Solving the 1-D Heat/Diffusion PDE: Nonhomogenous PDE and Eigenfunction Expansions

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In this video, I give a brief outline of the eigenfunction expansion method and how it is applied when solving a PDE that is nonhomogenous (i.e. contains a source term).

Questions? Ask me in the comments!

Prereqs: The previous videos in this PDE playlist, especially the two lectures on separation of variables. Knowledge of the Sturm-Liouville Theorem is also helpful.

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Thanks for all the great videos, you should be proud that you are helping so many people!

jam-es

Single male, likes to solve Partial Differential Equations when he's bored and lonely, looking for partner to become an integral part of the equation.

sajateacher

Been trying really hard to understand what's going on with the bn's and fn's and their realtionship to the Sturm-Liouville theorem. Finally got it. Big thanks!

Tobblatzius

I don't watch your videos and solve PDEs because I'm sad and lonely in life but because I need career prospects but thank you :) great video overall

boblabaugh

Very brief, concise and conceptual video

talhanaeemrao

Funny. I thought I didn't realize my process for solving it was the same thing all along. Somebody told me it was different. Thanks!

UnforsakenXII

love the robot voice, lol. Thanks for the video, heres to hoping it helps me understand my final project better

mizzmusicthief

This was really good. I think I finally am starting to understand this stuff.

I was curious if for the eigen-function expansion that you have to use the basis found for the homogeneous case, or if you could use any eigen function expansion so long as it fit the strum-liouville theorem?

I guess my question is more on if the choice in eigen-function expansion is unique?

corydiehl

Thank you very much! It seems helpful! Could you make a video of the hydrogen atom problem, that can be interesting and useful?

Oksana-ywfq

why in this case u is a sum of T_n*X_n and not a linear combination of these ? i mean, why thres isnt a unknow coefficient b_n like in the homogeneus case ? and what is the reason by wich X_n doesnt change but T_n does (respect to the homogeneus form)?

CarlosRamos-txer

very helpful, are you going to make finite difference/element videos ?

zawette

Do you have an example for 2-D Heat Equation? Considering a nonhomogeneous PDE?

vitoralbuquerque

Is the f(t) only a constant there? Since it is defined as a definite integral. Shouldn't it be just a constant?

zhongyuanchen

to be honest the eignfunction expansion method is ultimately hard for me .... my question is ... could we solve 1 dimensional parabolic non-homogenous PDE's using Laplace transforming method please ?

mohamedhussein

Would you follow the same procedure for a constant source term?

majormaki

What if F depends on u? Following this method, that would mean that all the coefficients f_n depend on u, which makes no sense, since they are used in the solution of u.

TimeTraveler-hkxo

How could anyone be bored when there are so many math problems to solve (if an answer exists.)

ntvonline

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