Laplace Transforms for Partial Differential Equations (PDEs)

In this video, I introduce the concept of Laplace Transforms to PDEs. A Laplace Transform is a special integral transform, and when it's applied to a differential equation, it effectively integrates out one of the independent variables to make the differential equation a simpler equation. Once we solve this simpler equation, we can take the inverse Laplace Transform (with the help of tables) and obtain the solution to the original differential equation.

After introducing Laplace Transforms, I apply the method of Laplace Transforms to a simple example involving the heat equation on a semi-infinite domain. After some computation, we end up with a complimentary error function as our solution.

I'm also pleased to announce that after several infuriating months of trying to find a way to display the cursor on my recording, I have finally achieved success. The cursor can be seen as the yellow dot, and I hope that it will make my videos easier to follow. Please be sure to congratulate me on this achievement by writing 'thank mr cursor' in the comments section.

Special thanks to my Patrons:
- Tom
- Jennifer Helfman
- Justin Hill
- Jacob Soares
- Yenyo Pal
- Chi
- Lisa Bouchard

[huehue]

I'm glad someone can explain this in 12 minutes while my teacher couldn't do it in a month.

[beeckthurman]

I love the efficiency of these videos! So much understanding in such a condensed form. I literally use them as an online reference guide whenever I get stuck while doing physics or chemistry. You're doing the world a great service Kahn.

[rubetz]

Wow man, you really shine with the aid of Mr Cursor! Thank you both for the very lucid and informative lesson!

[michaelgich]

Thanks for this, I have gained more in 12 minutes than in 3 weeks of lecturing

[sjat]

Really...well explained...very helpful

[juanucedaperez]

Thank you, very much. Now, I will be able to understand the mathematical calculus in unsteady diffusion and decay of a pulse.

[crosby]

Cool video. Doing some videos on priori/energy estimates would be cool. Maybe harmonic analysis?

[tharindusathischandra]

simply explain!d a difficult content. Thank you sir.

[marcoskrupiczer]

Can you use Laplace transform in finite domian problems?

[rommelchinas]

I've seen before the limits of integrations of the erf defined from 0 to y. Is there a reason for writing the limits from y to infinity in this case? Are both integrals equivalent?

Greetings and thank you for uploading these excellent videos.

[aligenc]

What about the functions with 3 variables

[hakeemcanonio]

What if u(x, 0) is not zero, leading to a non homogenous equation? Yp and Yc?

[nathanteig]

How did you know U(x, s) has to be bounded in x->infinity? I understand everything except how you would jump to knowing this, which leads to C1 being equal to zero.

[hajramughal]

Assalamoalaikum Sir, kindly upload video regarding integral transform solution of partial differential equations (Fourier). please.

[aniap]

Is the table with Laplace Transform solutions and inverses available anywhere?

[antejurcevic]

What's the name of the program he is writing equations in?

[ibrahimeltaleb]

I have a similar problem with linear flow diffusivity equation but the inner boundary condition is (u_0) is unknown at zero? it goes zero but it's not defined at x=0

[hassanaftab]

On what type of PDEs we can use Laplace transform. Kindly explain

[ambershehzadi]

Can u provide this table? Laplace transform nd its inverse...

[kuldeepmalik]

Sir can we apply laplace on product of two function u(x, t).v(x, t)