Read and understand the posted solution to HW 8. The goal here is to see another solution besides your own, to get coding ideas, and to think about things that your or I could do better. Just state if you did this.
Integrate the following equation using the Trapazoid Rule, Simpson's rule, and Gauss-Legendre quadrature from x=0 to x=1:
$$f(x) = x\cdot\tanh(d\cdot(x-0.5))+1.$$Use d=10. Make a log-log plot of the relative error of the methods versus the number of grid points. To get the "exact" integral, you can use the built-in integrator.
In python, use the following:
from scipy.integrate import quad
(Ie, abserr) = quad(f,0,1)
When varying the number of points, because we are plotting on a log scale, and because the error goes as a power, we want the number of points to increase as a power. I did something like:
npoints = 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9, 2^10,
for 10 runs. Then, when you plot on a log scale you'll have an even spacing of points on the x-axis.
Use your results to verify that the trapazoid and Simpson methods have convergence rates $O(\Delta x^2)$ and $O(\Delta x^4)$, respectively.
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import quad
This problem is from Hoffman's book: Chapter 4 problem 106 (b).
The following temperature and rate data were measured:
T(K) | K$_f$ |
---|---|
1000 | 7.5E15 |
2000 | 3.8E15 |
3000 | 2.5E15 |
4000 | 1.9E15 |
5000 | 1.5E15 |
Find parameters $B$, $\alpha$, and $E/R$ for the model
$$K = BT^{\alpha}\exp\left(-\frac{E}{RT}\right).$$Plot the model with the best coefficients with a smooth line (use lots of T points). Also plot the measured points using data markers (not lines).
Evaluate the function $f(x)=\exp(4x)$ at $x=0.55$ using a cubic spline fit to points $x=0,\,0.2,\,0.4,\,0.8,\,1.0.$ Also report the relative error at $x=0.55$.