Partial Differential Equations Overview
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- čas přidán 19. 05. 2024
- Partial differential equations are the mathematical language we use to describe physical phenomena that vary in space and time. Examples include gravitation, electromagnetism, and fluid dynamics.
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This video was produced at the University of Washington
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0:00 Overview of Partial Differential Equations
4:28 Canonical PDEs
15:23 Linear Superposition
22:15 Nonlinear PDE: Burgers Equation - Věda a technologie
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](/img/n.gif)
Really really grateful to be here! Thank you Professor!! Keep inspiring all of us and please never get tired of teaching :)
Am I the only one who's stunned by how easly he can write fliped text?? insane party trick!
The video is horizontally flipped, but it still looks impressive
I like how you make these relatively complicated concepts look simple.
I realized the other day that if I wanted to accomplish my dream of being a chemical engineer then I would need to master this subject ,,, then I started reading the textbook. Yikes! Why is it that math texts post-calculus our always incomprehensible? After this video there is a glimmer of hope. Thanks Professor
Love the videos so much. You just highlight all the most important points and weave them together so that things make actual sense! Thank you!
Really hoping to see Green's functions method of solving PDEs explained, Prof. Brunton.
And everything Fourier related (yes, I know there are tons of videos on F. transform on your channel already, the more the better though :)
Thank you professor! I am an engineer in semiconductor laser design. It is very useful and important to understand vector calculus and PDEs! The numerical method to solve a lot of related physics problem in our field is based on what you are teaching now. I wish I could have met you 10 years ago!
clear, concise, excellent! thank you!
Thanks for watching!
Great overview, thanks for the intuition and insights!
Thanks for your efforts, I find your videos to be a big help.
Thank you professor.. All these years I have been not enjoying calculus because it wasn't intuitive.. This is really fun
Thanks Professor for such an interesting video. I generally get a bit tensed when it comes to PDEs but being a CAE engineer I have to deal with it. But now I developed interest in it and eagerly waiting for more videos on PDEs
WOW! I'm sold on Linearity! Thanx for that clear explanation! 😊
Sir..Very lucid, informative, and esoteric presentation.
Great and informative, thank you so much professor.
Dear Prof, thank you very much for all your great videos, super!!
I love pdes and excited about this video. Thanks Prof.
Nice one, thanks for your efforts.
Awesome introduction, good pacing!
Thank you professor, great effort ..
oh my goodness this is amazing. thank you.
Thank you for your videos !
I subscribe immediately! What a great teacher
Amazing video! Well done🤩
Its about time (pun here) someone done a neat and thoughtful presentation on PDE's.
Man, this is one topic I really wish I understood well. I struggled through it during engineering school, but never was really comfortable.
Today lecture look like... review class for mid term exam paper...tq Prof.
great instructor !
Nice lecture. I think you should do a mini-series on approximating PDEs in Python-there's good resources online for this, but I think they all rely too much on packages, rather than the math.
if people in the past were canonized for bringing understanding or enlightment to the masses, shouldn't people like Dr. Brunton be canonized today
that is incredibly kind -- thank you!! =)
Awesome, you have replaced my PDEs teacher. Greetings from Spain.
Thanks for the video
great lecture, thank you
Excellent educating
Thank you sir!
Hello Prof, your videos helped me pass my control paper.... can you make one one the mathematical treatment of quantum mechanics... specifically quantum entanglement
Prof., could you please comment on the connection between the heat equation and the Schroedinger equation?
Sehr gut
Do you have a personal recommendation for a book on PDEs and on Numerical methods for PDEs?
What does the term "canonical" here or in general mean? Thanks again
I wish my PDE professor were you
Usually, U_t = U(t), du/dt is usually denoted by u'_t, where d^2u/dx^2 is usually denoted by u"_x (with dpuble quotes),
Thank you for the video Professor. Yet another amazing lecture and can't wait for more.
I have one question about the way you write the Burger's Eq. Can you write the convective term (u . ∇u) like this or it has to be like this (u . ∇)u when u is a vector because that way you're taking gradient of a vector.
