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Computational Mathematics Applied to Fluid Motion

Integrals: An Introduction to Research in Mathematics

Computational Mathematics Applied to Fluid Motion

Seminar Leader/Research Advisor: Ricardo Cortez, Tulane University

Prerequisites: A course in differential equations and basic physics. Some experience in numerical analysis or computational software will be helpful.

Overview:  All liquids and gases are considered fluids.  For this reason, understanding the motion of fluids is a key component in almost every physical phenomenon.  Blood flow in the body, the motion of parachutes, the formation of bubbles, the swimming motions of organisms and the air flow around airplanes are all examples in which fluids play an important role.  Very interesting problems come from the study of flows in the biological sciences, where organisms with flagella or cilia perform certain motions in order to propel themselves.  Other problems include the motion of cells and flows in capillaries.

The equations that describe the motion of a fluid and its interaction with a flexible membrane or filament are too difficult to solve analytically.  However, one can develop numerical methods to approximately solve the equations with the use of computers and visualize the motions. Some of these numerical methods, called Particle Methods, are used to model not only fluid flow but also planetary motion, chemical transport and the spreading of pollutants in the air.  The main idea behind particle methods is to consider a continuum (such as a fluid) as a collection of many particles moving under the influence of one another.  The problem of computing the trajectories of all particles can be written as a system of ordinary differential equations.  This is an area of research that has increased in activity in the last two decades.

             Simulation of an organism propelling itself 
                through a fluid by moving in a helical motion.

Seminar Description: The seminar will include lectures, a computer lab and research projects.  During the lectures, the students will learn background material on differential equations, diffusion, solution by the method of characteristics and aspects of fluid dynamics.  The computer lab will prepare the students to write computer programs to solve the necessary equations and to modify other programs written in advance by the seminar leader or associate.  In addition, students will learn the software MATLAB and possibly some basic FORTRAN. Four groups of three students will work on different assigned research projects on topics related to biological fluid motion.

Some of the material will be taken from notes developed for a first-year graduate course in Applied Mathematics at Tulane University and also from standard textbooks in partial differential equations and numerical analysis, such as Partial Differential Equations by Lawrence C. Evans and Introduction to Theoretical and Computational Fluid Dynamic by Costas Pozrikidis.

Sample Projects: The goal of the projects is to give the students a taste of research in computational mathematics.  Some tentative projects include: (1) the analysis and evaluation of three related deterministic diffusion methods, (2) the investigation of the swimming speed of "tubular" organisms as a function of their thickness, (3) the extension of an existing method based on forces to a method for problems in which the fluid motion is driven by torques, (4) a comparison of different types of flows around solid bodies using particle methods.

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Integrals: An Introduction to Research in Mathematics

Seminar Leader/Research Advisor: Victor H. Moll, Tulane University

Click here for a PDF file containing the description below.  

Required Background: Integral Calculus (usually second semester calculus) and basic familiarity with a computational language like Mathematica of Maple.  A basic course in number theory would be helpful, but not essential.

Seminar and Research Description:  The evaluation of integrals is one of the topics that every undergraduate calculus course covers in some detail.  The goal of this project is to employ integrals as a bridge to more sophisticated mathematics.  The central question to be explored is this: if somebody gives you an integral to evaluate, can you do it?  If you can, what did you learn from the process and the answer?  If you do not succeed, is it because it cannot be done or you have seen the right method? What does it really mean that an integral cannot be evaluated?

I hope that the examples below illustrate these points. 

1. Consider the simplest of all examples:

The proof of this is elementary: it is equivalent to the fact that

The proof of this simple fact will remind you of the binomial theorem, and it should inspire you to think about the definition of  .  What does  mean?  Or even worse, what is ?

The definite integral analog to the integral above is

 

Now if you differentiate this with respect to the parameter n you get 

 2. During the last twenty years there has been an interesting development of symbolic languages.  These are languages that allow you to compute a large number of integrals.  Another interesting use of these languages is to create a large amount of data that will help you generate interesting conjectures.  We illustrate the idea with a proof of Wallis' formula:

The first approach is by blind evaluation.  This simply means to ask the machine to do the problem for you.  If you ask Mathematica for the value of , it will tell you that 

It is unclear what is going one here.  What is the gamma function that appears in the answer?  Is there a better way to find this integral?  What is  doing in the answer?

If you ask for specifics, then things get better:

Now you see that the answer is a rational multiple of p, the question is how to guess the exact form of it.  One the methods that will be discussed in SIMU 2002 is that of recurrences.  For example, integration by parts will show you that 

From here we can guess the answer and prove it!

3. During SIMU 2000 we discussed the following problem.  It is not hard to show that

 

where

is a polynomial in a with rational coefficients.  A sequence of numbers is called unimodal if they increase up to a point and then they decrease.  We had observed that  is, for fixed m, a unimodal sequence.  A new proof by one the SIMU 2000 groups became a paper in the Electronic Journal of Combinatorics.  The stronger fact that   are log concave: is still an open question.

4. Consider the iterated indefinite integrals of ln(1 + x):

Define  the function that appears at the nth step.  Then it is not hard to show that 

where and  are polynomials in x.  It is easy to figure out the value 

The value of  is very mysterious.  Write 

where   has integer coefficients and 

It turns out that

Why? 

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