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When Can I learn Partial Differential Equations?

  1. Nov 21, 2015 #1
    Is my background enough to learn partial differential equations? I have completed up to calculus 2 and linear algebra. I am currently taking Cal 3 and Ordinary Differential Equations. I am doing well in both courses. I would like to learn PDE and a bit more Linear Algebra, during the winter break.

    Do I have to learn more mathematics for PDE? If not, what would be good textbook recommendations.
     
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  3. Nov 21, 2015 #2

    micromass

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    Yep, you should be good to go. Of course, everything depends on how theoretical you want it. PDE can be extremely theoretical, and then you might require things like functional analysis and differential geometry. Still, the basics can be done right now.

    As for books, I have two recommendations for you:
    1) Strauss https://www.amazon.com/Partial-Differential-Equations-An-Introduction/dp/0470054565
    Very good book. But also very applied. If you want to see several cool methods of solving equations, this is the book for you. If you're looking for some deeper theory, this might not suit you completely.

    2) Bleecker, Csordas https://www.amazon.com/Partial-Differential-Equations-Bleecker-University/dp/1571460365
    Also a very good book. More theoretical than Strauss. Discusses the main types of PDE's (heat equations, etc.)
     
  4. Nov 21, 2015 #3

    Krylov

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    Often I agree with this user's book recommendations, but here I don't. I would not recommend this title, mainly because in my opinion it compromises too much on formalism and, as a consequence, does not manage to connect well to the more abstract concepts that the audience may already know from linear algebra. Also, I recall it as being frequently sloppy. (I do applaud discussions of applications, think they are actually quite essential, but in my opinion this should not come at the expense of mathematical clarity.) Because of these reasons, as a student I found the book off-putting. It instilled in me a dislike of PDE that lasted for quite a while.

    Allow me to recommend Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis:

    https://www.amazon.com/Functional-Analysis-Differential-Equations-Universitext/dp/0387709134

    Its level may be a bit too high for you at the moment, but it's a beautiful motivated introduction to functional analysis for the study of PDE. The presentation is friendly. Understanding this type of material brings you quite up to speed with the modern applied literature. You could give it a try. If it's too far fetched for now, no money is lost, for you will surely enjoy it later. (Besides, you could ask questions here on the forum.)
     
  5. Nov 22, 2015 #4
    Partial Differential Equations for scientists and engineers (a Dover book, so it's not like you're losing anything just by buying it) was a pretty good introduction to the subject for me. All it really took to get into was ODEs and a very small amount of calculus 3.
     
  6. Nov 22, 2015 #5

    jasonRF

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    MidgetDwarf, I'm not sure exactly what you are looking for - there are a wide variety of treatments of the subject. If you have access to a university library I would start browsing what you can find to see what works for you. I will comment on a few books:

    The book Jack476 recommended,
    https://www.amazon.com/Differential..._UL160_SR104,160_&refRID=0K3FNWGKYZJCC2RN925Q
    is the simplest book I have ever seen on the subject. It is nice for getting some of the basic idea, but has almost no theory of any kind. I really like it, and for a quick first look over a winter break it would be comfortable to read and do problems from. Farlow is the PDE equivalent of the easiest calculus book you can imagine. If you primarily want theory it is not for you.

    According to the preface, the book by Brezis assumes a knowledge of real analysis at the level of Royden's book, so you are unlikely to benefit from that level of treatment. I could of course be wrong, but certainly do not buy it without a chance to look through it first. You can see a preview on Amazon:
    https://www.amazon.com/Functional-A...lev+spaces+and+partial+differential+equations

    If you are interested in applying PDEs to physics type problems, I Iike the book by Haberman, which is good with explaining techniques and has significantly more theory than Farlow.
    https://www.amazon.com/Elementary-D...ords=haberman+elementary+applied+differential

    There are many other resources, of course. Again, check the PDE section of your library if possible. If you are looking for a more applied approach some "advanced engineering mathematics" or "math methods for physics" type books may help. good luck.


    jason
     
  7. Nov 22, 2015 #6

    Krylov

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    You may be right, although this by itself is of course an excellent motivation to learn real analysis :smile:
     
  8. Nov 22, 2015 #7
    I go to a community college, so the level of matchbooks is very limited. I'm thinking Farlow would be to easy. Not sure what I want to do with PDE's. I'm taking a ODE class at the moment and just wanted to learn a bit more. Ofcourse, I would later like to learn it in a more mathematical manner, and the book by Brezis seems like something I will enjoy in a year or two.

