Numerically Solving Partial Differential Equations

In this video we show how to numerically solve partial differential equations by numerically approximating partial derivatives using the finite difference method. We develop an algorithm that will allow direct numerical simulation of a partial differential equation.

Topics and Timestamps:
0:00 – Introduction
1:04 – Fokker-Planck equation
7:45 – Verifying and visualizing the analytical solution in Mathematica
15:29 – The Finite Difference Method
57:57 – Converting a continuous PDE into an algebraic equation
1:07:57 – Boundary conditions
1:14:29 – Math Joke: Star Wars error
1:15:53 – Implementation of numerical solution in Matlab

A sample Mathematica notebook that accompanies this lecture is located at


Additional videos in this series:

Associated videos on software tools relevant to PDEs include:

You can see all the videos related to partial differential equations at

#PDEs

[iremerkan]

AE501: The animation was very clear to me and it clearly shows the difference between numerical and analytical solution when we change certain parameters. Really interesting topic!

[danielahn]

AE501: Great explanation of Numerically solving partial differential equations to finish the rest of my homework. Thank you

[KevinCastaneda]

AE501: I can most definitely appreciate the art or skill it takes to choose the correct parameters for a numerical solution to a PDE. In my mind its in the form of composing a song or a dance where the timing and the steps have to be correct so that the dance falls in rythem. This really sets the stage for some interesting question down the road for in a FEM course and the challenges that will come with it. Great lecture thank you professor.

[lutherbrown]

You are a fantastic teacher. Very logical, clear, and easy to follow.

[mostafaahmadi]

Just incredible. What I learnt in 2 hours might have taken me days to learn from a book. Thank you very much for your time and effort.

[tarynz]

AE 501 - Thanks for stepping through the derivation with clear explanations and graphics. The demo at the end really showed how sensitive the solution is just by tweaking the step size slightly.

[sabrinusglaucomys]

Thanks, this was great! I'm taking a class where we need to use this method, but I haven't taken a numerical methods class or anything beyond the barest intro to differential equations. I looked through a lot of other resources trying to understand quickly, but none of them made sense at my level of background knowledge. I've come to really appreciate teaching styles like this where you include enough information so people who haven't been immersed in this stuff recently can still learn what they need to know––it turns out extensive prereqs actually aren't needed to get a working knowledge of a lot of concepts and techniques! And it's great in video form b/c if something is too basic, you can always skip ahead.

[carlhitchon]

Excellent presentation. Your careful step by step approach and the precision of your wording made this comprehensible! Thank you.

[jackpascho]

AE 501: Glad we got to cover this, fun to get to mess around with the different values while getting a useful script template for future use!

[RajChhabraUW]

AE501: Great introduction to numerical solutions to PDEs!

[evansteele]

AE501: This seems to be a well rounded conclusion to our discussion of PDEs for the scope of this class. Numerically solving these types of problems is definitely a practical area to practice and implement. I know these types of solutions are their own field of study, so good job condensing the topic into this video!

[astall]

AE501: A really powerful lecture. To be honest, I was dreading the numerical portion of PDE's in this class but your explanation of Finite Difference Methods was practical and straight forward. Thank you.

[connorbaldwin]

AE 501 This was a nice way to have all of our PDE work come together. I can see how people spend their careers working with PDEs

[glopez]

Not all unicorn, glitter and sprinkles. Very powerful technique. A lot of art. Thanks for explaining complicated PDE solns. (AE501).

[jasonwhyte]

Really great visual representation of the finite difference method! Gives a good visualization of how CFD tools are doing the heavy lifting.

[michaelcook]

AE 501 - Michael Cook. Thanks Professor, I find the videos where we build the matlab scripts with the explanations at each step EXCEPTIONALLY helpful, really helped solidify the text/chalkboard explanation of numerical solutions to matlab practice and also visualizing what these approximations are doing.

[mattpierucci]

AE 501 - It has been a long time since my numerical methods class so I really appreciated this refresher. It's nice to have a better understanding of what is going on under the hood when I'm using FEA software to analyze a bit of structure at work. I think this was probably the most interesting lecture for me thus far, probably because I can clearly see how it relates to my work on a daily basis.

[sajeew]

Professor, this is really helpful to understanding the partial differential equations by numerically. After coming back to school after long time working going to be little hard but these vidos helpful distance learners like us. Thank you

[charlesharmon]

Good to have a practical alternate method to solve more complicated systems. Thank you for the tutorial!

[philipcasey]

AE501 - At 24:40, really like the 3D diagram and the projections, very helpful for visualizing what's going on.