the algorithm for using the Runge-Kutta method of order $ 4$ . This is a generalization of the R-K method of order $ 2$ to higher order methods. Without getting into analytical details, we state the R-K method of order $ 4$ . It is widely used algorithm. For the IVPs ([*]), set $ x_i = a + i h, $ for $ i=0,1,2, \ldots, n$ where $ h = \displaystyle \frac{b-a}{n}$ . Also, we set $ y(a) = y_0$ . The for $ k = 0,1,2,\ldots, n-1$ , we define

$\displaystyle y_{k+1} = y_k + \frac{k_1 + 2 k_2 + 2 k_3 + k_4}{6},$

where

$\displaystyle k_1 = h f(x_k, y_k), \; k_2 = h f(x_k + \frac{h}{2}, y_k + \frac{... ...{h}{2}, y_k + \frac{k_2}{2}) \; {\mbox{ and }} k_4 = h f(x_k + h, y_k + k_3).$
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Tuesday, January 11, 2011

Regula falsi method

Find the real root of the equation x log10 x – 1.2 = 0 correct upto four decimal places

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