Solving PDEs with the Laplace Transform: The Heat Equation

This video shows how to solve Partial Differential Equations (PDEs) with Laplace Transforms. Specifically we solve the heat equation on a semi-infinite domain.

@eigensteve on Twitter

This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Overview and Problem Setup
7:03 How Classic Methods (e.g., Laplace) Relate to Modern Problems
9:17 Laplace Transform with respect to Time
15:10 Solving ODE with Forcing: Homogeneous and Particular Solution
19:12 The Particular Solution and Initial Conditions
28:20 The Homogeneous Solution and Boundary Conditions
31:52 The Solution in Frequency and Time Domains

[revcdevs]

The funny thing about all of the great lectures on youtube is that as the series continues, the number of views decreases quickly.
Almost nobody completes the valuable lectures as a whole. Lectures are just lectures. We even need to read a book after watching lectures for comprehensive understanding.
I am now proud of myself because I understood all of this when I self-studyed this subject and I even more clearly refresh the subject from the greatest professor Steve.

[hoseinzahedifar]

Thank you very much...
In time 36:40:
The inverse laplace of exp(-sqrt(s)*x) is equal to

[Mutual_Information]

I was recently learning about the Laplace's transform on Wikipedia. It answered some questions, but not until I watched this video did I really get the utility. This channel is such good education.. seriously I've watched many episodes by this point and it's never been a waste.

[michaelburleson]

Thank you so much for taking the time to make these videos. Your explanations are incredibly well thought out, informative, and easy to understand. You have been a lifesaver for me in my Fourier Analysis class.

[rivieredevoue]

Great lecture! Although, I can’t wrap my head around the fact the B.C. and and I.C. are not the same at u(0, 0)…

[cerbahsamir]

This is so interesting. At my school we sadly didn't go through mush PDEs which is weird since they are so important in engineering.
Thanks for the video, super helpful!

[azimonti]

Thank you for the lecture!

As for the typo in the solution, on the board there is the inverse transform of *e(-sqrt(s))*, you missed the *x* when inverse transforming 😊.

As *x* it is a constant in the frequencies domain, the fix is immediate, there is an extra *x* as the numerator of the fraction and *x^2* in the numerator of the exponential (with _x>0_ which is the case by construction of the problem).

[superuser]

I’ve been waiting to see how you present this! Excellent

[robin]

Can anyone help me understand the seeming conflict between the Boundary Condition that u(inf, t)=0 and the Initial Condition u(x, 0)=Sin4x? There's a similar mismatch between IC and BC at u(0, 0). [Sin(4*0)=/=e^(-0)]
Incredibly helpful series, much appreciation for the effort to create it! Extremely well explained!

[johnsinclair]

Thank you for this -- beautiful explanation. Great

[drskelebone]

Weird and dumb question: have any links for that shirt? That looks comfy and easy-to-wear.

Also, I only passed calc 4/diffEq class by memorizing the table of Laplace transforms so I could "cheat" my way through the final. Received a "this is correct, but there are other ways to solve it."

[korawichbikedashcam]

Someone in the comment asks about forcing terms. I think of it like as if you subject this metal rod to some kind of external input heat, which is 0 when this metal rod is free from any external heat source, then a particular solution is when you put it in a stove with some wavey fire. Steve, correct me if I am wrong.

[dominicellis]

Could you make a video on the inverse Laplace transform of exp(-sqrt(s)). This is exactly where I got stuck when doing this problem independently.

[TriThom]

Why do we consider a semi-infinite domain? If the bar just has some fixed length, could the domain not also be fixed? I assume it does not matter, as the domain just has to be one sided, but not necessarily semi-infinite?

Additionally, can you solve using this method without necessarily having a prescribed boundary condition? For example, what if the input at the boundary is allowed to take any (real) value at any point in time.

[balajisriram]

Hello sir. I am Balaji from IIT kanpur, India. I wanted to ask if for the problem of 1D transient diffusion (I solve it using RK4 time integration and with a FDM discretization), does diffusion take place on the boundaries or does it only happen inside the material (apart from boundary nodes)?. PS: I have an influx boundary condition at the boundary nodes (Neumann's boudnary).

[TriThom]

I have a question not related to this video directly, but kind of blending the underlying idea with this paper "Distributed Control of Partial Differential Equations Using Convolutional Reinforcement Learning" by you and some collaborators.

When modeling environments for reinforcement learning described by PDEs, what was your approach? I have been thinking about this and if you use the method of lines let's say, you end up with a high dimensional set of ODEs that can be stiff and thus difficult to solve quickly. In terms of RL, this seems like it wouldn't work, as you could be waiting too long for the solution of these ODEs to go to the agents as the current state of environment. So, I am curious how you and your collaborators chose to simulate these PDEs such that when an action is taken, the new environment state can be generated almost immediately.

I really appreciate your time! This is a really cool paper and I am not sure many have an answer to a question like this.

[theojunming]

hmmm is it right to just remove e^(positive value) without also remove sin(4x) from U(x, t) . U(inf, t) is still unbound as sin(4x) is still inside and lim=sin(inf) doesn’t exist.

[JoshtMoody]

Hi Steve, I have an idea for a high confidence model for a transition from the current economy to a two-tiered sustenance and cultural economy. Would you be interested in trying to help me model this, or know some experts we can talk to?

[drskelebone]

Ugh. Wolfram/Mathematica and bad syntax.
$x / \left(2 \sqrt{\pi} t^{3/2} \right) * \exp{-x^2 / (4*t)}$
why is mathjax not the default in yt comments? I am angy at all the corporations tonight.

[ipideilsasha]

its eather "x/..." instead of "1/..." or "-x*x/4t" instead of "-1/4t" or both or i completely wrong, dont listen to me :) (quick google search failed me this time)