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Final Exam (20232)

Final Exam (20232)

FIZ228 - Numerical Analysis
Dr. Emre S. Tasci, Hacettepe University

05/06/2024

import numpy as np
import scipy.optimize as opt
import pandas as pd
import matplotlib.pyplot as plt

1

Consider the data given in the FIZ228_20232_Final_data.csv file. It contains 100 measurements taken for various \(x\) values.

Two models have been suggested for the mechanism behind the event: Gaussian and Lorentzian, defined respectively as:

\[\mathcal{G}(x;\mu,\sigma) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]\]
\[\mathcal{L}(x;x_0,\gamma)=\frac{1}{\pi}\left[\frac{\gamma}{(x-x_0)^2+\gamma^2}\right]\]

Fit the data into these two models, calculate the errors and decide the best model for the data by stating your justification.

Bonus: Can you come up with a better model? (Not talking about a polynomial fit of 100. order!) If you can, propose your model, fit it and calculate the error. (Hint, plotting the optimized Gaussian, Lorentzian and the data might give you an idea ;)

2

Solve the following ODE for the given conditions:

\[\begin{split}y''-y'-2y=e^{3x}\\ y(0) = 1.8500,\quad y(1) = 25.9714,\quad x\in[0,1];\quad h\le0.01\end{split}\]

Bonus:

Its analytical solution is: \(y(x) = -1.3e^{-x}+2.9e^{2x}+\frac{1}{4}e^{3x}\)

Calculate the total RMS error defined as:

\[\sqrt{\frac{1}{N}\sum_i{(y_i-t_i)^2}}\]

where \(y\) is the model estimate and \(t\) is the analytical solution’s value (i.e., true value).

3

Solve the following ODE for the given conditions:

\[\begin{split}y' = y -x\\y(1)=4.71828,\quad x\in [1,2];\quad h\le0.01\end{split}\]

Bonus:

Its analytical solution is: \(y(x)=e^x+x+1\)

Calculate the total RMS error defined as:

\[\sqrt{\frac{1}{N}\sum_i{(y_i-t_i)^2}}\]

where \(y\) is the model estimate and \(t\) is the analytical solution’s value (i.e., true value).

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