Derivation and Solution of Laplace’s Equation

[@ChristopherLum]

In case it is helpful, here are all my PDE videos in a single playlist kzbin.info/aero/PLxdnSsBqCrrFvek-n1MKhFaDARSdKWPnx. Please let me know what you think in the comments. Thanks for watching!

[@MatthewSWilson-UW]

AE501: Awesome video. I used to get so confused about how to solve PDEs with Fourier series until I watched your videos.

[@EfremNickel]

AE501: Clear and easy to understand lecture as always. Interesting to learn that there are different names for different boundary conditions!

[@KarlaPkva]

AE501 - The explanation on the Dirichlet problem for Laplace's Eq. was good and straightforward.

[@anthonyz2719]

Laplace's equation is everywhere! Thank you for showing how these concepts connect!

[@carlydunford2546]

AE 501: The last step, setting the f(x) equation equal to u(x,y) was a bit confusing to me, but after watching you do the same steps in the heat equation video and other videos, it makes sense. The repetition really helps.

[@jacobgivens2000]

AE501: This was a really clear and straightforward lecture, thank you.

[@timproby7624]

[AE 501] 4:59 Just to clarify, the 1D heat problem in the last video was not a steady state solution. We assumed we had an initial temperature gradient across the bar and then it was changing with time. Whereas this case, we are just trying to determine the temperature gradient in a spatial sense but the temperature does not vary with the time.

[@ChristopherLum]

Right, be careful as the 1D moniker refers to 1 spatial dimension. There was a temporal dimension so there were 2 independent variables, thereby creating a PDE. In the HW we examine a steady state solution so there is no temporal dimension but there are 2 spatial dimensions so in terms of number of independent variables, the two systems are similar.

[@glopez7967]

Great video, and excitement. Interesting how the hyperbolic function of sin comes in and how it relates to the Fourier constants, (AE501).

[@charlesharmon9065]

Very helpful to see the step-by-step processes by which this and other PDEs are derived and solved!

[@alisoncaprioli4861]

Another helpful lecture for the homework! Being thorough in these videos is so helpful. Thanks!

[@keyshawnb4845]

AE 501: This explanation was clear and easy to understand :)

[@enigma5627]

AE501: Jesse Perez - I always find your videos easy to follow. The derivations are straightforward and make lecture notes much more understandable. In this case, at 3:00, the 2D heat problem was simplified nicely with the listing of most terms and with the elimination of redundant components such as the thermal diffusivity due to the steady state solution. Thanks!

[@connorbaldwin5845]

AE 501 Your explanations as always make complicated derivations make sense. I wish this was taught with the same structure during my undergrad. Thank you

[@lienchang8391]

Thank you professor, the overview of deriving and solving PDE's is getting more clear to me. It seems like there's a general step in solving each case...

[@ChristopherLum]

You are welcome. Keep in mind that these clear steps are only good for the situation we are investigating. Not all PDEs work with the separation of variables.

[@paxtonschipper1926]

AE501: Very helpful video, connected the dots and made the concepts easy to understand

[@KennethWright-k2h]

AE501: After watching the 1D wave and heat equation videos, doing the separation of variables solution process is a piece of cake now. Thanks!

[@TuanTran-fs8ll]

AE501-Tuan Tran: I was wondering why u(x,y) has An coefficient instead of An star, but it was corrected toward the end. Thanks professor for very thorough derivation.

[@jasondorn3514]

Thanks for the video. The reference to the wave equation here 17:34 was great to help differentiate the material

[@LorraineB_0911]

Thanks! Armed and ready to tackle the HW! Great video.

[@daniellerogers5959]

Always helpful walking through these derivations of equations.

[@nathanlipshutz3140]

AE501 32:07 It's interesting how all these cases boil down to Fourier Analysis. I guess they are all very specific cases of PDEs. As you mentioned during the class meeting, as soon as they become non-linear they can only be solved numerically.

[@matthewbajamundi8774]

This video really help me understand how the Laplace equation can be used to solver certain systems.

[@thomasireson1427]

Always glad to see the derivation of the equations we use.

[@mitchellhubbard5727]

Great video, lecture notes would help me follow along much better than writing it out myself.

[@HIEPNGUYEN-wb7yg]

Thanks for the very well explanation video! Yes, lecture notes and a summary slide would be helpful.

[@ahmedashmaig]

AE501: Thanks for such an informative video on the application of Laplace's equation!

[@iremerkan725]

AE501: The last example on the rectangular plate for the calculation of An was really helpful and easy to follow.

[@aimeepak717]

AE 501: The derivation for the solution of a Dirichlet problem was very easy to follow. Thanks!

[@juicydangler207]

AE501: Seconding Aimee's comment. Thank you for the video! - Elizabeth Sampley

[@sunnysarkar7643]

AE 501: the construction of relating Fourier to the equation was very clear and easy to follow.

[@GiovanniRiggi-j2q]

AE501B - Johnny Riggi. Cool to see the usage of the LaPlace equation. Everything builds on each other in math!

[@melissawells8133]

Very clear progression through the derivation. Thank you!

[@elijahleonen198]

AE501: Wonderful lecture and helped me understand the concepts for Laplace's equation.

[@garrettdavis4665]

Helpful seeing the simplification explained

[@Tamanaaaa1]

AE 501: Was very helpful! Thank you!

[@NataliaErmolaeva-i8q]

AE501: Thank you Professor for the example with the rectangular plate, the video of pdf was helpful for Problem 2. -Natalia Ermolaeva

[@karlbalitaan]

This was a great video to help me out on the homework!

[@RudiJacobson]

A E 501: At the end (32:38) you mention that this is a solution to the heat transfer equation with a Dirichlet boundary condition, but since the three sides have u = 0 and the top has u = f(x), the temperature is not the same along all sides?

