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Chapter 22: Problem 1

Use order of \(h^{8}\) Romberg integration to evaluate $$\int_{0}^{3} x e^{x} d x$$ Compare \(\varepsilon_{a}\) and \(\varepsilon_{t}\)

Short Answer

Expert verified
To evaluate the integral \(\int_{0}^{3} xe^{x} dx\) using the order \(h^{8}\) Romberg integration, first, define the function and limits. The next step is to calculate the true value of the integral analytically, which is around 40.185. Then, apply the Romberg method up to \(h^{8}\) by successively calculating \(R_{n,m}\) values starting from \(R_{1,1}\) using the trapezoidal rule and performing Richardson extrapolation at each level. Once the \(h^{8}\) approximation is obtained, calculate the absolute error (\(\varepsilon_{a}\)) as the difference between the approximations at orders \(h^{8}\) and \(h^{6}\). The true error (\(\varepsilon_{t}\)) is calculated from difference between the true value of the integral and the approximation at order \(h^{8}\). Finally, compare the errors to assess the accuracy.

Step by step solution

01

Define the function and limits

First, let's define the function that we need to integrate: \(f(x) = xe^{x}\). The given limits of integration are \(a = 0\) and \(b = 3\).
02

Calculate true value of the integral

To compare the error of our result, we need to find the true value of the integral analytically: \[\int_{0}^{3} xe^{x} dx = e^{x}(x-1)\Big|_{0}^{3} = e^{3}(3-1) - e^{0}(0-1) = 2e^{3} + 1\approx 40.185\]
03

Apply Romberg method

Now, let's apply the Romberg method to find the \(h^{8}\) approximation: 1. Calculate \(R_{1,1}\) using the trapezoidal rule: \[R_{1,1} = h(f(a) + f(b))/2\] 2. Calculate \(R_{2,1}\) using the trapezoidal rule with twice as many intervals: \[R_{2,1} = R_{1,1}/2 + h \sum_{k=1}^{N/2} f(a+(2k-1)h)/2\] 3. Calculate \(R_{2,2}\) using Richardson extrapolation: \[R_{2,2} = (4R_{2,1} - R_{1,1})/3\] 4. Repeat this process increasing the order of the approximation until reaching the desired order \(h^{8}\), by calculating \(R_{3,1}, R_{3,2},\) and so on, and applying Richardson extrapolation at each level.
04

Calculate and compare errors

After applying the Romberg method up to \(h^{8}\), we can calculate the absolute error (\(\varepsilon_{a}\)) and true error (\(\varepsilon_{t}\)). Absolute error (\(\varepsilon_{a}\)) is the difference between the approximations at orders \(h^{8}\) and \(h^{6}\): \[\varepsilon_{a} = |R_{4,4} - R_{3,3}|\] True error (\(\varepsilon_{t}\)) is the difference between the true value of the integral and the approximation at order \(h^{8}\): \[\varepsilon_{t} = |R_{4,4} - (2e^{3} + 1)|\] Compare the errors to assess the accuracy of our Romberg method result.

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Most popular questions from this chapter

Use Romberg integration to evaluate $$I=\int_{1}^{2}\left(2 x+\frac{3}{x}\right)^{2} d x$$ to an accuracy of \(\varepsilon_{s}=0.5 \%\) based on Eq. \((22.9) .\) Your results should be presented in the form of Fig. \(22.3 .\) Use the analytical solution of the integral to determine the percent relative error of the result obtained with Romberg integration. Check that \(\varepsilon_{t}\) is less than the stopping criterion \(\varepsilon_{s}\)

The amount of mass transported via a pipe over a period of time can be computed as $$M=\int_{t_{1}}^{t_{2}} Q(t) c(t) d t$$ where \(M=\operatorname{mass}(\mathrm{mg}), t_{1}=\) the initial time \((\min ), t_{2}=\) the final time \((\min ), Q(t)=\) flow rate \(\left(\mathrm{m}^{3} / \mathrm{min}\right),\) and \(c(t)=\) concentration \(\left(\mathrm{mg} / \mathrm{m}^{3}\right) .\) The following functional representations define the temporal variations in flow and concentration: $$\begin{array}{l} Q(t)=9+4 \cos ^{2}(0.4 t) \\ c(t)=5 e^{-0.5 t}+2 e^{0.15 t} \end{array}$$ Determine the mass transported between \(t_{1}=2\) and \(t_{2}=8\) min with Romberg integration to a tolerance of \(0.1 \%\)

Develop a user-friendly computer program for the multiplesegment (a) trapezoidal and (b) Simpson's \(1 / 3\) rule based on Fig. \(22.1 .\) Test it by integrating $$\int_{0}^{1} x^{0.1}(1.2-x)\left(1-e^{20(x-1)}\right) d x$$ Use the true value of 0.602298 to compute \(\varepsilon_{t}\) for \(n=4\)

There is no closed form solution for the error function, $$\operatorname{erf}(a)=\frac{2}{\sqrt{\pi}} \int_{0}^{a} e^{-x^{2}} d x$$ Use the two-point Gauss quadrature approach to estimate erf(1.5). Note that the exact value is 0.966105

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