- Видео 244
- Просмотров 308 196
Jason Bramburger
Канада
Добавлен 13 мар 2020
I am an assistant professor in applied mathematics at Concordia University with a passion for teaching. I am also actively researching at the interface of dynamical systems, machine learning, and computation. I have broad interests in the application of dynamical systems theory, including pattern formation, chaos theory, and data analysis. I received my PhD in applied mathematics from the University of Ottawa in 2017 under the supervision of Benoit Dionne and Victor LeBlanc. I have also held postdoctoral fellowships at Brown University, the University of Victoria, and the University of Washington. I was named a fellow of the Institute of Advanced Study in Guildford, UK for the 2021-2022 academic year.
The Heat Equation on an Infinite Line - Partial Differential Equations | Lecture 33
For the duration of this lecture series we have looked at PDEs on finite (bounded) domains. Mathematically, we typically study very large domains as being infinite to exclude finite-size effects. However, this infinite domains lead to new issues that have to be carefully dealt with. In this lecture we derive the solution to the heat equation on an infinite line using separation of variables. We see that there are now uncountably many eigenvalues of the associated Sturm-Liouville problem, leading to a solution written as the superposition of a continuum of eigenfunctions.
Lectures series on differential equations: ruclips.net/video/UDUH58w4-5w/видео.html
More information on the instructor: h...
Lectures series on differential equations: ruclips.net/video/UDUH58w4-5w/видео.html
More information on the instructor: h...
Просмотров: 47
Видео
Eigenfunction Expansions and Resonance - Partial Differential Equations | Lecture 32
Просмотров 22214 дней назад
With our theory of solutions to the Helmholtz equation we are able to apply it to solving inhomogeneous heat and wave equations in higher spatial dimensions. In this lecture we perform an eigenfunction expansion on the PDE model for a vibrating membrane to obtain an explicit series solution. In particular, we focus on the case of a time-periodic forcing applied to the membrane and demonstrate t...
Vibrating Circular Membranes - Partial Differential Equations | Lecture 31
Просмотров 17714 дней назад
In this lecture we apply our study of the Helmholtz equation in the previous lecture to the wave equation in two spatial dimensions. In particular, we solve the PDE of a vibrating circular membrane using separation of variables. Using polar coordinates, we show that the solution can be represented in terms of Bessel functions. Bessel functions are extremely important special functions that aris...
The Helmholtz Equation - Partial Differential Equations | Lecture 30
Просмотров 27921 день назад
Now that we have mastered Sturm-Liouville theory, we turn to its higher-dimensional analogue. In this lecture we present the Helmholtz equation which naturally arises from separation of variables applied to both the heat and wave equations in more than one spatial dimension. We present a theorem detailing the solutions to the Helmholtz equation which is similar, but contains measurable differen...
Inhomogeneous Boundary Conditions - Partial Differential Equations | Lecture 29
Просмотров 16621 день назад
In the previous lecture we were introduced to eigenfunction expansions for solving inhomogeneous PDEs. In this lecture we extend the method of eigenfunction expansions to include inhomogeneous, or forced, boundary conditions. It is demonstrated that some care must be taken since the solution expansion can never satisfy the inhomogeneous boundary conditions, but remains valid within the domain a...
Eigenfunction Expansions - Partial Differential Equations | Lecture 28
Просмотров 22828 дней назад
In this lecture we leverage Sturm-Liouville theory to solve inhomogeneous partial differential equations. We demonstrate with a worked example of the heat equation with a space and time dependent heat source. We show that the solution to the PDE can be expanded as an infinite sum of the Sturm-Liouville eigenfunctions from an associated homogeneous problem for which we then obtain ordinary diffe...
Vibrations of a Nonuniform String - Partial Differential Equations | Lecture 27
Просмотров 124Месяц назад
With the previous lecture better introducing us to the Rayleigh quotient, we can now put our theories to use to better understand vibrations of a nonuniform string. In this lecture we solve a general wave equation. While the form can only be written abstractly in terms of Sturm-Liouville eigenfunctions that are unknown explicitly, we use the Rayleigh quotient to better understand the eigenvalue...
The Rayleigh Quotient - Partial Differential Equations | Lecture 26
Просмотров 211Месяц назад
An important, but difficult to grasp, aspect of Sturm-Liouville theory is the Rayleigh quotient. In this lecture we work to better understand the Rayleigh quotient by first deriving it and then introducing the minimization principle associated to it. The minimization principle turns out to be a powerful tool in understanding the smallest eigenvalue of a Sturm-Liouville problem, as demonstrated ...
