Resources for self-studying PDEs

[Somaiyah]

Summary:: I'm looking for some resources to study PDEs.

Hello everyone,

I'm a sophomore majoring in Physics and this semester I am taking a course on Mathematical Methods focusing on PDEs and I'm really struggling in the course. Can someone suggest some resources to self-study PDEs? The textbook we are using in class is Mathematical Methods for Physics and Engineering and I don't find it easy to follow since I don't have a proper understanding of special functions like Bessel Function and Legendre Polynomials and so forth which appear as solutions for the DEs. I can't seem to find anything on YouTube either.

Thank you!

 

[fresh_42]

First hit on a Google search for "PDE + lecture notes + pdf"
https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf

 

[Orodruin]

Did you cover the chapter before the PDE chapter, which deals with eigenfunctions to linear differential operators?

If you did and still struggle, then it is time to look for an alternative exposition (or ask questions here, we are happy to help).

Edit: Full disclosure, I authored a competing textbook, so I am biased in terms of offering actual recommendations.

 

[Orodruin]

First hit on a Google search for "PDE + lecture notes + pdf"
https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf

While a set of lecture notes on PDEs, they hardly cover what the OP is looking for, which is a deeper insight into things like Bessel functions and Legendre polynomials. In fact, Bessel functions are just stated to be solutions to Bessel’s DE without further explanation (kind of what the OP finds difficult) and “Legendre” does not appear at all in a document search.

What is needed to cover the OP’s request is in essence a treatment of function spaces, linear differential operators on those spaces, and Sturm-Liouville theory (with the subsequent application to Bessel’s and Legendre’s DEs). Based on the text OP’s class is using, a physics and application oriented text may also be more appropriate.

 

[fresh_42]

Well, then add Bessel, Legendre, and physics to the search key, e.g.
http://www.math.iitb.ac.in/~gopal/MA207/ma207_book.pdf
The web is full of all kinds of explanations, especially when it's about standards.

If you did and still struggle, then it is time to look for an alternative exposition (or ask questions here, we are happy to help).

... which means a far more detailed approach to actual gaps and misunderstandings.

 

[Orodruin]

I believe this is still not the kind of text that the OP is looking for. It is easy to find basic information about a topic like Bessel functions online, even the Wikipedia page contains a lot of information. Finding a text that covers what you need to understand the subject with the right angle for you may be non-trivial and not just a Google search away.

But maybe we should hear from the OP before discussing this further.

 

[fresh_42]

Finding a text that covers what you need to understand the subject with the right angle for you may be non-trivial and not just a Google search away.

Sure, but this requires more details in the question, which is why I added your quote to my post.

Even if you have half a dozen of books on the topic, it might still be the case, that the best one for an individual isn't among them. I used to listen to recommendations from professors I knew a) whether they know the subject well enough and b) what kind of books they prefer. I relied more on the person than on the book. However, this approach cannot be followed on a public internet forum, so the second-best is to sight as many sources as possible and find an example that is written in an appropriate style and necessary detail. It also matters which goals are pursued: insights, exams, overviews, application-oriented, or whatever, and which amount of time is available. It is easier to buy a pair of sneakers than a suited textbook.

 

[Twigg]

Sorry for the disorganized reply, didn't have time to revise

For this material, I would recommend not trying to understand everything all in one sitting. Just focus on some examples and practice problems about solutions to a particular PDE. For example, try just focusing in one Laplace's equation in 2D with rectangular symmetry (e.g., voltage inside a box where the walls are at fixed voltages). Do some practice problems like that until separation of variables feels comfortable to you. If you're having trouble finding practice problems in Boas, see if one of your friends has Griffith's E&M (ch3 should have some I think). If you still can't find any practice problems, I'm sure other members here can direct you.

