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201-Lect 8 - The central difference scheme is more accurate...

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MECH201 Engineering Analysis (2009) p. 11 The central difference scheme is more accurate than the other twos as it is of order Δ x 2 . See figures 23.1 to 23.3 for higher order representation of derivative of a function. The higher order or accuracy can be achieved by using more points in the neighborhood of x i . Note that 1 i i x x x ± = ± Δ . For second derivative of f(x), consider two additional points on each side of x i : 1 1/ 2 1/ 2 1 1 1 ; ; ; & 2 2 i i i i i i i i i x x x x x x x x x x x x x - - + + = - Δ = - Δ = + Δ = + Δ Using central difference scheme: 1 1/ 2 i i i f f df dx x - - -   =   Δ   and 1 1/ 2 i i i f f df dx x + + -   =   Δ   2 1 1 2 i i i df df d f dx dx dx x + -     -         = Δ 1 1 2 1 1 2 2 2 i i i i i i i i f f f f f f f d f x x dx x x + - + - - -   -   - + Δ Δ   = = Δ Δ Numerical differentiation of experimental data can easily lead to error if the data is not smooth. As slight deviation of the data point from a smooth curve will render large error in the differentiated results. See Figure 23.5 for graphical explanation.
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MECH201 Engineering Analysis (2009) p. 12 4 SOLUTION OF ORDINARY DIFFERENTIAL EQUATION. 4.1 The Ordinary Differential Equation Pages 697-705 The ordinary differential equation (ODE) is commonly found in the governing equations of many engineering system. For example, the damped simple harmonic motion equation: 2 2 0 d x dx m c kx dt dt + + = It can be in the form of a system of equations. In the case of a 2-D projectile we have two equations: 2 2 d y m g dt dx m V dt = - = The word "Ordinary" that precedes the "Differential Equation" Implies that the differentiations are all "total differentiation" In contrast to "partial differentiation" which will be the topic In latter sections. We shall see that the methods of solving ordinary differential equations Is different to that used to solve partial differential equations. The aim of this section is to consider four commonly used methods for solving ODEs. These are: the Euler method, the Huen method, the mid-point method and the Rung-Kutta method. 4.2 The Euler Method Section 25.1 Consider an ordinary differential equation: ( ) , dy f y x dx = and the value of y at x = 0 is given, that is y(0) = y 0 . To find the value of y at x = x 1 where x 1 -x = Δ x We may use the Taylor’s series to obtain the value of y( Δ x) form y(0): ( ) ( ) 2 2 2 0 0 1 0 ....
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