Sturm-Liouville Eigenvalues and Eigenfunctions - Partial Differential Equations | Lecture 25
Просмотров 225Месяц назад
In Lecture 23 we were introduced to Sturm-Liouville theory. Now, in this lecture we analyze the eigenvalues and eigenfunctions of Sturm-Liouville problems in more detail. In particular, we introduce the concept of a self-adjoint linear operator, provide Lagrange's identity, and Green's formula. These concepts are used to show that all eigenvalues are real, while their eigenfunctions are orthogo...
Heat Flow in a Nonuniform Rod - Partial Differential Equations | Lecture 24
Просмотров 422Месяц назад
Now that we have learned Sturm-Liouville theory, we can apply it to more complicated partial differential equations. In this lecture we solve the heat equation on a nonuniform rod with Dirichlet boundary conditions at one end and Dirichlet at the other. Although the exact eigenvalues and eigenfunctions are unknown, we can appeal the theories from the previous lecture to gain valuable insight in...
Sturm-Liouville Theory - Partial Differential Equations | Lecture 23
Просмотров 909Месяц назад
In this lecture we learn about some of the most important results in partial differential equations theory. Here we present Sturm-Liouville eigenvalue problems which generalize those that we saw in the previous lecture and show that eigenvalues always exist. Moreover, Sturm-Liouville theory guarantees that each problem has infinitely-many eigenvalues, all of which are real, and the eigenfunctio...
Eigenvalues Problems in PDEs - Partial Differential Equations | Lecture 22
Просмотров 384Месяц назад
We have seen that applying separation of variables to the heat and wave equations results in two ordinary differential equations to be solved. Moreover, both differential equations involve a single parameter that is identified by the solution to the equation involving the spatial variable. In this lecture we identify these types of problems as eigenvalue problems involving linear second-order d...
The Wave Equation in Higher Dimensions - Partial Differential Equations | Lecture 21
Просмотров 510Месяц назад
We can derive the wave equation in higher dimensions to describe vibrating membranes, i.e. 2-dimension strings. Here I simply provide the multi-dimensional wave equation and discuss some important features and aspects of it. As with the heat equation, in higher space dimensions we have to pay particular attention to the boundary conditions put on the PDE. We further discuss an important kind of...
Solving the Wave Equation - Partial Differential Equations | Lecture 20
Просмотров 536Месяц назад
In this lecture we solve the wave equation with Dirichlet (pinned) boundary conditions. We use separation of variables to obtain a Fourier sine series representation of the solution, much like for the heat equation. The major difference is that we now find oscillations in time, as opposed to the exponential decay of heat flow. We focus on the physical interpretation of the solution, particularl...
Boundary Conditions of the Wave Equation - Partial Differential Equations | Lecture 19
Просмотров 224Месяц назад
Now that we have derived a new PDE, we need to consider appropriate boundary conditions. In this lecture we discuss and derive boundary conditions for the 1-dimensional wave equation describing vibrations of a string. An interesting set of boundary conditions comes from considering one end of the string attached to a vibrating spring-mass system. In this case we find that the boundary condition...
The Wave Equation - Partial Differential Equations | Lecture 18
Просмотров 6082 месяца назад
The Wave Equation - Partial Differential Equations | Lecture 18
Complex Fourier Series - Partial Differential Equations | Lecture 17
Просмотров 4632 месяца назад
Complex Fourier Series - Partial Differential Equations | Lecture 17
Integrating Fourier Series - Partial Differential Equations | Lecture 16
Просмотров 3952 месяца назад
Integrating Fourier Series - Partial Differential Equations | Lecture 16
Differentiating Fourier Series - Partial Differential Equations | Lecture 15
Просмотров 3322 месяца назад
Differentiating Fourier Series - Partial Differential Equations | Lecture 15
Sine and Cosine Series - Partial Differential Equations | Lecture 14
Просмотров 6222 месяца назад
Sine and Cosine Series - Partial Differential Equations | Lecture 14
Fourier Series - Partial Differential Equation | Lecture 13
Просмотров 4852 месяца назад
Fourier Series - Partial Differential Equation | Lecture 13
Qualitative Properties of Laplace's Equation - Partial Differential Equations | Lecture 12
Просмотров 2702 месяца назад
Qualitative Properties of Laplace's Equation - Partial Differential Equations | Lecture 12
Fluid Flow Outside a Cylinder - Partial Differential Equations | Lecture 11
Просмотров 3162 месяца назад
Fluid Flow Outside a Cylinder - Partial Differential Equations | Lecture 11
Laplace's Equation on a Disc - Partial Differential Equations | Lecture 10
