What are applications of Partial differential equations?
Channel: MechanicaLEi
Date: 2018-12-05

Welcome back MechanicaLEi, did you know that unlike ordinary differential equations which deal with one dimensional dynamics systems, Partial differential equations can solve multi dimensional dynamic systems? This makes us wonder, What are applications of Partial differential equations?
Before we jump in check out the previous part of this series to learn about what Crank-Nicolson Method is? 
Now, Consider the transverse vibrations in an elastic membrane. The membrane subjected to a transverse load (perpendicular to the plane of the frame) of magnitude q of x comma y, where x comma y is a system of Cartesian coordinates in the plane of the unloaded membrane. We are interested in calculating the transverse deflection w at w of x comma y, corresponding to an equilibrium configuration. The transverse vibrations of a tensed membrane are given by solving the PDE equation w xx plus w yy equals to minus q upon T plus rho h upon T into wtt. Here, h is the thickness of the membrane. 
One another application of PDEs was the heat flow equation which is given by dou u of x comma t upon dou t equals to delta u of x comma t. where delta operator equals to summation of dou squared upon dou x squared i, from i equals one to three. The heat flow equatin was based on the Newton's Law of Cooling. Hence, we saw applications of PDE, first in transverse vibrations of an elastic membrane and then in the  Heat equation for heat flow equation.
In the next episode of MechanicaLEi find out what two and three dimensional Laplace equations are?


Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library