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MWR for the first- and third-order differential equations

Hi all,

In engineering sciences, we usually end up using either the second- or the fourth-order differential equations, and the MWR (the method of weighted residuals) works pretty well for them.

The question is: how about the first- and the third-order differential equations? Why don't we see any applications of MWR for these odd-ordered differential equations? What gives?

If not equations of real application importance, then at least some hypothetical situations or toy applications involving the first- and third-order (and possibly fifth-order as well!) could have been taken up, just to illustrate the method itself. However, that is never found done, not at least in the introductory text-books/course notes. They only mention the issue, if they at all do, but they mostly remain silent about the mathematical reasons behind it.

So, here is a request: If you know of any nice introductory passages in a book (or course-notes, or articles) dealing with this issue in sufficient detail, and, preferably, at the final-year undergraduate level, then please leave comments giving references to them. (Treatments at more advanced level too are welcome.)

Ditto, if you don't know any references but could work it out and explain the issue to me. (I really don't know the reason. I have tried to think through the issue, but have ended up deriving nothing but some guesses. These guesses could not only be plain naive, they could also easily turn out to be completely besides the point if not outright wrong. And, so, this request. (If you wish, I could easily share my guesses right here; just let me know.))

Thanks in advance for your replies.


PS: Since these days I check blogs etc. at most only once a day or so, I might be a bit late in coming back posting replies, and if so, please bear with me a bit. Thanks! 



Hi, Ajit R. Jadhav,

    I know there is a example of  first-order differential equations.

    In slip lines theory for 2D plasticity problem,  the differential  equations are:

σx /∂x+  σxy /∂y=0;

  σxy /∂x+  σy /∂y=0; 

[(σx -σy )/2]2+(σxy )2=k2;

this system should belong to first-order differential equations.


 Institue of Rock and Soil Mechanics, The Chinese Academy of Science, Wuhan

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