MTH 621,-2,-3: Differential and Integral Equations of Mathematical Physics (2005-06)
Partial differential equations arise as basic models of flow and transport,
diffusion, and vibration and will be treated by classical means,
generalized functions and variational principles in Hilbert space.
Topics include first and second order partial differential equations
and classification, particularly the wave, diffusion, and potential
equations, their origins in applications and properties of solutions,
characteristics, maximum principles, Green's functions, eigenvalue
problems, and Fourier expansion methods.
Prerequisite:
6 credits of senior-level analysis. Must be taken
in order. May be repeated up to 6 credits.
Textbook: | R. Guenther and J. Lee, Partial Differential
Equations of Mathematical Physics and Integral Equations.
For the year 2005-2006 this 3-term sequence will be organized as
follows. The three terms will emphasize, respectively, (1)
boundary-value problems and their realization as operators in function
spaces, (2) classical techniques for the representation of solutions
of partial differential equations, and (3) the dynamics of
initial-boundary-value problems of transport and vibration.
MTH 621 is aimed at variational
principles in Hilbert Space and Fourier expansion theory for
boundary-value problems. These lead naturally to generalized solutions
in function spaces. We introduce these concepts in an elementary but
useful setting, and our development will include the theory of compact
self-adjoint operators. In the variational setting we shall develop
tools to establish well-posedness of boundary-value problems and to
obtain useful numerical approximation schemes.
Note: This first term serves as an introduction to basic topics of
Hilbert space. The variational principles lead to finite-element
methods, which are the focus of the Numerical Analysis course MTH 655
to be offered in Winter, 2006.
MTH 622 will be an introduction to
partial differential equations in two variables, including the
first-order transport equation, the second-order wave equation and the
diffusion equation in one spatial dimension. Each of these equations
is introduced as a model of basic convection, diffusion or vibration
processes, and their role in applications is emphasized. The
discussion includes the classification of equations, properties of the
solutions of each type, and construction and representation of
solutions of initial-value problems in the half-plane and of the
simplest initial-boundary-value problem in the quarter plane and in
the rectangle.
MTH 623 will begin with a treatment of
the Laplace - Poisson equation and then develop various models of
flow, transport, and vibration in higher dimension. We develop useful
representations and Hilbert space techniques for the corresponding
initial-boundary-value problems and the associated boundary-value
problems. Further properties of the various types of equations will be
developed and compared.
For additional information, contact
R.E. Showalter , Kidder 368,
show@math.oregonstate.edu
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