Solving the 1-D Heat/Diffusion PDE by Separation of Variables (Part 1/2) 

@@seabasschukwu6988 I’m still uploading don’t worry! Just check out my Videos tab!

sinh = 0.5*(exp(x)-exp(-x)), cosh = 0.5*(exp(x)+exp(-x)). They're the same thing in that cosh and sinh are just linear combinations of two exponentials. Hope that helps!

Thank you! It usually does have a constant coefficient, like alpha/the thermal diffusivity (so your PDE would be u_t = alpha*u_xx). However, it's possible to nondimensionalize the heat equation (convert everything to dimensionless variables) to get rid of the alpha. My video just used the nondimensionalized/simplified form (i.e. I'm treating alpha as 1). Even with the alpha present, you can still do separation of variables, so it doesn't make a difference in how you solve the PDE. With separation of variables, the PDE/boundary conditions just have to be linear and homogeneous. A constant multiplying a term is okay, but a constant being added to a term is not. So if it were like u_t = u_xx + alpha, you can't directly use separation of variables there.

I don't personally have a video on solving this first-order ODE, but yes, it's an ODE thing. The solution to dy/dx = k*y (k is a constant) is y = A*exp(k*x), where A is the constant of integration; this is just prerequisite knowledge I assume you know for this video.

Wait now I see, you had defined AC=0 just prior due to the IC. Awesome.

it changes from equation to equation. every equation dont need to satisfies that u=x(x)t(t).

>Are you sure that you find the complete set of solutions when using separation of variables? Yes, I'm sure. This is because the solutions to the Sturm-Liouville Problem form an orthogonal basis (see: en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory). If you watch part 2 of this solution (ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-5AkjTUD6TDw.html), you'll see that the ODE for X(x) represents a Sturm-Liouville Problem. Because the solutions form an orthogonal basis, pretty much any nice enough function (i.e. continuous, differentiable, etc.) can be expressed as a linear combination of the solutions to Sturm-Liouville. In this case, we've got sine functions as the solutions of the Sturm-Liouville Problem, and you probably know from Fourier's Theorem that any periodic function (or function over a finite interval) can be expressed as a sum of sines and cosines. That's why my solution works. It also works because it satisfies existence and uniqueness, so the solution I eventually find is indeed a unique solution. Finally, I use lambda squared because lambda^2 is guaranteed to be positive for a real lambda. As a result, I can force it to be either negative (-lambda^2) or positive (lambda^2) just by manipulating the sign out front. This is just a matter of convention though: if you end up using just lambda, it'll also work out, but you might have to deal with square roots in your final answer which can be a bit annoying. Hope that helps!

​@@FacultyofKhan the ODE for X becomes a sturm Liouville equation only when you assume the solution to be the product of X & T in the first place. But how we can assure that the solution to the PDE is in the form of the product of X & T? Do we simply experiment by taking such a form or is there any explanation for us taking such a form of the solution?

Thank you!

I have a problem with that too... @FacultyofKhan ?

I think i got it! its C=0, because AC=0 and A is not 0 and that would imply B=0, since AB=0.

*ABL=0 and if A isn´t 0 and L isn´t 0 then B=0.

He used the propertties of hyperbolic trig functions. e^x = cosh(x) + sinh(x) and e^-x = cosh(x) - sinh(x) so both solutions are the same

Depends: if you've specified a set of boundary and initial conditions, then the solution is unique. Otherwise, there are many possible solutions.

Heat equation with convection? I'm working on that right now!

Interesting. Are there any parts that are difficult to follow?

Faculty of Khan From searching Intro To PDE this video shows (having never seen PDE) Nothing explained the symbols and why operations performed were done. I found a more simplistic approach that gives more rudimentary supplement I need. Perhaps when I have a bit more understanding I'll watch these videos

Ah okay. Though this video wasn't meant to be an intro to PDEs for newcomers. It's more of a lesson for solving equations by separation of variables. If you'd like to check out the intro to PDEs I made, this is the relevant video that might help you get up to speed: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-O3ahEHAX-KU.html Also, if you want to follow things in sequence, then I recommend watching my PDE playlist in the order presented: ru-vid.com/group/PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD And thank you for the feedback! Hopefully it cleared up the confusion and I'm glad that you responded!