Lec # | Topics |
---|---|
Introduction | |
1-3 | Fredholm Alternative, Exactly Solvable Integral Equations (IE); Elementary Nonlinear IE; Bifurcations |
4-5 | Volterra IE; Rigorous Theory; Iteration Scheme; Separable Kernels; Laplace Transform; the Tautochrone Problem |
Green’s Functions | |
6-8 | Conversion of ODEs to IEs; Potential Scattering; Mechanical Vibrations; Propagation in Nonlinear Medium; Born Approximation and Iteration Series |
Fredholm IEs and Fredholm Theory | |
9-10 | Iteration Scheme; Resolvent Kernel; Fredholm Determinant; Examples |
11-12 | Exactly Solvable Cases; Fourier Series and Transforms |
13-14 | Hilbert-Schmidt Theory for Symmetric Kernels; Kernel Eigenvalues; Bounds for Eigenvalues; Kernel Symmetrization; Connection to Sturm-Liouville System |
Wiener-Hopf (W-H) Technique | |
15-16 | Introduction; W-H IE of 1st and 2nd Kind; W-H Sum Equations; Examples; Basics of Solution Technique; Analyticity in Fourier Domain; Liouville’s Theorem |
17-19 | Application to Mixed Boundary Value Problems for Partial Differential Equation (PDEs); Application to Laplace’s Equation; Application to Helmholtz’s Equation; the Sommerfeld Diffraction Problem; Dual Integral Equations |
20-22 | Introduction to Theory of Homogeneous W-H IE of 2nd kind; Kernel Factorization; the Heins IE; General Theory of Homogeneous W-H IE; Definition of Kernel Index |
23-24 | General Theory for Non-homogeneous W-H IE (2nd kind); Significance of Index; Connection to Fredholm Alternative |
Singular Integral Equations of Cauchy Type | |
25-26 | Cauchy-type IE of 1st Kind; the Riemann-Hilbert Problem |
27 | IE of 2nd Kind (Non-homogeneous) |
28 | Kernels with Algebraic Singularities; Kernels with Logarithmic Singularities; the Carleman IE |
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