Once you've built up confidence with rectangular symmetry, try spherical symmetry (with no azimuthal dependence). Work on some practice problems. If you get stuck, try asking on the forums and I'm sure plenty of folks can help (and you'll get feedback tailored to you!). Get comfy with just Legendre functions (again, don't try to learn everything about all the special functions in Boas in one sitting, it just won't work). Once you feel good about Legendre, you'll notice the Bessel functions have the same kinds of properties as Legendre (orthogonality, recurrence relations) that are very similar with some differences in the details. With special functions, their properties and their usage is more important than actually knowing their definitions (don't bother trying to memorize how to derive them from scratch or their definitions as power series, it's not useful info in the long run).

Once you feel comfy with all of the above, that is a good time to try and read the entire chapter of Boas in one sitting, to get a big picture. That also might be a good time to peruse some introductory notes on Sturm-Liouville theory.

tl;dr: Focus on solving some simple boundary value problems as practice, and it'll start making more sense. Don't try to learn it all at once, it's not efficient.

Edit: whoops, the book was RHB not Boas. Thanks Orodruin for the correction. I am not familiar with RHB, but I still encourage the general approach outlined above.

 

[Orodruin]

If you're having trouble finding practice problems in Boas

She is using RHB, not Boas. Although similar advice applies.

 

[Astronuc]

I don't find it easy to follow since I don't have a proper understanding of special functions like Bessel Function and Legendre Polynomials and so forth which appear as solutions for the DEs.

See if these notes on Bessel's function are helpful.
https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch7/bessel.html

Note that there are two functions for the solution of Bessel equations. One blows up as the independent variable increases, so that can't possibly be a physical solution.

What types of physics problems (e.g., mass transport, heat transfer, diffusion, stress-strain, fluid dynamics, . . . .) are one encountering?

 

[Frabjous]

There‘s the cheap dover book Special Functions for Scientists and Engineers by Bell
https://www.amazon.com/dp/0486435210/?tag=pfamazon01-20

Sneddon also wrote a book which is available used
https://www.abebooks.com/servlet/Se...lts&an=Sneddon&tn=Special+functions&kn=&isbn=

 

[Orodruin]

Note that there are two functions for the solution of Bessel equations. One blows up as the independent variable increases, so that can't possibly be a physical solution.

When the independent variable decreases. Also, it should be kept in mind that the other solution is only physically impermissible if $r = 0$ is part of the domain. If not, such as when solving the Laplace equation on an annulus, it will generally be necessary to include both solutions.

 

[Astronuc]

When the independent variable decreases. Also, it should be kept in mind that the other solution is only physically impermissible if r = 0 is part of the domain.

Yes, thanks for the correction. I was thinking of the first kind J_n(x) and mixed the modified first kind I_n(x) with Y_n(x).

It is important to consider the boundary conditions. Physical problems are normally bounded.

https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
https://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
https://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

 

[Orodruin]

It should also be mentioned that an important special case where $Y_0$ is not only permissible, but required, is that of a point source at the origin for which the function needs to diverge.

 

[Zexuo]

Assuming you've diagnosed the problem correctly, you already have a handle on the derivation of the Laplacian in Cartesian, cylindrical, and spherical coordinates, separation of variables in PDEs, Fourier series, and solving PDEs with rectangular symmetry.

Vis à vis PDEs, Legendre and Bessel functions are sets of solutions to their respective equations, which result from the separation of variables under different symmetry conditions in different coordinate systems. To use them in solving PDEs you'll want to learn to recognize the equations and understand the orthogonality conditions on the functions that solve them so you can combine them properly. Just as you fit sets of solutions in PDEs with rectangular symmetry [sin nx, exp(ny), etc] to the boundary conditions therein, you will fit sets of Legendre and Bessel (and related) functions to problems with spherical or cylindrical symmetry.

Your textbook, while light on exercises, has the derivations for Legendre and Bessel functions and their relevant properties in chapter 18, sections 1-3 and 5-6.

 

[Mondayman]

I was going to mention a few Dover books. The first two are quite elementary, but very good.

Partial Differential Equations for Scientists and Engineers - Farlow

Special Functions for Scientists and Engineers - Bell

These ones are more advanced.

Partial Differential Equations of Mathematical Physics - Tikhonov

Special Functions and Their Applications - Lebedev

There's a ton of info on solving PDEs in these books.