On mild solutions of the p-Laplacian fractional Langevin equations with anti-periodic type boundary conditions
The python codes here are to perform the numerical experiments:
- Example 4.1, (a) and (b)
- Example 4.2
- Example 4.3
- (a) without noise to input
- (b) with random noise to input
To do, we have 5 tasks. Three first tasks should be done at first and only once. We will focus only on two last ones.
a) Windows:
b) Linux:
c) MacOS:
a) Open a Command Prompt (Windows) or Terminal (Linux)
b) Run the following commands
conda install numpy
conda install scipy
conda install matplotlib
Download https://codeload.github.com/tranqv/GCOM-2021-0117/zip/refs/heads/main
and extract the zip file. One should have GCOM-2021-0117-main folder.
There are three subdirectories:
Three examples are prepared with their main programs as follows:
- main_ex01a.py: Example 1a,
- main_ex01b.py: Example 1b,
- main_ex02a.py: Example 2,
- main_ex03a.py: Example 3a, without noise,
- main_ex03b.py: Example 3b, with noise to the input.
All the programs above perform Procedure (P) in Section 4 of the manuscript. Read comments included in and messages inside the print() function of the codes to gain information of the computation steps, and check with output to the screen when executing.
Settings for the examples are given in inc_ex1a.py, inc_ex1b.py, inc_ex2a.py, inc_ex3a.py, inc_ex3b.py.
Auxiliary procedures for the computation are defined in inc_sub2.py, while the ones for plotting are in inc_post.py.
We provide figures in JPG, which can be compared with ones output from the codes running on other computers.
We provide numerical results in CSV format, which can be used to check with the output on other computers.
a) Method 1: Using Jupyter Notebook (installed together with Anaconda)
Change directory to GCOM-2021-0117-main, and to code, then we click on one of the following files
- main_ex01a.ipynb: Example 1a,
- main_ex01b.ipynb: Example 1b,
- main_ex02a.ipynb: Example 2,
- main_ex03a.ipynb: Example 3a, without noise,
- main_ex03b.ipynb: Example 3b, with noise to the input.
They are copied from the corresponding file main_ex???.py to run in Jupyter Notebook.
Click to choose a file, move to the new tab, then click on Kernel and Restart & Run All to execute the program.
b) Method 2: Using Anaconda Prompt (Windows) or Terminal (Linux)
-
Change directory to GCOM-2021-0117-main, and to code
-
Inside code, run the examples using the following commands
python3 main_ex01a.py
python3 main_ex01b.py
python3 main_ex02a.py
python3 main_ex03a.py
python3 main_ex03b.py
where the first command is for Example 1a, and so on.
- Inside the codes above, the variable lmax plays important role. It is to define the mesh sizes:
N_l = 10 * 2**l
for l = 0,1,2,...,lmax-1. Therefore, to run fully, define
lmax = 11But it will take very long time. To run partly for testing the codes, define
lmax = 4-
At the first time we run one of the above commands, two folders, i.e., figs and xout, will be created.
- figs saves figures from all the main programs in both JPG and PDF formats.
- xout saves the results (numerical mild soluions, error estimates) in ascii format, i.e., .txt and .csv.
-
The main_ex03a.py (or main_ex03a.ipynb) must be excecuted in advance of main_ex03b.py (or main_ex03b.ipynb) since the latter program needs the output from the preceeding one, while the others (for Examples 1a, 1b, 2, and 3a) can run independently.
-
When running for Example 3b (main_ex03b.py or main_ex03b.ipynb) we must ensure that **luref ** is less than or equal to lmax-1, where lmax was mentioned previously.
-
Check messages output to your screen with the log_???.txt files.
-
Check results in figs and xout with ones in zfigs and zxout, respectively.
After we finish running the examples, we can run
main_post.ipynb
with Jupyter Notebook (read 4. Method 2), or invoke the command
python3 main_post.py
in Anaconda Prompt (Windows) or Terminal (Linux) to plot the results of Examples 1a, 1b, 2, 3a. Note that, in main_post.py (or main_post.ipynb), we must set the values of fprefix and lmax accordingly.
All figures for Example 3b were already made when we ran main_ex03b.py (or main_ex03b.ipynb).
-
For Examples 1a, 1b, 2, and 3a, we should run main.ipynb or main.py.
After opening this file (i.e. with Jupyter Notebook for main.ipynb or with a text editor for main.py)
- set the value of lmax moderately, e.g., lmax = 4,
- assign the value of fprefix as it was prepared inside the code, e.g.,
#fprefix = "ex01a" #fprefix = "ex01b" fprefix = "ex02a" # remove the comment "#" to run Example 2 #fprefix = "ex03a"
- run the code as in 4a) Method 1 or 4b) Method 2.
-
For Example 3b, we should run main_ex3b.py or main_ex3b.ipynb.
- set the value of luref to be small, e.g. luref = 3. (We must ensure that luref <= lmax-1)
Make sure that the folder xout does already exists and includes the results of Example 3a, i.e. files with the prefix ex03a.
At the end of the computation process, some warnings about "divide by zero encountered" apprear. The reason is U_N = Uexact = 0 for all N, therefore all related error estimates are also zero. These messages won't crash the program. So, ignore them.
Mittag-Leffler function:
- [1] https://github.com/khinsen/mittag-leffler
- [2] https://github.com/tranqv/Mittag-Leffler-function-and-its-derivative
- [3] https://se.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function
Python distribution platform:
Python libraries:
- [5] https://numpy.org/doc/stable/reference/
- [6] https://docs.scipy.org/doc/scipy/reference/interpolate.html
- [7] https://docs.scipy.org/doc/scipy/reference/special.html
- [8] https://matplotlib.org/
Contact: [email protected]