Мое понимание решения этой PDE.

[drzewo]

(Оффтоп)

and that is miserable view indeed

[Knight2023]

I guess I have made it.

$$ \begin{align} u(x,y) = u \left( x(x',y'), y(x',y') \right) \\ x(x',y') = ax' \\ y(x',y') = bx' + y'\\ \partial u = \frac{\partial u}{\partial x} \delta x + \frac{\partial u}{ \partial y} \delta y \\ u_{x'}= u_x \frac{\partial x}{\partial x'} + u_y \frac{\partial y}{\partial x'} \\ u_{x'} = a ~u_x + b~u_y = 0\\ \implies u = f(y') = f( y-b/ax) \end{align} $$

Still, the mystery is how did you come up with such a nice substitution for variables.

--------------------------------------------------------------------------------------------------------------------

As His words are already mentioned in this thread, I too must not keep myself from remembering

Resist not evil

[пианист]

Knight2023 в сообщении #1651298 писал(а):

the mystery is how did you come up with such a nice substitution for variables.

Some practice with variable substitution will help you.

[vpb]

1)

Knight2023 в сообщении #1651178 писал(а):

(Возникновение этой идеи связано с книгой Вальтера Штрауса)

Note that

W.Strauss писал(а):

The main prerequisite is a solid knowledge of calculus, especially multivariate.

At the moment, your knowledge is not sufficient, as the colleague already said to you.

2) By the way, the changing of variables proposed by пианист may not work (namely, in the case $a=0$).

[vpb]

P.S.

Knight2023 в сообщении #1651298 писал(а):

I guess I have made it.

Your calculation is correct, but not accurately enough written. In particular, write $y-(b/a)x$, not $y-b/ax$, because $y-b/ax$ may mean either $y-(b/a)x$, or $y-b/(ax)$ (I usually think in such cases that it means $y-b/(ax)$).

[Knight2023]

Red_Herring в сообщении #1651274 писал(а):

Knight2023 в сообщении #1651265 писал(а):

I couldn't advance any further. Can you please help with your hint a little more?

It would be completely useless. The crap you wrote indicates that you would miserably fail in the first test of Calculus II .

Honestly, I suspect that you "mastered" Calculus I exactly the same way. I am dealing all the time with students who failed Calculus II or ODE and managed to persuade administrative assistants to let them enrolled in PDE or who failed 3rd year PDE or Complex variables and managed to persuade administrative assistant to enrol into 4th year class requiring those as prerequisites.

My 50+ year experience teaching math tells me that I may live long enough to see humans landing on Mars but not to see you getting BSci in Math from any reputable institution.

Red Herring (Prospectus?)

You wrote this a few months ago, and now it is time. Knight have completed the following topics (which seem to dominate the exam questions): Family of surfaces in three dimensions and formulation of partial differential equations, Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

Knight can solve three famous second order PDEs, i.e. Heat Equation, Laplace Equation and the Wave Equation in three different ways: 1. Change of variable, 2. Fourier Series 3. Fourier Transform. And the proof of convergence of Fourier Series and and the notion behind Fourier Cosine transformation can also be given.

Here are some sample problems from previous years paper:
https://www.mathematicsoptional.com/upl ... tions-(PDE)-Previous-year-Questions-from-2020-to-1992.pdf

And if you personally want to test him on these topics (as they form the syllabus, though some little topics are omitted due to the issue of time and importance) you're very welcome, he will attempt those on real-time camera be it Zoom, Meet or Teams.

Knight read Joseph Fourier's The Analytical Theory of Heat in original and D'Alembert's method of solving the wave equation through Arnold's lectures.

Your experience and age is certainly respected, but Knight have only this question from you:

"Did you really mean that Knight won't ever be able to complete his PDE course? Or it was just that you wanted to sound derogatory and old?"

Knight wanted to tell you something, though he had a little hope of any positive reply from you. He wants to say that he hails from land of Rishis, the Himalayas and the Ganges, and in that land the teachers or preceptors used to love their students (or disciples) so much so that that they would even take some or all of their sins for their spiritual upliftment (very much like Jesus Christ), and if you don't believe that you might need to read about Sri Yukteshwar ji (though if I'm not wrong Madame Blavatsky was Russian and interested in these things, and ISCKON is also prevalent in Russia I guess, so you might get some idea). But you possess 50+ years of experience and can't penetrate who will do what and how. (It could be the case that spirituality is funnier to you than any comedy available on YouTube, but does that change anything? Would your disrespect for spirituality would render its effects null?)

Yes it is true, you know certainly more than Knight and you also have more influence on this forum (you might get Mods to ban him or suspend him, but that's no surprise, hatred is more common than a sunrise, there might be days when sun won't rise, maybe North Pole is in that state right now, but wherever there are humans hatred will pre-exist that place), but that's not going to be permanent, you know well.

Knight have enough respect for you as a senior, as a Professor Emeritus, as a human and this respect will continue to be so regardless of any condescending words from your side.

Knight is going to prepare for Complex Analysis. Please let your views be posted again!

[Red_Herring]

Knight2023 в сообщении #1663734 писал(а):

Did you really mean that Knight won't ever be able to complete his PDE course?

I have no idea what "his PDE class" means. And do not care to know.

[drzewo]

Knight2023 в сообщении #1663734 писал(а):

Rishis, the Himalayas and the Ganges, and in that land the teachers or preceptors used to love their students (or disciples) so much

in ancient Greece teachers loved their students much stronger

-- 05.12.2024, 18:38 --

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