What are Partial Differential Equations?
Channel: MechanicaLEi
Date: 03/12/2018

Welcome back MechanicaLEi, did you know that Partial Differential equations can be used to describe a variety of phenomenon such as heat, fluid dynamics, sound etcetra? This makes us wonder, What are Partial differential equations?
Before we jump in check out the previous part of this series to learn about what Parsvel's identities are? 
Now, A differential equation involving partial derivatives of one or more dependent variables with more than one independent variable is called a partial differential equation. The order of the highest derivative term in the equation is called the order of the PDE. Above equations are all of the second order. It is also a common practise to use subscript notation in writing partial differential equations. For example the Laplace Equation in three dimensional space dou squared u upon dou x squared plus dou squared u upon dou y squared plus dou squared u upon dou z squared equals to zero can be written as u x x plus u y y plus u z z equals to zero.
First method to find a solution for PDE's is the Bender-Schmidt Explicit Method. Consider, a general equation of the form dou u upon dou t equals to alpha squared into dou squared u upon dou x squared. The explicit formula is given by the following equation; where lambda equals to k upon a h squared. The explicit formula is only valid for values of zero less than lamba less than or equal to one by two. If lamba is equal to one by two then the explicit formula is reduced to ui comma j plus one equals to one by two into u i plus one comma j plus lambda into u i minus one comma j. This formula is called Bender-Schmidt formula. The values of i and j are then filled in a table to present the solution.
Hence, we first saw what Partial Differential Equations are and then went on to see how Numerical Solution of PDE are obtained using Bender-Schmidt Explicit Method?
In the next episode of MechanicaLEi find out what Crank-Nicolson Method is?

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