Lamm's Equation/Confluent Hypergeometric Equations

Summary

2011-07-28T09:06:00.000-07:00

I have decided that the simple way to resolve the problem is simply to take the two complex solutions (probably 13.2.37 or 13.2.35) indicated by +- and simply add them to cancel out the complex form. This allows the usage to these functions where the Wronskian won't go to zero. I don't know why the complex form is consider basic/standard when generating the real-complex forms seperately is so easy.
Of course, as always, I should have realized it earlier.
Unless somebody shows an interest this blog will probably die at the end of the year; cleanup time.

Ray

Progress

2010-08-19T07:18:00.000-07:00

I finally figured out a satisfactory solution set; with a foundation. I probably should use complex integration on Pochammer's contour but I looked at it; and decided that a root canal was preferable. Nonetheless I have had several done, so I might tackle it in the future.
These results mostly reference the NIST DLMF section on Confluent HyperGeometric functions. I have the earlier book but the new version readily available on the www.
There were a couple of problems that irritated me; and I think are resolved.
1) DLMF 13.2.42 requires "b" be not an integer. While this can probably be gotten around by LHopital's theorem, but then one needs the derivatives. This problem can be gotten around by using 13.2.41.
2) Scanning a=[-inf,0] for boundary condition satisfaction (boundary matrix determinate = 0); the range of solutions of the Confluent HyperGeometric differential equation corresponding to negative/decaying time constants. This physical condition is imposed during separation of variables. Usage of 13.2.41 has problems when "a" is a negative integer; but using 13.2.7 allows an avoidance of this.
3) The new GSL code handles this okay, but blows up at b=1. This can be handled by a simple shell script though. I submitted a bug and example of how to avoid it.
4) So we are on our way with the Wronskian 13.2.37 or 13.2.35; and equation 13.2.41.
5) But another annoyance appears; the U() function is complex. Consider the solution form
Y=A*M(a,1,z)+B*U(a,1,exp(+-i*\pi)*z) This is basically two solutions when the +- is carried through 13.2.41. The complex part of U() can not be taken out B complex since the phase of U() is not constant in z. Since we know that the solution is real; what is going on? A different form illustrates how this makes sense.
Y=A*(M(a,1,z)+(B/A)U()) now the boundary conditions set B/A and the amplitude variable A (for each harmonic) can be complex and yield a real result. I haven't proved this but based on some calculations am sure it's true.
6) But we can eliminate the complex U() by taking the real part. This is proved by proving conjugate(U(a,1,exp(i*pi)*z))=U(a,1,exp(-i*pi)*z) , and then the sum of solutions are solutions thus using U'()=U(a,1,exp(-i*pi)*z)+conjugate(U(a,1,exp(i*pi)*z))
Yields a solution that has real values.
7) Unfortunately we still have computational problems. For Archibald's case and the reasonable requirement of about 1% accuracy we need values for a< -1000 . In the direct manner this fails because of floating point implementations. Fortunately there exist scaling and recursion formulas such that the numerical values stay in range. I might post a link to a paper that proves the dominant/recessive properties for such a scaling.
8) Having the explicit forms in hand we might consider a different approach. Considering recursion in "a" and state space formulation we might be able to avoid the U's and M's altogether, and just drop down to some calculation for -1
We thus have a solution set that is real and can be computed without mystery. I might finish formally writing this up; or not. Nobody seems interested in this subject/blog so why bother patching up all the holes and writing an explanatory paper. All the mystery is removed for me, so I might move on.

2010-05-24T09:15:00.000-07:00

As is usual I have gotten sidetracked. Hopefully shortly I will add some work.

Addition

2009-06-28T10:23:00.000-07:00

Added an old discussion of mine
lame_exact.pdf under "My Files"
I have a lot more thoughts not in there, but I will update it.
Incidently I have a Maple file to explore Lamm but there is still a kludge I haven't gotten around.

Online References

2009-06-26T10:41:00.000-07:00

Added NIBIB and Wikipedia links.
The Wikipedia entry is a very sparse; I will probably improve it.
The NIBIB gives an excellent discussion of solving the Lamm's equation and general centrifugal separation equations numerically. Short thrift for exact solutions that I am interested in.
Added two more similar links
Sedfit link It seems to work (one crash) under Wine in Linux FC9. I don't have my xp machine up and running yet; and I won't bother the Vista machine, it's busy doing something to itself.
Ultrascan II Linux program. I haven't tried it.
Enough on the Online references.
I do have a copy of Foundations of UltraCentrifugal Analysis, Fujita --If you want to study UltraCentrifugal analysis this quite good. I covers Lamm's equation from an applied standpoint and a lot of other topics.
I also have copies of Archebald's "The Integration of the Differential equation of the Ultracentrifuge" . I'm impressed; particularly how he managed to extract a new Confluent Hypergeometric solution and did the Eigenvalues without a computer. I don't really understand how he got the solution; I have only read it 4 times. Don't be put off, except for the contour integration it's very readable. Actually that would probably be readable if I was better at complex contour integrations.

Introduction&Purpose

2009-06-26T09:01:00.000-07:00

I am a retired EE with a mathematical bent.
The purpose of this blog is to record and make available some thoughts on Lamm's equation. More generally solving the differential equation and associated problems. The plan is to make a tutorial on solving a partial differential equation. I haven't like the organization of the books I have.
An alternate title would have been a Engineering study of the Confluent HyperGeometric equation and solutions. Several problems arise from applying it. It does provide a vehicle to examine the process of solving Partial Differential equations.
I hope to provide "motivation" behind some techniques. Comments are welcome. Hopefully some nice graphics, but I have never found any so I will have to fabricate some.
I have a few books and some others are available on the web. They provide a scattering of techniques, introduction, and results. I think a collection spot might be useful. If I manage to arrange a coherent paper I will probably submit it to Planetmath or some such.
Occasionally I will lift GPL content to avoid continuous cross referencing; giving due credit.
The underlying organization will be a paper (or series of papers) together with gripes and groans in the blog proper. I'm old and find a lot of things to complain about.