Просмотров 2582 месяца назад
Laplace's Equation on a Disc - Partial Differential Equations | Lecture 10
Laplace's Equation on a Rectangle - Partial Differential Equation | Lecture 9
Просмотров 3763 месяца назад
Laplace's Equation on a Rectangle - Partial Differential Equation | Lecture 9
Separation of Variables III: Periodic Boundaries - Partial Differential Equations | Lecture 8
Просмотров 3493 месяца назад
Separation of Variables III: Periodic Boundaries - Partial Differential Equations | Lecture 8
Separation of Variables II: Neumann Boundaries - Partial Differential Equations | Lecture 7
Просмотров 2453 месяца назад
Separation of Variables II: Neumann Boundaries - Partial Differential Equations | Lecture 7
Separation of Variables I: Dirichlet Boundaries - Partial Differential Equations | Lecture 6
Просмотров 4863 месяца назад
Separation of Variables I: Dirichlet Boundaries - Partial Differential Equations | Lecture 6
Linearity of the Heat Equation - Partial Differential Equations | Lecture 5
Просмотров 2693 месяца назад
Linearity of the Heat Equation - Partial Differential Equations | Lecture 5
The Heat Equation in Higher Dimensions - Partial Differential Equations | Lecture 4
Просмотров 3573 месяца назад
The Heat Equation in Higher Dimensions - Partial Differential Equations | Lecture 4
After taking your Dynamical Systems course, it seems that dynamical systems can run forward and backward in time. But the forward (in time) Heat Equation and the backward (in time) Heat Equation are different. Is this true?
Amazing lecture, thanks a lot!
Thanks for watching!
Really cool. Thanks from Brazil
Thanks for watching! (from Canada)
thank you for this! I was looking for this proof for my numerical analysis course but couldn't find a video that made sense to me, until now
Great Series! Fantastic! Thank You for your time, effort, and expertise!
Great job. I jumped into the lecture on poincare maps, as a refresher. Have also watched a few others as well. You are an excellent communicator, and the content is spot on.
Thanks for watching! I'm glad I can help!
Bro you're MY Genie ❤❤
Which book would you recommend , very nice video
In the intro video I mention Strogatz's book
I'm up to Lecture 23 and I have to say Fantastic Lecture Series and Thank You!!!
Thanks for watching!
is this direction promising now？
In what way?
This is gold, thank you for these videos
Thanks for watching!
Gonna do a Linear Algebra series in the future?
Not anytime soon. There are plenty of other resources on YT that do linear algebra quite well and I don't want to compete with them.
17:18 doesn't that sign from 0 to -infinity say y is increasing?
Yes, that is what is drawn in the figure
As an electrical engineer I am fascinated by this equation and know for a fact the utility of it. Do you know if there's any material that explains the relationship of this equation properties with the electromagnetic fields?
No precise definition of linearity for DEs is given...
My brain is hamburgered.Amzing job btw
r is always positive! So the only problems are the singularity at r=0 of q(r) and the condition at r=0.
Higher order Sturm Liouville ;-) I like that wording 😀. End thx for presenting those N D related theories. I think this is maybe the only one I have seen so far. And I have seen the solutions in my linear DE equation book but no such kind of explanation. thx 🙂
Not yet watched but beauty topic :-)
Is the video complete? It just ends so abruptly!! Great video anyways, this lecture series has been thus far very educating
No that's it... I just forgot to say bye at the end...
@@jasonbramburger Ok great, I was scared I missed something...
Ive just finished the whole course. Thank you Jason, you're the king!
Thanks for watching! Glad I could help!
I don't always get an opportunity to watch your videos, but I cannot thank you enough for making this high quality material openly available for everyone.
Thanks for watching/supporting!
Thank you so much for your numerous, fantastic videos. I learnt a lot through them. I am especially interested in modeling macroscopic biological systems (ecosystems, neuronal systems, social interactions etc) I would be very happy If you could do dome videos in this direction, maybe also a "go through" for important papers in this field ? Something like that would be nice !
hey, a great video but i can't understand how the step at 29:00 came to be?
You just divide both sides by the integrating factor and integrate, so it'll be something like: (mu)'/(mu) = quotient which then becomes: ln(mu) = integral of quotient.
Tau method: the better version of polynomial expansion of eigenfunctions, when monomial expandion doesnt work/numerically diverges. Chebfun and related orthogonal polynomials. And all linear algebra. Not collocation but Tau. If pde is still linear. Spectral methods rock
very underrated channel. Keep up the amazing help!
Thanks for the support!
Amazing lecture. Thank you.