[@jasonfranklin4614]

AE501-Very clear and bitesize derivation. Thanks for putting this together.

[@mrassbandit23]

This was very helpful for understanding the homework problem!

[@fn5002]

AE501: this is really clear and really easy to follow. Farouk Nejah

[@sethwhittington28]

AE501: At the 20:00 timestamp, how did you leap to the general solution for G(y)?

[@joewalker5753]

Definitely a good refresher

[@minaanderberg884]

Definitely getting the hang of when to convert certain variables to An* or Bn*

[@TriMartz-f4x]

[AE501] I hadn't heard of these names of the different boundary conditions before, (Dirichlet, Neumann, Robin). Very interesting!

[@Chuan-YuTsai]

AE501: Good explanation of the separation of variables. I hope there is an example for the second and third boundary value problem.

[@ryancazin5930]

Great video! I agree with the summary slide idea.

[@omarkhalil4676]

Thank You, It is a simple and wonderful explanation

[@ChristopherLum]

I'm glad ti was helpful. I have other videos on PDEs at kzbin.info/aero/PLxdnSsBqCrrFvek-n1MKhFaDARSdKWPnx. Thanks for watching!

[@paulpietrowicz3683]

AE501- What other applications of this initial derivation are there, other than the example you showed of a single sided heat input to a surface?

[@solomondawit8866]

AE501: Thanks for this great video. Very helpful!

[@kylejohnson8447]

when you say n is a member of all real #s or all integers, you are including 0 which is not the case because if n=0 then k=0 but we know k>0. Besides that, great explanation thanks

[@sajeew]

Thank you for showing laplace to heat equation using variables to Bn Star.

[@Richard_Le]

AE501 Why isn't the thermal conductivity of the plate be factored in? Are we assuming the plate will behave identically at steady-state despite being different materials?

[@atruon17uw]

[AE501] I find it useful when there is a video I can reference when I am going through the homeworks.

[@orkidaan6470]

AE501: So we were only able to substitute that first "sinh" term for a constant because our initial condition gave us a constant to plug in for y right? Will there be instances where we will not get a constant in for the initial condition?

[@ChristopherLum]

Orkida, do you have a timestamp regarding this question, it would help me answer more completely, thanks.

[@jenniferjohnson9827]

[AE 501 JENNIFER JOHNSON] This video was really helpful for understanding deriving Laplace's Equation and for my homework.

[@RajChhabraUW]

AE501: Great derivation of solution to Laplace's Equation using Dirichlet BC

[@KarolOrtizSolar]

AE501: Thanks for the great video!

[@ryoonoue1076]

AE501. Very helpful video to solve the 2D heat distribution problem. Thanks

[@mylesdeluca8789]

Thanks for the video!

[@jasonwhyte4162]

Great video!

[@renowhitcomb5505]

AE 501 Fall 2020. Thanks for the helpful video

[@connorbaldwin5845]

A E 501 Thank you for this great lecture on deriving Laplace's Equation

[@robertdelossantos1936]

The video made the material digestible, especially with the prior examples of heat and 1D wave equation to back it up.

[@SScook206]

Great info! thanks

[@ryanmeinhardt81]

AE501 : Easy to follow derivation. Would also be interested to see how the solution would change if the BCs were allowed to be non-zero on the other edges

[@tonykuenzli2182]

AE501 - I found it pretty interesting that the steady state solution for the rectangular plate only depends on the size of the plate and the boundary condition at the edge, especially in comparison to the transient 1D heat transfer case we looked at earlier where the material properties also play a role in the solution.

[@bryanmasserang5190]

Thanks for the explanations

[@marissalamadrid8190]

Clear explanation!

[@MagdaleneRachelleShelton]

AE501: Laplace is mentioned in so many things, woohoo late 1700s French dude still being useful to society today! -Maggie Shelton

[@ChristopherLum]

He really did have a lot of influence in math and physics

[@jesusespinoza2349]

Great video

[@JW-ps8vp]

omg tysm, u saved me!

[@XBI0xDiMZzX]

great video!

[@ChristopherLum]

Thanks for watching!

[@dorismnama9084]

Always is wonderful.

[@ThienNguyen-vx1td]

AE501: Great proof very useful.

[@AurashFilsoof]

AE:501 Thank you professor

[@kylekearsley800]

Good video, thanks!

[@calebwatson6331]

AE501: Reviewing this is helpful, I didn't;'t have the best lecturers in undergrad for heat transfer

[@jonathonlaird6279]

thank you for the video!

[@adamyousri4903]

AE 501: Get video! Goes over the material well

[@ardacetken9090]

AE501: Arda Cetken - Thank you professor

[@calebtuw]

AE501, Caleb T thanks for help with the hw!

[@zakery-pablo]

[A E 501] Thank you Chris!

[@MrThegonz1]

great video, This is Martin Gonzalez, credit plz

[@trevorubrown]

Great vid

[@BennettBoyd-h1f]

AE501: Clear explanation of 2D steady state heat equation

[@helentat121]

I think a summary slide at the end would be helpful

[@ChristopherLum]

There were a few requests for this but I'm a little unclear what type of summary would be helpful, can you provide more details on what you'd like to see?

[@alexsteahr8458]

All this practice is making it a little easier to do stuff without referring to notes.

[@LilanieAlfredaAbdur-Rahman]

AE501: I mistakenly watched the 2D wave lecture 1st. This one takes quite a detour from the others in the series.

[@jakegalleryn6200]

Cool!

[@malachimorris1649]

AE501 - Malachi Morris

[@davel7037]

The african kids are dancing in the ad

[@yiwang2537]

Thanks! Great video

[@christophersanchez2256]

Thanks for the video!

[@masontelander3847]

Great video