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Numerical Integrations

  1. Trapezoidal method rule
    According to trapezoidal rule for evaluating a definite integral,
    510.png = 515.png 
  2. Simpsons 1/3rd rule
    520.png = 525.png530.png  
  3. Simpsons 3/8th rule
    535.png = 540.png [(y0 + yn) + 3(y1 + y2 + y4 + ... + yn–1)+ 2(y3 + y6 + y9 + ... + yn–3)]

Solution of Differential Equations

  1. Runge-Kutta method
    The rule for finding the increment k of y corresponding to the increment h of x.
    545.png= f (xy) ; y (x0) = y0
    Calculation is done as follows :
    k1 = h f (x0, y0), k2 = 550.png
    k3 = 555.png, k4 = hf (x0 + hy0 + k3)
    k = 560.png (k1 + 2k2 + 2k3 + k4) =565.png (k1 + k4) + 570.png (k2 + k3)
    The approximate value, y1 = y0 + k
  2. Taylor’s series method
    Given 575.png = y′ = f (xy) and f (x0) = y0
    By taylor’s theorem the series about a point x = x0
    y = y0 + (x – x0) (y′)0 + 580.png
    From this equation we can find y1 of y for x = x1 and then y, y, y″′ are found out.
  3. Euler’s method
    Given 585.png = y′ = f (xy) and y(x0) = y0
    when h  0, by Taylor’s series
    y(x + b) = y (x) + h y′(x) = y(x) + h f (xy)
    Thus, yn+1 = yn h fxnyn)
    where, h = 590.pngi.e. (xn = x0 + nh)




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