Linear Operators and their Adjoints

This lecture is part of a series on advanced differential equations: asymptotics & perturbations. This lecture introduces the concept of linear operators and their adjoins for solving systems of the form Lu=f where L is a differential operator with its associated boundary conditions.

[olivermechling]

your lectures should have millions of views! by far the best content on Diff. Equations!

[lambdadotjoburg]

This guy has always been the f*kn BOSSS!

[HassanKhan-csho]

Much much gratitude for making this phenomenal series of lectures on advanced differential equations. The intuition behind each and every concept explained is what makes this the most beautiful series on this topic

[debbahisaad]

fourth like first comment, ouahh! functional space and Adjoints lectures are definitely among the most welcome ones...these have always been a serious source of headache....when dealing with optimal transient energy growth in fluids. waiting for more lectures, thank you!

[eyalofer]

I have never understood Fredholm alternative until now. Thx !!!

[DM-slhp]

Great lecture. I wish I'd had a pause and rewind button in school to pause and let things sink in or for those, " wait, what?" moments.

[horimekaer]

I wished assist your courses... so clear, deep and beneficial...

[Fysiker]

The equation at 22:12 seems to add an extra assumption or is using an additional result. Earlier we defined the adjoint operator of L (call it L^T) using <Lu, v> = <u, L^T v>. But at 22:12, it seems as though we assume (L^T)^T = L.

[HariKrishnan-wils]

excellent exposition, keep up the good work

[sunghyunkim]

Typo from 29:49 on the forth line of RHS, sign issue

[inothernews]

Great lesson, I enjoyed it thoroughly! Though I think there are some sign errors for the boundary terms @29:52. All three terms should be negative. Fortunately, the adjoint boundary conditions are still correct.

[adamtaylor]

Great video! One comment: using a, b for the boundary points AND the function coefficients for your differential equation made for slightly confusing notation in the integration by parts example for the general 2nd order differential operator ;)

[comment]

Is there a recommened sequence for watching these videos ... something like a playlist?

[jsmdnq]

Function spaces are vector spaces so no doubt it will have the same algebraic mechanics. This underlying idea of operator solvability stems from the group theoretic aspect of vector spaces as the same mechanics are found in the Normal subgroups(and I'm sure there is a more general theory in categorical terms). All these coincidences seem quite amazing until one realizes they all actually contain the same underlying structure.