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            <div style="font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%">Computational Seismology</div>
            <div style="font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)">Numerical derivatives based on the Fourier Transform</div>
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Seismo-Live: http://seismo-live.org

##### Authors:
* Fabian Linder ([@fablindner](https://github.com/fablindner))
* Heiner Igel ([@heinerigel](https://github.com/heinerigel))
* David Vargas ([@dvargas](https://github.com/davofis))
---

## Basic Equations
The derivative of function $f(x)$ with respect to the spatial coordinate $x$ is calculated using the differentiation theorem of the Fourier transform:

\begin{equation}
\frac{d}{dx} f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} ik F(k) e^{ikx} dk
\end{equation}

In general, this formulation can be extended to compute the n−th derivative of $f(x)$ by considering that $F^{(n)}(k) = D(k)^{n}F(k) = (ik)^{n}F(k)$. Next, the inverse Fourier transform is taken to return to physical space. 

\begin{equation}
f^{(n)}(x) = \mathscr{F}^{-1}[(ik)^{n}F(k)] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} (ik)^{n} F(k) e^{ikx} dk
\end{equation}



```python
# Import all necessary libraries, this is a configuration step for the exercise.
# Please run it before the simulation code!
import numpy as np
import matplotlib.pyplot as plt

# Show the plots in the Notebook.
plt.switch_backend("nbagg")
```

#### Exercise 1

Define a python function call "fourier_derivative(f, dx)" that compute the first derivative of a function $f$ using the Fourier transform properties. 


```python
#################################################################
# IMPLEMENT THE FOURIER DERIVATIVE METHOD HERE!
#################################################################
```

#### Exercise 2

Calculate the numerical derivative based on the Fourier transform to show that the derivative is exact. Define an arbitrary function (e.g. a Gaussian) and initialize its analytical derivative on the same spatial grid. Calculate the numerical derivative and the difference to the analytical solution. Vary the wavenumber content of the analytical function. Does it make a difference? Why is the numerical result not entirely exact?


```python
# Basic parameters
# ---------------------------------------------------------------
nx = 128
x, dx = np.linspace(2*np.pi/nx, 2*np.pi, nx, retstep=True) 
sigma = 0.5
xo = np.pi

#################################################################
# IMPLEMENT YOUR SOLUTION HERE!
#################################################################

```

#### Exercise 3

Now that the numerical derivative is available, we can visually inspect our results. Make a plot of both, the analytical and numerical derivatives together with the difference error.   


```python
#################################################################
# PLOT YOUR SOLUTION HERE!
#################################################################
```

97f8b509bb37d5a49ab29ec324f2485fb2c0676e
ipynb
Jupyter Notebook
pseudospectral/ps_derivative.ipynb
cheshirepezz/PDE
75e829c4f52a570d2551574b97396f32cc9fb893
MIT
2020-12-11T14:43:22.000Z
2020-12-12T09:15:32.000Z
pseudospectral/ps_derivative.ipynb
cheshirepezz/PDE
75e829c4f52a570d2551574b97396f32cc9fb893
MIT
pseudospectral/ps_derivative.ipynb
cheshirepezz/PDE
75e829c4f52a570d2551574b97396f32cc9fb893
MIT
2022-03-15T22:15:42.000Z
2022-03-15T22:15:42.000Z
Qwen/Qwen-72B
1. YES
2. YES

