18.307 | Spring 2006 | Graduate
Integral Equations


Lec # Topics
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
Course Info
As Taught In
Spring 2006