Both are equivalent! ∇u can be interpreted as a second-order tensor (basically a square matrix with style) whose columns are the partial derivatives of the vector u. Taking u . ∇u is then a matrix multiplication, as u . ∇u = (u^T)∇u, assuming u is a column vector. Performing the calculations all the way out is a bit messy, but one can show this is equivalent to the result from applying the scalar operator (u . ∇) to u.
0D = (point):
[Math; Geometry]
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple ( , , ..., ) consisting of. coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors.
[Math; 4D quaternion algebra]
A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.
What do we mean by tuple?
In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair.
In mathematics, a versor is a quaternion of norm one (a unit quaternion). The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory.
How do you make a quaternion?
You can create an N-by-1 quaternion array by specifying an N-by-3 array of Euler angles in radians or degrees. Use the euler syntax to create a scalar quaternion using a 1-by-3 vector of Euler angles in radians.
[Biology]
Points, conjugate. (Science; Microscopy) The pair of points on the principal axis of a mirror or lens so located that light emitted from either point will be focused at the other. Related points in the object and image are located optically so that one is the image of the other.
(See: polarizing element)
(Time) = length, breadth, depth:
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn't correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton's picture of a universally ticking clock.
Does time exist without space?
Time 'is' as space 'is' - part of a reference frame in which in ordered sequence you can touch, throw and eat apples.
Time cannot exist without space and the existence of time does require energy.
Nice
is it,s being linear what allows us to deal with each variable separately when doing partial derivatives?
Laplacian, can we consider it the basic laws of physics where "energy/matter is neither created not destroyed, it just converts indifferent forms"?
At 1:56 don't you think a variable for time would be of the upmost importance as well? So, shouldn't there be four components, not just three? Cause the distribution of air within any given room does not stay constant, in other words, the air doesn't just stay still as time progresses, it is constantly moving throughout the room.
The most impressive part of this video is that the professor is writing in mirror image. How is he doing that?
I am confused why alpha and "c" are squared if they are constants. If they are any constant isn't the square redundant? I remember in Intro to DEs we would get rid of the square when solving.
11:33
Question: how many sides are this to this plate ?
2 ?
4 ?
6 ?
I find the description in the segment between 11:33/11:40 ambiguous .
Would you you please clarify ?
Thank you so very much for your consideration.
Think of it as a long metallic rectangular plate, which is thermally insulated in one pair of opposite sides. It is constantly heated at some spot or end.
The spatial distribution of heat energy will eventually agree with _Laplace'' eqn_ (which is just a special case of the _Fourier heat eqn_ part ii, with u(t) -> 0).
Actually could be any shape, its just easier to solve for rectangular laminae.
@@tomctutor Thank you for your reply.
I am trying to ellicit a more explicit statement on the thickness of the plate. In my convoluted way of thinking, the descriptive language employed in the cited segment leaves room for assuming , presumably incorrectly, an infinitely thin 2D plate.
@@lgl_137noname6 if you'd like to have a physical system that can be described with a 1D PDE, you may think at a system with one dimension much larger of the other two, as an example a rod or a beam structural elements with length much larger than the linear dimensions of its section
I think you added the wrong video to your vector calc and PDE playlist(as of Jul 5, there is a video called "The Nightmare Artist")
Thanks for catching this!
You say potentials away from charges! Why not near them?! Thanks
good
I'm sorry are you writing backwards? That must hard to keep track of lol props
How does he manage to write everything backwards? That's insane
They probably just mirror flip the video. No reason not to
@@krystofjakubek9376 but it doesn't make sense, you can see his hand movements as he writes, he's really writing things backwards
Lol I'm at UW and chatted with some of his students: he totally just flips the video. But you have a sharp eye! 😄
Hold on, you are able to write backward ??
This is a special table, mirror table with its electronics
Instead of watching stranger things, the boys, minions... I'm watching this. We are Not the same
thanks if guide how to solve or solve some step I would be great full Use the linear nite-dierence method studied at the class to approximate the solution and
construct a table with the maximum of errors of the approximate solution obtained by the FDM
with several step sizes given by hi =
0.1
2
i
, i = 0, 1, 2, 3, 4. Comment the results.