Thank you for the support!
Thank you for the video!
Thank you for watching!
Guys look like messi
I am
"Oh shit"... Finally, I have something in common with a mathematician.
Yeah actually, the higher the eigenvalues the similar the eigenfunctions are to sine and cosine especially for bigger x. I guess thats for many but not all depending on p and q
What do you mean by similar to sine and cosine? How is this measured? Many famous eigenfunctions are polynomial, so not similar at all since sine and cosine are transcendental functions.
@@jasonbramburger eigenvalues are similar for some of them. My experience. Has something to do with the number of zeros on the interval and the further to the right boundary the more eigenfunction resembles sine or Cousine. I just think why this is .... maybe only when q term is Zero? What I Studied was Diffusion covection with quadraticly in space dependence.... dy2dx2 -3x2 dydx + lambda y =0
@@sschmachtel8963 Here is an example of eigenvalues and eigenfunctions that are very different from sine and cosine examples: www.global-sci.org/oldweb/jms/freedownload/v50n2/pdf/502-101.pdf
@@jasonbramburger thx. Yeah I just remembered when I was deriving some analytical solutions for convection diffusive systems and how rich all the SL theory and also generally orthogonal polynomials, including general fourier series is. And that eigenvalue similarity was probably only for some specific systems, anyway using those eigenvalues for lambda100 or more gave a much better reconstruction of the initial condition. Really interesting is also fitting of transients, where you have the sum of exponentials. Didnt yet quite understand why DMD seems to not like those, but works better for complex eigenvalues or periodic in time functions. That I namely also tried to do:: Fit eigenvalues out of measured data, but that failed horribly, I remember reading it somewhere that this is very difficult because lamdas can interswitch, and I dont exactly know if constraints in the form lamda1<lamda2 etc help, which I think supposedly it would.... Quite interesting but yet very complex what you can do with these linear PDEs of SL type, and quite a petty that this theoretic background is mainly for 1D and very little for 2D. It would be extremely nice to have something like orthogonal polynomials for triangles or even 3D and then very accurate solutions also there. Many SL solutions are very beautiful, but actually for triangles eigenfunctions are even more beautiful .... 🙂
Super excited that the algorithm helped me find you! Haven’t been this excited since finding Eigen Steve Brunton.
as somebody just starting calculus 1 what in the world is going on here
Great stuff. Thanks a lot! :)
Ok this is brilliant, super duper clear. I read a few proofs and was struggling to understand why, if an interval had no finite subcover, that one of the halves of that interval must also have no finite subcover. But your explanation at 27:49 cleared that up for me :)
great series, but i came looking for lyapunov-schmidt reduction method and it seems like i can't find anything related to that on youtube, would be great if you could make a video about it with a couple examples.
Going into my first year of a maters and I'm finding these lectures very useful. thanks
Glad I can help!
thank you very much.
Gonna teach my kid this
goat
Thank you for your teaching method and emphasizing things. You taught us mesmerizing subject.
outstanding lecture
Hey, thanks for the video. But a part which difficult to me to understand is that in the sinusoidal diagram some parts have negative solpe which means their value is decreasing, but you've mentioned that the whole part despite the negative slope can be considered as increasing, I'm talking about the interval between the zero to pi on the x axis
Negative slope indicate that the rate of change of x is decrease(That not means that rate of change of x is negative). You can think it like that derivative of x is the velocity of particle and x is the position of particle. From zero to pi, for first half velocity is increase( at x = 0, velocity is zero then increase as value of sinx is increase) and in second half, velocity is decrease but as velocity is positive always in that range , the value of x will increase and at x = pi , the velocity is zero and particle will not move(Fixed point).
really great video!!
Get sick of solving PDE’s? Ha! Yeah right.
Beautiful presentation. Affine space is my favorite kind of space.
thank you so much for this lecture!
Can you do one on Markov Processors please including Semi Markov Process. Maybe after a series in Stochastic Processors.
Hi Jason, I thijnk you got the part at 20:32 wrong. You reasoned that since |fn(x)-f(x)|<epsilon/2 works for all x, it is uniform convergent. However, isnt that just pointwise convergence? It never implied uniform convergence right? Hope you answer, thank you!
Or do you mean maybe for each epsilon, you choose a single N WITHOUT fixing x such that |fn(x)-fm(x)|<epsilon/2 for every x simultaneously. Then in this case if n>m>N and m--> inf, then for every epsilon chosen, there is an N such that |fn-f|<epsilon/2, so it is uniformly convergent
I enjoyed this.
this is just great, hopefully RUclips, bring this to more people