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18.307

Integral Equations

Graduate · Historical course

Spring 2006

Topics

  • Fredholm Alternative, Exactly Solvable Integral Equations (IE); Elementary Nonlinear IE; Bifurcations
  • Volterra IE; Rigorous Theory; Iteration Scheme; Separable Kernels; Laplace Transform; the Tautochrone Problem
  • Conversion of ODEs to IEs; Potential Scattering; Mechanical Vibrations; Propagation in Nonlinear Medium; Born Approximation and Iteration Series
  • Iteration Scheme; Resolvent Kernel; Fredholm Determinant; Examples
  • Exactly Solvable Cases; Fourier Series and Transforms
  • Hilbert-Schmidt Theory for Symmetric Kernels; Kernel Eigenvalues; Bounds for Eigenvalues; Kernel Symmetrization; Connection to Sturm-Liouville System
  • 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
  • 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
  • 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
  • General Theory for Non-homogeneous W-H IE (2nd kind); Significance of Index; Connection to Fredholm Alternative
  • Cauchy-type IE of 1st Kind; the Riemann-Hilbert Problem
  • IE of 2nd Kind (Non-homogeneous)
  • Kernels with Algebraic Singularities; Kernels with Logarithmic Singularities; the Carleman IE

Readings

  • Masujima, M. Applied Mathematical Methods of Theoretical Physics - Integral Equations and Calculus of Variations . Weinheim, Germany: Wiley-VCH, 2005. ISBN: 3527405348.
    Listed reading
  • Stakgold, I. Green’s Functions and Boundary-value Problems . New York, NY: Wiley, 1998. ISBN: 0471610224.
    Recommended
  • Kanwal, R. P. Linear Integral Equations: Theory and Technique . Boston, MA: Birkhauser, 1996. ISBN: 0817639403.
    Recommended
  • Drazin, P. G., and R. S. Johnson. Solitons: An Introduction . Cambridge, UK: Cambridge University Press, 1989. ISBN: 052133389X; 0521336554 (paperback).
    Recommended
  • Jereys, H., and B. S. Jereys. Methods of Mathematical Physics . Cambridge, UK: Cambridge University Press, 2000. ISBN: 0521664020.
    Listed reading
  • Coddington, E. A., and N. Levinson. Theory of Ordinary Differential Equations . New York, NY: McGraw-Hill, 1955.
    Listed reading
  • Hellwig, G. Partial Differential Equations . New York, NY: Blaisdell, 1964.
    Listed reading
  • Bergman, S., and M. Schier. Kernel Functions and Elliptic Differential Equations in Mathematical Physics . New York, NY: Academic Press, 1953.
    Listed reading
  • Kellogg, O. D. Foundations of Potential Theory . New York, NY: Dover, 1954.
    Listed reading
  • Courant, R., and D. Hilbert. Methods of Mathematical Physics . Vol. I. New York, NY: Wiley, 1989. ISBN: 0471504475.
    Listed reading
  • Feshbach, H., and P. M. Morse. Methods of Theoretical Physics . New York, NY: McGraw-Hill, 1953.
    Listed reading
  • Tricomi, F. G. Integral Equations . New York, NY: Dover, 1985. ISBN: 0486648281
    Listed reading
  • Mikhlin, S. G. Integral Equations and their Applications to Certain Problems in Mechanics, Mathematical Physics, and Technology . New York, NY: Pergamon, 1957.
    Listed reading
  • Smithies, F. Integral Equations . New York, NY: Cambridge University Press, 1958.
    Listed reading
  • Pipkin, A. C. A Course on Integral Equations . New York, NY: Springer-Verlag, 1991. ISBN: 0387975578.
    Listed reading
  • Hochstadt, H. Integral Equations . New York, NY: Wiley, 1973. ISBN: 047140165X, 0471504041.
    Listed reading
  • Weinstein, A., and W. Stenger. Methods of Intermediate Problems for Eigenvalues . New York, NY: Academic Press, 1972. ISBN: 0127424504.
    Listed reading