2. For the boundary value problem
y
′′(x) − q(x)y(x) = r(x), x ∈ [a, b] (1)
y(a) = y(b) = 0 (2)
use a dierence scheme of the form
uj+1 − 2uj + uj−1
h2 − (α1q(xj+1)uj+1 + α0q(xj )uj + α−1q(xj−1)uj−1) = α1r(xj+1) + α0r(xj )uj + α−1r(xj−1),
j = 1, 2, ..., N − 1, (3)
with u0 = uN = 0, h =
b−a
N
and uj ∼ y(xj ), j = 0, 1, ..., N.
(a) Determine α0, α1, α−1 such that the truncation error is O(h
4
). We assume that y
(vi)
,
q
(iv)
and r
(iv)
are continuous. Note that the solution of (1)-(2) is such that
y
(iv)
(x) − (q(x)y(x))′′ = r
′′(x).
(b) If q(x) ≥ Q∗ > 0, ∀x ∈ [a, b], then show that for su ciently small h:
|uj − y(xj )| ≤ h
4
720
2M6 + 5N4 + 5R4
Q∗
,
where M6 = max
x∈[a,b]
|y
(vi)
(x)|, N4 = max
x∈[a,b]
|(q(x)y(x))(iv)
| and R4 = max
x∈[a,b]
|r
(iv)
(x)|.
(c) Implement the nite dierence scheme (3) and apply to an example in order to illustrate
the result proved in b).
0D = (point); exact location only; zero size;
non-composite substance.
Not a thing. Not a part. Monad. Soul.
'Represented' by a dot in a theoretical circle.
1D = line; two points; beginning and ending (see 1D, 4D, 7D Symmetry); composite substance; physical
1st four dimensions are 0D, 1D, 2D, 3D ✅.
1st four dimensions are not 1D, 2D, 3D, 4D 🚫.
Human consciousness, mathematically, is identical to 4D algebra unit quaternions with w, x, y, z being (0D, 1D, 2D, 3D) and i, j, k being contingent (1D xi, 2D yj, 3D zk).
'Time' is an illusion.
1D-9D 'contingent' universe has "conscious lifeforms" (1D xi, 2D yj, 3D zk)..."turning" 'time'. We're unit quaternion "turners", "to turn".
[Contingent Universe]:
3 sets of 3 dimensions. The illusory middle set (4D, 5D, 6D) is temporal. Id imagine we metaphysically create this middle set similar to a dimensional Venn Diagram with polarized lenses.
1D-3D set/7D-9D set creating the temporal illusion of 4D-6D set.
1D, 2D, 3D = spatial composite
4D, 5D, 6D = illusory temporal
7D, 8D, 9D = spectra energies
1D, 2D, 3D line, width, height
4D, 5D, 6D length, breadth, depth
7D, 8D, 9D continuous, emission, absorption
Symmetry:
1D, 4D, 7D line, length, continuous
2D, 5D, 8D width, breadth, emission
3D, 6D, 9D height, depth, absorption
Gravity is flawed.
Center of contingent universe 1D-9D is 5D. All things are drawn to the center, the whole. That's "Gravity".
Our universal constants have convoluted answers. Leibniz Law of Sufficient Reason fixes this.
FUNDAMENTALS > specifics
Leibniz > Newton
Yo up your mic dude, I can barely hear you
Hey Steve! What's with the microscopic writing? It's scandalous! Who lied to you? Stop believing everything you think in your head and start paying attention to those in your audience that don't possess 20/20 vision or own an 85" top of the line monitor/tv for which, these days, we will be paying twice or three times the original price due to our recession economy , plus it's not like you don't understand our position since most of the intelligenstsia in the world, much like you, wear the thickest glasses humans con tolerate, it's already a disadvantage for all of us. You just cannot have people squinting at the looking glass it distracts from whatever information you're trying to convey. You looking glass doesn't have to be as good as say "The Organic Chemistry Tutor" but will it kill you to put a little zoom on the text and numbers. P.S.We're skeptical that you care enough to make the changes from microscopic text and number to reasonable sized text and numbers but yeah! There it is! You'll hear from us again.