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Laplace Transforms for Partial Differential Equations (PDEs)
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- čas přidán 13. 01. 2018
- 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.
Prerequisites: Basic knowledge of Laplace Transforms from ODEs (though I've tried to give a sufficiently thorough review without getting too thorough) and the first 3 videos of this playlist: • Partial Differential E...
Lecture Notes: drive.google.c...
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thank mr cursor
I'm glad someone can explain this in 12 minutes while my teacher couldn't do it in a month.
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.
Wow man, you really shine with the aid of Mr Cursor! Thank you both for the very lucid and informative lesson!
Our pleasure!
Really...well explained...very helpful
Thanks for this , I have gained more in 12 minutes than in 3 weeks of lecturing
best applied maths channel on YT !!
Thank you, very much. Now, I will be able to understand the mathematical calculus in unsteady diffusion and decay of a pulse.
Very useful method, cheers!
simply explain!d a difficult content. Thank you sir.
Can you use Laplace transform in finite domian problems?
Cool video. Doing some videos on priori/energy estimates would be cool. Maybe harmonic analysis?
Sure, I'll add it to my to-do list.
What if u(x,0) is not zero, leading to a non homogenous equation? Yp and Yc?
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.
Because the error function is not the same as the complementary error function. The difference is precisely the integration limits.
Best explanation found , thanks !
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.
Thank you so much! It helps a lot!!
Glad to hear that!
What about the functions with 3 variables
Thank you sir
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
Is the table with Laplace Transform solutions and inverses available anywhere?
Can u provide this table? Laplace transform nd its inverse...
thank mr cursor (Y)
Great sir....
Assalamoalaikum Sir, kindly upload video regarding integral transform solution of partial differential equations (Fourier). please.
I'll add it to my to-do list!
What's the name of the program he is writing equations in?
Smoothdraw; you can also just use MS Paint.
3:05 I know how laplace transforms work, they translate your diffeq into a simpler alebraic equation that can be solved and inverted to find a solution. But, what do you mean by "integrating out"? As i understand it, integration creates independent variables whereas derivation eliminates them eg. f(x) = 4x, then f'(x) = 4, and f''(x) = 0.
When you solve a differential equation (DE), you basically 'integrate' it. That may not be technically true (i.e. you usually don't actually integrate the equation), but functionally, that's what happens. That's why you see arbitrary constants in the solution to your DE before applying boundary/initial conditions; those arbitrary constants are integration constants.
What I'm trying to say in the video is that the Laplace transform is an *integral* transform, and you can intuitively think of it as *integrating* out one of the variables. For instance, when I apply the Laplace transform to du/dt = d^2 u/dx^2, it's as though I'm *integrating* out the t (i.e. solving the DE so that there's no more derivatives in t), resulting in a simple ODE in x. Does that clarify things?
Faculty of Khan Yes, i realize my understanding is limit since we are talking about pde's instead of ode's. So how exactly does a laplace transform change a time domain singal to a frequency? I hear this in every explanation of an LT, yet i have little intuition about it.
It's because when you look at the formula for a Laplace Transform:
L[f(t)] = integral f(t)e^(-st) dt,
we can see the 's' next to the 't' in the exponential. Since the argument of the exponential must be dimensionless, 's' must have units of 1/time because it's multiplying the time t. That's why when we take the Laplace Transform, we go from an independent variable in time (i.e. t) to an independent variable in 1/time, or frequency (i.e. s). Hope that helps!
Thank you
Sir can we apply laplace on product of two function u(x,t).v(x,t)
On what type of PDEs we can use Laplace transform. Kindly explain
Could you kindly share the table? Can’t seem to find on web
hi how so you solve this ODE how do you get the exponential
i think he used laplace boi to get a 2nd order ODE in the s variable world, solved for it thats why it has exponentials
sorry its not a good answer, u can use the rule directly for the 2nd order diff equations with a positive discriminant, or use laplace transform on a similar looking equation you will get an exponential like answer
Where did you get your table?
PDE用LT如果沒表的話太難了。
you are khan?
Thank god no Indian accent.
lol
thank mr cursor