This is the second semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CAM.
1. Preliminaries (topology and Lebesgue integration)
2. Banach Spaces
3. Hilbert Spaces
4. Spectral Theory
6. The Fourier Transform (3 weeks)
o The Schwartz space and tempered distributions.
o The Fourier transform.
o The Plancherel Theorem.
o Fundamental solutions of PDE's.
7. Sobolev spaces (3 weeks)
o Basic Definitions.
o Extention Theorems.
o Imbedding Theorems.
o The Trace Theorem.
8. Variational Boundary Value Problems (BVP) (3 weeks)
o Weak solutions to elliptic BVP's.
o Variational forms.
o Lax-Milgram Theorem.
o Galerkin approximations.
o Green's functions.
9. Differential Calculus in Banach Spaces and Calculus of Variations (4 weeks)
o The Frechet derivatives.
o The Chain Rule and Mean Value Theorems.
o Higher order derivatives and Taylor's Theorem.
o Banach's Contraction Mapping Theorem and Newton's Method.
o Inverse and Implicit Function Theorems, and applications to nonlinear functional equations.
o Extremum problems, Lagrange multipliers, and problems with constraints.
o The Euler-Lagrange equation.
o Applications to classical mechanics and geometry.
10. Some Applications (if time permits)
1. R. A. Adams, Sobolev Spaces, Academic Press, 1975.
2. J.-P. Aubin, Applied Functional Analysis, Wiley, 1979.
3. C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, 1982.
4. E.W. Cheney and H.A. Koch, Notes on Applied Mathematics, Department of Mathematics, University of Texas at Austin.
5. L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.
6. G.B. Folland, Introduction to Partial Differential Equations, Princeton, 1976.
7. I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, 1963; reprinted by Dover Publications.
8. J. Jost and X. Li-Jost, Calculus of Variations, Cambridge, 1998,
9. A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover Publications, 1970
10. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978.
11. E.H. Lieb and M. Loss, Analysis, AMS, 1997.
12. J.T. Oden & L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.
13. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
14. M. Reed & B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
15. W. Rudin, Functional Analysis, McGraw Hill, 1991.
16. W. Rudin, Real and Complex Analysis, 3rd Ed., McGraw Hill, 1987.
17. H. Sagan, Introduction to the Calculus of Variations, Dover, 1969.
18. R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, available at World Wide Web address http://ejde.math.txstate.edu//mono-toc.html.
19. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton, 1971.
20. K. Yosida, Functional Analysis, Springer-Verlag, 1980.