Paul DuChateau B.S.: Purdue University Specializations: Applied mathematics, partial differential equations |
Professor DuChateau received his B.S. in
Engineering Science from
Professor DuChateau's research interests lie in the area of partial differential equations, particularly in inverse problems arising in modeling flow through porous media. His research in these areas has been supported in the past by National Science Foundation Engineering and by the Office of Naval Research. This research has resulted in the publication of more than 50 papers on inverse problems and partial differential equations.
Professor DuChateau has taught a wide range of
courses at various levels, including calculus for biological sciences, applied
mathematics, real and functional analysis as well as ordinary and partial
differential equations. He has authored eight textbooks from the intermediate
to advanced levels on such topics as advanced calculus, complex variables,
Fourier and vector analysis and partial differential equations.
PROGRAM DESCRIPTION: INVERSE PROBLEMS IN PARTIAL DIFFERENTIAL EQUATIONS
Many physical systems can be modeled by
partial differential equations and if all the necessary inputs for the problem
are known, then the solution can be computed and used to predict how the system
will behave under various conditions. The necessary inputs include such
information as initial or boundary data, forcing terms, coefficients and even
the shape and size of the domain, and if any of these ingredients is unknown,
then it is not possible to use the model for studying the physical system. On
the other hand, it may be possible to experimentally measure certain outputs
from the system and use this information together with the inputs that are
known in order to recover the missing input information. This is called an
inverse problem.
For example, the coefficients in a partial
differential equation model are generally related to the physical properties of
the system that is modeled. The equation is viewed as describing an entire
class of systems rather than a specific system. In order to describe a specific
system in the class, the coefficients must be found which characterize physical
properties of the system we wish to model. In simple situations, these physical
properties can be determined directly from some sort of experiment and the
results used to tie the model to a specific physical system. This process is
referred to as calibrating the model. In
more complicated situations it may be difficult or impossible to measure the
physical property associated with a coefficient in a model equation. In such
cases it may be necessary to proceed indirectly, i.e., to formulate and solve
an inverse problem for the missing information. The identification of unknown
coefficients in differential equations leads to a variety of interesting
mathematical problems having applications in many important areas. Professor
DuChateau's research has focused particularly on the applications of inverse
problems to modeling flow in a porous medium. In recent years several students have completed Master’s projects relating
to such problems and there has been one PhD completed.
Notes for M345:
Ordinary Differential Equations
Notes on Partial Differential Equations:
Introduction to Nonlinear PDE's