    I already ordered the Strauss book, and will be getting the second Micromass suggested. Does this book go over fourier series or would the book by Tolsov, "Foureir Series," will also be great to use together with Strauss?

    Yes I will be learning Real Analysis, I can't start University until Spring 2017, due to financial reasons. I will have a year of from school, as I will be done with all my coursework at the community college this semester. Need to get 2 maybe 3 jobs to have money for school.

    Wanted to review old material as well as learn new material. I purchased two books on Real Analysis, Sherbet and Lay. Hopefully, I will work on analysis in January. Looking to complete these books so I can tackle Rudin/Apostol.
     
  9. Nov 22, 2015 #8

    Krylov

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    It is very nice to see such dedication to learning about analysis and PDE, especially given your circumstances. I hope you will enjoy yourself and find it a satisfying experience. Surely the field has a lot to offer.
     
  10. Nov 22, 2015 #9
  11. Nov 22, 2015 #10

    micromass

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    Strauss and Bleecker will contain the necessary Fourier theory to get through the book. Fourier theory is a very huge topic though, and it is also very beautiful. So it's of course very understandable that you wish to know more about Fourier theory. For this, you should start with the very beautiful book by Folland "Fourier theory and its application". This contains proofs, but it is not as difficult as an analysis text. It is written in order to be rigorous, but so that physicists could still understand it comfortably.

    Once you know analysis, there will be more options, including the very beautiful first volume of Stein-Shakarchi.
     
  12. Nov 22, 2015 #11
    Thanks, You have always recommended great books. I have learned a lot from being a member of physics forum. I think it was either you or Mathwork that pointed me to a geometry book by Edwin E. Moise. Had great fun with this little book.
     
  13. Nov 22, 2015 #12
    Is spending 3 months on most text, average time? Or am I just a slow learner?
     
  14. Nov 22, 2015 #13

    micromass

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    That was probably me. It's not the type of book mathwonk would recommend I think. He would say to just study Euclid directly. I know I really liked Moise though. A good follow-up (assuming you know linear algebra up to diagonalization, and the very basics of groups) would be Brannan, Esplen, Gray's geometry. Very good theoretical exposition. The exercises might be a bit too easy though. http://www.cambridge.org/be/academic/subjects/mathematics/geometry-and-topology/geometry-2nd-edition

    And then there is always baby Hartshorne's Euclid and beyond.
     
  15. Nov 22, 2015 #14

    micromass

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    I'd say that's actually very fast. You spend 3 months on an entire book? I wish I could do that...
     
  16. Nov 22, 2015 #15
    of course, there not books at the level say Spivak and Apostol Calculus. Although, I think it's time for me to start building mathematical maturity ( I feel more comfortable in reading textbooks). KK Mechanics took me almost 9 months.
     
  17. Nov 23, 2015 #16

    f95toli

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    One thing worth remembering that most "useful" PDE's can't really be solved analytically (and when you can you might get the solution as an infinite series, not very useful),. Hence, if you are learning PDE's because you want to be able to solve problems in physics I would also advice you to familiarize yourself with some basic methods for solving PDEs numerically (e.g. how to solve the wave equation with different boundary condition). Note that I would suggest doing this at the same time as learning analytical methods, this is how I was taught and it really helped me because playing around with numerical solutions is a good way to get some intuition about the behavior of the solutions and the limitations of analytical methods. You can also tackle some more "fun" problems numerically.
     
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