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MATH 560 (1998) Outline of Topics

1.

Linear algebra on 2#2.

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Eigenvalues, eigenvectors, diagonalization

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Adjoint, self-adjoint, orthogonal diagonalization

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Fredholm alternative

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Least Squares

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Moore-Penrose Inverse

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Singular Value decomposition

2.

Linear algebra on vector spaces

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Change of bases

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Gram-Schmidt orthogonalization

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Similarity transformations

3.

Manifolds

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Linear manifolds

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Invariant manifolds for 4#4

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Closed linear manifolds

3.

Hilbert Spaces

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Approximation in Hilbert spaces

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Complete (orthonormal) sets

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Bessel's inequaility

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Parseval's equality

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Orthogonal complements (closed)

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Projection theorem

4.

Sturm Liouville Theory

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Regular S-L problems (defn)

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Basic existence of complete set theorem

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Eigenvalue problems and equations

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Separated and nonseparated boundary conditions

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Regular SL 5#5 orthogonality of eigenfunctions

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Regular SL 5#5 real eigenvalues

5.

Operators

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Nullspace, range

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Bounded, continuous

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Adjoint of a bounded operator, Riesz Representation Theorem

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Fredholm alternative for bounded operators with closed range

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Compact operators and properties

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Compact operators spectral theory

6.

Integral Equations

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Fredholm (separable and nonseparable-defns)

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Volterra

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Solution of Volterra eqns. by Laplace transform

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Solution of separable Fredholm eqns.

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Eigenvalue problems

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Conversion of BVP's to Fredholm integral equations

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Hilbert-Schmidt operators

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separable and nonseparable kernels

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Resolvent and pseudo resolvent kernels and operators

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Fredholm alternative for 6#6 with K compact

7.

Distributions and Green's functions

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Regular and singular distributions

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Differentiation, algebraic manipulations, properties

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distributional solutions of distributional equations

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7#7

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8#8 sequences

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Green's functions for ordinary differential equations

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Solutions of BVP using Green's functions

References

1.

Principles of Applied Mathematics: Transformation and Approximation, J. P. Keener, Addison-Wesley Pub., 1995.

2.

Green's Functions and Boundary Value Problems, 2nd ed., Stakgold, 1998. (Alternate source for 2. above)