Visualizations of some specific solutions of different differential equations.

Overview

Diff_sims

Visualizations of some specific solutions of different differential equations.

Heat Equation in 1 Dimension

(A very beautiful and elegant explanation for the heat equation and its solution is given here

  • The heat diffusion equation gives us the temperature at every part of a conducting 1 dimensional material (specific to the current discussion), given we specify the initial condition, and the boundary conditions.
  • The thing to be noted here is that the differential equation has a doble derivative with respect to position and a single derivative with respect to time.
  • Hence, one can say that the curvature of the position function and the slope of the time function (of course, assuming one is able to write the solution as independent functions of position and time, which we can) are the key playing factors which determine how the temperature gets diffused, so to speak.
  • The co-efficient D is the thermal diffusivity constant, and here, can be thought of to be some positive constant. (Let's henceforth assume D = 1)

heat_eqn

The Gaussian

  • If the initial temperature distribution were to be in the form of a gaussian:

gauss

This function looks like a bell shaped curve centred at the origin. In other words, it tells us that at the very centre of the rod, the temperature is the highest, and as we move along in either direction it decreases very fast, and some distance from the centre, the temperature is almost zero. This ditribustion might be due to a source which is in contact with the conductor initially, exactly at the centre.

Laying off rigor (we haven't discussed about the boundary conditions here, but a gaussian converges rather rapidly as the magnitude of x increases, so essentially we can assume that the function tends to zero at its boundaries), the solution can be written in the following manner (setting certain constats to 1):

Soln

The denominator in the exponential part, actually tells us how much spread out the bell shape is, the higher it's value, more spread out it will be. The factor multiplied in the beginning is a normalizing factor, which makes sure that the energy remains constant at all times.

Hence, intuitively, we can see that as time increases, the temperature function smooths out.

  • The above solution is by no means accurate but it helps us understand the behavior.

  • One more thing to be noted, is that as the heat starts spreading, it takes more and more time for it to move farther. This is because temperature is an axponentially decaying function of time. Link to simulation here

A curated list of awesome Dash (plotly) resources

Awesome Dash A curated list of awesome Dash (plotly) resources Dash is a productive Python framework for building web applications. Written on top of

Luke Singham 1.7k Jan 07, 2023
Interactive plotting for Pandas using Vega-Lite

pdvega: Vega-Lite plotting for Pandas Dataframes pdvega is a library that allows you to quickly create interactive Vega-Lite plots from Pandas datafra

Altair 342 Oct 26, 2022
a plottling library for python, based on D3

Hello August 2013 Hello! Maybe you're looking for a nice Python interface to build interactive, javascript based plots that look as nice as all those

Mike Dewar 1.4k Dec 28, 2022
A python visualization of the A* path finding algorithm

A python visualization of the A* path finding algorithm. It allows you to pick your start, end location and make obstacles and then view the process of finding the shortest path. You can also choose

Kimeon 4 Aug 02, 2022
Pretty Confusion Matrix

Pretty Confusion Matrix Why pretty confusion matrix? We can make confusion matrix by using matplotlib. However it is not so pretty. I want to make con

Junseo Ko 5 Nov 22, 2022
Data Analysis: Data Visualization of Airlines

Data Analysis: Data Visualization of Airlines Anderson Cruz | London-UK | Linkedin | Nowa Capital Project: Traffic Airlines Airline Reporting Carrier

Anderson Cruz 1 Feb 10, 2022
Python library that makes it easy for data scientists to create charts.

Chartify Chartify is a Python library that makes it easy for data scientists to create charts. Why use Chartify? Consistent input data format: Spend l

Spotify 3.2k Jan 04, 2023
Bcc2telegraf: An integration that sends ebpf-based bcc histogram metrics to telegraf daemon

bcc2telegraf bcc2telegraf is an integration that sends ebpf-based bcc histogram

Peter Bobrov 2 Feb 17, 2022
Render tokei's output to interactive sunburst chart.

Render tokei's output to interactive sunburst chart.

134 Dec 15, 2022
cqMore is a CadQuery plugin based on CadQuery 2.1.

cqMore (under construction) cqMore is a CadQuery plugin based on CadQuery 2.1. Installation Please use conda to install CadQuery and its dependencies

Justin Lin 36 Dec 21, 2022
Define fortify and autoplot functions to allow ggplot2 to handle some popular R packages.

ggfortify This package offers fortify and autoplot functions to allow automatic ggplot2 to visualize statistical result of popular R packages. Check o

Sinhrks 504 Dec 23, 2022
Create 3d loss surface visualizations, with optimizer path. Issues welcome!

MLVTK A loss surface visualization tool Simple feed-forward network trained on chess data, using elu activation and Adam optimizer Simple feed-forward

7 Dec 21, 2022
Plotting library for IPython/Jupyter notebooks

bqplot 2-D plotting library for Project Jupyter Introduction bqplot is a 2-D visualization system for Jupyter, based on the constructs of the Grammar

3.4k Dec 30, 2022
I'm doing Genuary, an aritifiacilly generated month to build code that make beautiful things

Genuary 2022 I'm doing Genuary, an aritifiacilly generated month to build code that make beautiful things. Every day there is a new prompt for making

Joaquín Feltes 1 Jan 10, 2022
Plot toolbox based on Matplotlib, simple and elegant.

Elegant-Plot Plot toolbox based on Matplotlib, simple and elegant. 绘制效果 绘制过程 数据准备 每种图标类型的目录下有data.csv文件,依据样例数据填入自己的数据。

3 Jul 15, 2022
A high performance implementation of HDBSCAN clustering. http://hdbscan.readthedocs.io/en/latest/

HDBSCAN Now a part of scikit-learn-contrib HDBSCAN - Hierarchical Density-Based Spatial Clustering of Applications with Noise. Performs DBSCAN over va

Leland McInnes 91 Dec 29, 2022
Colormaps for astronomers

cmastro: colormaps for astronomers 🔭 This package contains custom colormaps that have been used in various astronomical applications, similar to cmoc

Adrian Price-Whelan 12 Oct 11, 2022
Friday Night Funkin - converts a chart from 4/4 time to 6/8 time, or from regular to swing tempo.

Chart to swing converter As seen in https://twitter.com/i_winxd/status/1462220493558366214 A program written in python that converts a chart from 4/4

5 Dec 23, 2022
Missing data visualization module for Python.

missingno Messy datasets? Missing values? missingno provides a small toolset of flexible and easy-to-use missing data visualizations and utilities tha

Aleksey Bilogur 3.4k Dec 29, 2022
Application for viewing pokemon regional variants.

Pokemon Regional Variants Application Application for viewing pokemon regional variants. Run The Source Code Download Python https://www.python.org/do

Michael J Bailey 4 Oct 08, 2021