A Survey of Discrete Calculus Plus Gronwall’s Lemma

Posted on October 14th, 2009 by

MCS Seminar by Prof. John Holte
Wednesday, 21 October 2009 11:30AM
Olin 321

I have long been interested in the parallels between “continuous” calculus and discrete calculus.  Finite difference formulas look a lot like familiar derivative formulas–with the right tweaking and summation formulas likewise resemble familiar integration formulas. Furthermore, Newton’s interpolating formula based on higher differences can be seen as analogs of Taylor’s polynomial.  In this seminar I shall present a survey of these parallels and others.  Also I’ll present a result I think is new: a discrete analog of Gronwall’s Lemma, a useful result in the theory of differential equations.  The discrete version then has implications in the theory of finite difference equations.

Pizza & beverage will be served.

 


2 Comments

  1. Gronwall’s Lemma, Taylor’s Theorem, and so on have all been extended from the continuous case to the general time scales setting, where a time scale is any closed subset of the real line, which includes the uniformly discrete case (difference equations). Please see the monograph

    Martin Bohner and Allan Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston (2001).

  2. John Holte says:

    Thanks to Doug Anderson for the reference. I included a reference to a related paper in my talk to the MAA section meeting. The proofs given in these references require the reader to know time-scale calculus, a new form of calculus first formalized in 1988. The proofs I gave do not even require calculus and are accessible to a student in MCS 256, Discrete Calculus and Probability.