Grant Keady's Differential Equations Research.

Introduction

I imagine that sometimes my d.e. research and CA work may come closer together in the future. In 1985 I did one "big sum" (big, for the time, and I still would not do it by hand) where REDUCE helped with manipulations prior to the application of maximum principles for elliptic p.d.e.

I'm afraid the range of problems I work on and have worked on has become very extensive. There are times when I am acutely aware that I am working on too many things at once, and that this is adversely affecting "productivity".

When I work by myself I discipline myself to work on second-order semilinear elliptic p.d.e.. However, I quite often get interested in other problems when I can work closely with other people. I particularly like problems which involve some combination of order-properties (Maximum Principles) and variational formulations. My published research papers give some indication of this.

Click here to see

• a list of my publications
• a list of electronically available technical reports and papers

d.e. work in progress, and related items

I have elliptic p.d.e. work in progress on the following.

• The elastic torsion problem.
  • I'm esp. interested in inequalities associated with solutions in convex domains.
  • I am also working on a list of exact solutions for the torsion problem implemented in CAS. The Mathematica Structural Mechanics Pack has some input from me, and there are a couple of conference papers describing some of this. An as-yet imperfectly realized aim for the larger project is exact solutions published in Mathematica on Web.
    

Next, the related items. I get to referee in some of these areas, and every so often I imagine I might have time to return to them too.

• Entry of a symmetric wedge into water initially at rest.
  (See links at reports.)
• Steady plane water waves.
  
  • Irrotational. The work here involves talking with people like Edward Fraenkel and John Tolland. The biggest unsolved problem is the wave of greatest height and whether it is convex between crest and trough. However, I have separate interests in CAS finding Stokes wave expansions. AXIOM might be able to find lots of terms but so far I have only used (REDUCE and) Maple: these get only around 10 terms, whereas Longuet-Higgins et al. got hundreds of terms. Old Maple code for around 10 terms is obtainable from links given in the file here
    . e-mail me for Maple Vr4 versions if you are really interested.
  • Rotational. I like trying to extend some work, esp. to the case of constant vorticity.
    
• steady planar vortex pairs
  I have published in this area and continue to follow it. Indeed some of my interests in the elastic torsion problem - inequalities for convex domains, near circular domains - are connected with my interests in planar vortex pairs with constant vorticity in the core. My most recent efforts include
  • A recent inequality of Banuelos and Carrol enables me to improve my 1987 paper on vortex patches.
  • I read up on preprints from people at Bath during my study-leave 1996-97 to England.
  • I worked, but with nothing submittable, on this problem were during a one week visit to UWA by Jianfu Yang in Nov. 95.
• Porous media flow. (See a 1990 paper involving variational inequality work in IMA Jnl of Appl. Maths.)

Research contacts in England, Study Leaves, etc..

I have significant research contacts with England. My most recent periods working there were during leaves Jul01-Jun02 and Jul96-Jun97.

Cooperation, memberships, etc.

I am an invited member of

• the Melbourne-based rgmia
  Old address: http://melba.vu.edu.au/~rgmia/
  Research Group in Mathematical Inequalities and Applications, my own interests being applications to elliptic p.d.e. problems,
• the USA-based International Society for Analysis, its Applications and Computation.

Research and teaching

Because the UWA School of Maths and Stats has, for some time, been in a rather penurious state, the 4th year (Honours) PDE course was abolished in the mid 1980s. (There is no earlier course on p.d.e.: all that the UWA undergrads now see is first examples of separation of variables for very easy linear p.d.e..) I attempt to keep my research and teaching at least partly related, and consequently have adapted my pde research to emphasise the following:

• Continuum mechanics applications
  • Fluid mechanics applications. (UWA still has advanced courses in Fluids. The Hons-level course in my Dept is under threat of closure, along with other Hons courses because of funding difficulties considering numbers of students involved. UWA Maths has sometimes shared its Fluids course at Honours level with Murdoch University Maths. UWA Maths may also be able to share its Fluids courses with some UWA Engineering Depts.) Even if the UWA Maths-and-Stats Dept Fluids courses are withdrawn, the mathematics of fluid mechanics is a worthwhile topic in UWA Engineering Faculty (and UWA Maths is in this) as there are many Engineering Postgrads, and staff, involved in fluid mechanics research.
    
  • Some other, e.g. elasticity applications, e.g. the torsion problem
• Complex variable, e.g. in order to get exact re-formulations, boundary-integral equations, for the elastic torsion problem, etc.. (Again, my UWA Dept still has advanced courses in this, and it seems very unlikely that the core material in this course will be removed before my retirement date in 2011. It could, of course, be re-packaged, taught at different levels, etc., etc., though, to the best of my knowledge really major changes are not being considered at present.)
  
• Computer algebra manipulations, as CAS are important in other areas of undergraduate teaching.
  

Useful links

• RGMIA inequalities. ( older URL: RGMIA inequalities)

Created 1995. Last major update 2002, minor update 2006.

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