Green's Function VIGRE
working group
Fall 2007 - Spring 2008
Time & location: All meetings are on Thursdays in Gibson Hall 400D from 2:00 to 3:15 P.M.
This seminar will run in the fall 2007 and spring 2008 semesters and is open to graduate students and postdocs at all levels. The topics will be presented by the participants without exception, sometimes individually and sometimes in teams. We expect to reach a stage where we discuss topics of current research and where group projects can be worked. This may lead to thesis topics for graduate students. We will meet once a week.
The goal of the seminar is to learn about Green's functions, starting from the
basics, and how to use them to solve differential equations that commonly
come up in applications in science and engineering.
DESCRIPTION: Green's functions are used to solve linear inhomogeneous differential equations (DE) in a given domain and with given boundary conditions. Typically, if one can find the corresponding Green's function for a given DE, say L u(x) = f(x), then the solution can be written as a convolution integral of the Green's function and the inhomogeneous term in the equations, u(x) = G*f, plus a boundary integral. For this reason , the Green's function can be considered the inverse of the differential operator which also contains information about the boundary conditions and the domain.
In cases when the differential equation is homogeneous, the result reduces to a boundary integral which may be solved numerically with careful computational methods. Perhaps the most common example is the Green's function for Laplace's equation in a ball whose solution is Poisson's integral formula. The theory is beautiful and leads to very interesting mathematical problems.
We will try to emphasize equally the theory and the applications.
For information, contact Prof. R. Cortez
Motion of a sphere near planar confining boundaries in a Brinkman medium by J. Feng, P. Ganatos and S. Weinbaum. J. Fluid Mech. (1998), v. 375, pp. 265-296.
Supplementary material from Pozrikidis' book.
A study of linearized oscillatory flow past particles by the boundary-integral method by C. Pozrikidis, J. Fluid Mech. (1989), v. 202, pp. 17-41.
Reading for 2/14/08 and 2/21/08
Image system for Stokes-flow singularity between two parallel planar walls by S. Bhattacharya and J. Blawzdziewicz. J. Math. Phys. (2002), v. 43, No. 11, pp. 5720-5731.
Image representation of a spherical particle near a hard wall by B. Cichocki and R.B. Jones. Physica A (1998), v. 258, pp. 273-302.
Reading for 4/10/08 and 4/24/08
On Green's Functions in the Theory of Heat Conduction in Spherical Coordinates by A. Lowan. Bull. Amer. Math. Soc. (1939), v. 45, pp. 310-315.
Corrections to "On Green's Functions in the Theory of Heat Conduction in Spherical Coordinates" by A. Lowan. Bull. Amer. Math. Soc. (1939), v. 45, pp. 951-952.
On Green's Functions in the Theory of Heat Conduction by H. S. Carslaw and J. C. Jaeger. Bull. Amer. Math. Soc. (1939), v. 45, pp. 407-413.
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Department Tulane University 6823 St. Charles Ave New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063 Last Updated:
October 2, 2007
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