Solving a partial differential equation using laplace transforms

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This was a great solution, but you were fortunate that the final expression was easy to inverse Laplace transform at the end. A different initial condition would have made this much more challenging.

michaelihill

I'd appreciate some more videos on differential equations, you don't see many math channels touching them beyond a surface level.

mcalkis

Great to see Laplace transform used to solve a PDE.
This said, if use the method of separation of variables with an added constant: u(x, t)=f(x)g(t)+c, we can very quickly arrive at the same solution. Things works out nicely because of the initial conditions.
So I wonder if given the initial conditions are defined in such a way the laplace method gives a simple solution is equivalent to using the separation of variable method ?

riadsouissi

The PDE I think is fun (and insightful) to some via Laplace transforms is u_xx = (1/c^2) u_tt + δ(x - v t), u(0, t) = 0, u(x, 0) = 0, u_t(x, 0), x >= 0. Take LT wrt t ( think). Then the cases v <c (subsonic), v = c (transsonic) and v> c supersonic are different. I hope I remembered everything right. And yes δ is the dirac delta function

Calcprof

You could also skip using the variation of parameters method if you introduced the Fourier transform of u/U, and do it totally algebraically. That would have brought you to about 10:54 much faster.
I agree, however, that variation of parameters method is an important one to know.

nadavslotky

Interesting that the boundary conditions create a situation where the homogenous solution doesn’t matter anymore and we only look the particular solution when need to invert it back.

SuperSilver

❗️❗️🗣️🗣️WE GETTING OUT OF THE ODES WITH THIS ONE🗣️🗣️❗️❗️

danielrosado

How would you do it with ft tho? I know how for R(1, 1) but not for R(1, 3), since my integrals always seem to diverge...

Qrudi

Can you plz explain how do you find the value of v1 and v2 in particular solution.

AttiaNaz-pwdo

Heaviside did this operationally, getting (as you did) e^Sqrt[s] and e^-Sqrt(s), but then interpreted s = d/dt, so we need to compute e^(d^1/2/dt^1/2), the exponential of the 1/2 derivative. See Electromagnetic Theory By O. Heaviside.

Calcprof

Can you do a playlist about the methods to solve the various types of PDE?

giuliogiacomelli

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