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Text: James P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Revised Edition.
1. Finite Dimensional Vector Spaces (6 lectures)
1.1 Linear Vector Spaces
1.2 Spectral Theory for Matrices
1.3 Geometrical Significance of Eigenvalues
1.4 Fredholm Alternative Theorem
1.5 Least Squares Solutions-Pseudo Inverses
1.6 Applications of Eigenvalues and Eigenfunctions
2. Function Spaces (6 lectures)
2.1 Complete Vector Spaces
2.2 Approximation in Hilbert Spaces
3. Integral Equations (6 lectures)
3.1 Introduction
3.2 Bounded Linear Operators in Hilbert Space
3.3 Compact Operators
3.4 Spectral Theory for Compact Operators
3.5 Resolvent and Pseudo-Resolvent Kernels
3.6 Approximate Solutions
3.7 Singular Integral Equations
4. Differential Operators (8 lectures)
4.1 Distributions and the Delta Function
4.2 Green's Functions
4.3 Differential Operators
4.4 Least Squares Solutions
4.5 Eigenfunction Expansions
5. Calculus of Variations (5 lectures)
5.1 The Euler-Lagrange Equations
5.2 Hamilton's Principle
5.3 Approximate Methods
5.4 Eigenvalue Problems
7. Transform and Spectral Theory (6 lectures)
7.1 Spectrum of an Operator
7.2 Fourier Transforms
7.3 Related Integral Transforms
7.4 Z Transforms
Tests: 4 lectures
Leeway: 2 lectures
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