  • Dunford, N., and J. T. Schwartz. Linear Operators . Vols. I-III. New York, NY: Wiley, 1988. ISBN: 0470226382.
    Listed reading
  • Carrie r, G. F., M. Krook, and C. E. Pearson. Functions of a Complex Variable: Theory and Technique . Ithica, NY: Hod Books, 1983. ISBN: 0070100896.
    Listed reading
  • Whittaker, E. T., and G. N. Watson. A Course of Modern Analysis . Cambridge, UK: Cambridge University Press, 1992. ISBN: 0521588073.
    Listed reading
  • Sveshnikov, A. G., and A. N. Tikhonov. The Theory of Functions of a Complex Variable . Translated from Russian by G. Yankovsky. Moscow, Russia: Mir Publ., 1974. ASIN: B0007APBKC
    Listed reading
  • Titchmarsh, E. C. The Theory of Functions . New York, NY: Oxford University Press, 1939.
    Listed reading
  • ———. Introduction to the Theory of Fourier Integrals . New York, NY: Chelsea Pub. Co., 1986. ISBN: 0828403244.
    Listed reading
  • Paley, R., and N. Wiener. Fourier Transforms in the Complex Domain . New York, NY: American Mathematical Society, 1934.
    Listed reading
  • Levinson, N. Gap and Density Theorems . New York, NY: American Mathematical Society, 1940.
    Listed reading
  • Noble, B. Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations . New York, NY: Chelsea Publishing Company, 1988. ISBN: 0828403325.
    Listed reading
  • Weinstein (Vajnshtejn), L. A. The Theory of Diraction and the Factorization Method (Generalized Wiener-Hopf technique) . Translated from Russian by P. Beckmann, Golem Press, 1969.
    Listed reading
  • Krein, M. G. Am Math Transl 22, no. 163 (1962).
    Listed reading
  • Reissner, E., J. Math Phys Mass Inst Tech 20, no. 219 (1941).
    Listed reading
  • Hopf, E. Mathematical Problems of Radiative Equilibrium . London, UK: Cambridge University Press, 1933.
    Listed reading
  • Wiener, N., and E. Hopf. S B Preuss Akad Wiss 696, no. 706 (1931).
    Listed reading
  • Levine, H., and J. Schwinger. Phys Rev 73, 383 (1948).
    Listed reading
  • McCoy, B., and T. T. Wu. The Two-dimensional Ising Model . Cambridge, MA: Harvard University Press, 1973. ISBN: 0674914406.
    Listed reading
  • Carrier, G. F., M. Krook, and C. E. Pearson. Functions of a Complex Variable: Theory and Technique . Ithica, NY: Hod Books, 1983, chapter 8. ISBN: 0070100896.
    Listed reading
  • Muskhelishvili, N. I. Singular Integral Equations . New York, NY: Dover, 1992. ISBN: 0486668932.
    Listed reading
  • Gakhov, F. D. Boundary-value Problems . New York, NY: Dover, 1990. ISBN: 0486662756.
    Listed reading
  • Lu, J.-L. Boundary-value Problems for Analytic Functions . River Edge, NJ: World Scientific, 1993. ISBN: 9810210205.
    Listed reading
  • Broberg, K. B. Cracks and Fracture . San Diego, CA: Academic Press, 1999. ISBN: 0121341305.
    Listed reading
  • Freund, L. B. Dynamic Fracture Mechanics . New York, NY: Cambridge University Press, 1990. ISBN: 0521303303.
    Listed reading
  • ———. J Mech Phys Solids 20 (1972): 129-140, 141-152.
    Listed reading
  • Willis, J. R., and A. B. Movchan. J Mech Phys Solids 43 (1995): 319-341.
    Listed reading
  • Hirth, J. P., and J. Lothe. Theory of Dislocations . New York, NY: Wiley, 1982, esp. chapter 8. ISBN: 0471091251.
    Listed reading
  • King, R. W. P. The Theory of Linear Antennas . Cambridge, MA: Harvard University Press, 1956. ASIN: B0006AUHPW
    Listed reading
  • Tyras, G. Radiation and Propagation of Electromagnetic Waves . New York, NY: Academic Press, 1969. ISBN: 0127056505.
    Listed reading
  • Collin, R. E. Field Theory of Guided Waves . New York, NY: Wiley-IEEE Press, 1990. ISBN: 0879422378.
    Listed reading
Sources