14 Aug

numerical methods engineering

Posted at 14:57h in 2014/11/9 by

1) For the parabola $f(x)=3+2x+3x^2$ , show that the numerical integration $int_{1}^{3}f(x)dx$ by Simpson's rule yields the exact analytical value, except for the round-off error. What effect would you expect an increase in the number of subdivisions to have on the accuracy of the numerical results? Explain.

2) Consider the expression for blackbody radiation given by Equation 4.62 $E_{blambda }(lambda ,T)=C_{1}/(lambda ^5)[e(^C2/lambda T) -1]$ . The integral of this expression over all wavelengths, that is, $int_{0}^{infinity}E_{blambda }dlambda$ , gives the total energy radiated by a blackbody per unit area and time. Using Simpson's rule, compute this integral at T=1000K as accurately as possible. The analytical result is given in the literature as $sigma T^4$ , where $sigma$ is known as the Stehan-Boltzman constant and has a value of 5.67×10^(-8) W/m^2K^4. Compare your numerical result with the analytical value at 1000K.

The RMS value of an electric current I(t), where I varies periodically with time t is given by the expression $I_{RMS}=frac{1}{t_{c}}sqrt{int_{0}^{t_{c}}}I^2dt$ where $t_{c}$ is the time period for one cycle in the variation of I(t). If I(t) is given as 5e^(-t)sin4pit, with $t_{c}$ =0.5 seconds, compute the RMS value using Simpson's Rule.

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