Will plug the book by Carl Bender on mathematical methods as a great intro to asymptotic methods in solving ODEs, evaluating integrals/sums, etc. An amazing monograph by Paris on Hadamard Expansions then goes into hyperasymptotic theory for evaluating integrals. Gives amazing intuition. Both books are problem/method based so very much "applied math" books, which I'm finding quite useful for doing theoretical physics.
Thanks for the insight and recommendation, Adam. I almost reached out to you during the making of the video but didn't want to bother you. Also, I actually did get that Hadamard Expansions book because of your recommendation! Thanks again.
One question... is there any limit to this method? Does the serie u=u_0+u_1+u_2 and so on always exist? Can the nonlinear diff. equ. be as complicated as we wish it to be?
As far as I understand it, perhaps the biggest limitation of the method is boundary conditions. The method is only appropriate to use with homogeneous boundary conditions. The series does not always exist. These are nice cherry-picked initial conditions to ensure somewhat easy calculations and a nice closed form solution. Sometimes, you'll get something that looks like it does not converge, but if you were to use software, you could get a nice closed form solution. Yes, the nonlinear DE can be as complicated as you want it to be, but this will change the Adomian polynomial that you'll be using. You can have two or more nonlinear terms. You can have a sin(u), e^u, etc. I strongly suggest you check out the Wazwaz book. It's just amazing. Really great questions here, I appreciate it! Edit: One more limitation is that the ADM will only ever find *one* solution to a DE. This is problematic because nonlinear DEs often have two or more solutions. Perhaps this is the biggest limitation.
It really depends on your knowledge, what your major or interests are, and what you want out of the subject. You can get very technical with a book like Evans on PDE. It requires no knowledge of PDE and is the best book for getting students prepared for research in PDE. I wouldn't recommend it unless you have a more mathematics oriented background. On the other end of the spectrum, you can try out Wazwaz, which has no theorems/proofs and is focused solely on problem solving. It also includes nonstandard methods for solving PDE and nonlinear PDE, such as the Variational Iteration Method (VIM) and the Adomian Decomposition Method (ADM). There are some books that have a more applied or physical approach, too, such as Farlow, which is a very popular book on the subject. Or something like Debnath, which is far more comprehensive and will take you from linear PDE all the way into nonlinear PDE and everything in between. Debnath has a lot of material and can be overwhelming, but not all of it needs to be read. I actually used Debnath, Farlow, and Asmar when I took my course on BVP. Perhaps a nice middle ground is the one by Bleecker and Csordas. It has a review of ODE and has all the standard material for a first course. More modern books on the subject (such as Evans or Salsa and Verzini) utilize lebesgue integration, some measure theory, and functional analysis, however. So, by that standard, all of the other books I've listed are technically outdated. This is not terribly relevant, and I personally don't even care all that much. I just thought that I'd give you the full picture as I understand it. EDIT: The book by Pivato is actually a modern yet accessible book on the subject. Might be as close as you can get as the "best."
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Nice to see you applying the methods from the books you discuss often.
Will plug the book by Carl Bender on mathematical methods as a great intro to asymptotic methods in solving ODEs, evaluating integrals/sums, etc. An amazing monograph by Paris on Hadamard Expansions then goes into hyperasymptotic theory for evaluating integrals. Gives amazing intuition. Both books are problem/method based so very much "applied math" books, which I'm finding quite useful for doing theoretical physics.
Thanks for the insight and recommendation, Adam.
I almost reached out to you during the making of the video but didn't want to bother you.
Also, I actually did get that Hadamard Expansions book because of your recommendation!
Thanks again.
One question... is there any limit to this method?
Does the serie u=u_0+u_1+u_2 and so on always exist? Can the nonlinear diff. equ. be as complicated as we wish it to be?
Well 3 questions actually :)
As far as I understand it, perhaps the biggest limitation of the method is boundary conditions. The method is only appropriate to use with homogeneous boundary conditions.
The series does not always exist. These are nice cherry-picked initial conditions to ensure somewhat easy calculations and a nice closed form solution. Sometimes, you'll get something that looks like it does not converge, but if you were to use software, you could get a nice closed form solution.
Yes, the nonlinear DE can be as complicated as you want it to be, but this will change the Adomian polynomial that you'll be using. You can have two or more nonlinear terms. You can have a sin(u), e^u, etc.
I strongly suggest you check out the Wazwaz book. It's just amazing.
Really great questions here, I appreciate it!
Edit: One more limitation is that the ADM will only ever find *one* solution to a DE. This is problematic because nonlinear DEs often have two or more solutions. Perhaps this is the biggest limitation.
Thank you for the book recomendation. Have a nice day and keep up the good work!
Thank you, you too! BTW, I just want to make sure you saw my edit on the biggest limitation of the method on my original reply!
I did :) 👍
What is the best book for a someone who want to learn pde for first time
It really depends on your knowledge, what your major or interests are, and what you want out of the subject.
You can get very technical with a book like Evans on PDE. It requires no knowledge of PDE and is the best book for getting students prepared for research in PDE. I wouldn't recommend it unless you have a more mathematics oriented background.
On the other end of the spectrum, you can try out Wazwaz, which has no theorems/proofs and is focused solely on problem solving. It also includes nonstandard methods for solving PDE and nonlinear PDE, such as the Variational Iteration Method (VIM) and the Adomian Decomposition Method (ADM).
There are some books that have a more applied or physical approach, too, such as Farlow, which is a very popular book on the subject. Or something like Debnath, which is far more comprehensive and will take you from linear PDE all the way into nonlinear PDE and everything in between. Debnath has a lot of material and can be overwhelming, but not all of it needs to be read. I actually used Debnath, Farlow, and Asmar when I took my course on BVP.
Perhaps a nice middle ground is the one by Bleecker and Csordas. It has a review of ODE and has all the standard material for a first course.
More modern books on the subject (such as Evans or Salsa and Verzini) utilize lebesgue integration, some measure theory, and functional analysis, however. So, by that standard, all of the other books I've listed are technically outdated. This is not terribly relevant, and I personally don't even care all that much. I just thought that I'd give you the full picture as I understand it.
EDIT: The book by Pivato is actually a modern yet accessible book on the subject. Might be as close as you can get as the "best."
@@MathematicalToolbox thank you alot sir❤️
Dark lettering on dark background is unreadable…
Thanks for the feedback. I'll keep that in mind for future videos. Sorry!