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18.307 Integral Equations

Spring 2006

Temperature profile around a plate.Temperature profile around a plate (red: hot, to blue: cold), which is found via a singular integral equation. (Courtesy of Dr. Jaehyuk Choi. Used with permission.)

Course Highlights

This course features complete set of assignments with solutions and an extensive reading list.

Course Description

This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.



Syllabus

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Prerequisite

Basic theory of one complex variable and ordinary differential equations (for example, either course Functions of a Complex Variable (18.112) or Advanced Calculus for Engineers (18.075) or Complex Variables with Applications (18.04)), or permission of Instructor.



Main Textbook

Amazon logo Masujima, M. Applied Mathematical Methods of Theoretical Physics - Integral Equations and Calculus of Variations. Weinheim, Germany: Wiley-VCH, 2005. ISBN: 3527405348.



Homework

There are eight homework assignments, each due 1.5 to 2 weeks after they are handed out. There are also two practice sets that will not be graded.



Exams

There are no exams in this course.



Grading

Grading is based entirely on the homework.





Calendar

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




Readings

Amazon logo When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more.

Below is a list of required and key recommended textbooks, and additional books and articles for further study.



Main Textbook

Amazon logo Masujima, M. Applied Mathematical Methods of Theoretical Physics - Integral Equations and Calculus of Variations. Weinheim, Germany: Wiley-VCH, 2005. ISBN: 3527405348.



Recommended Textbooks

Amazon logo Stakgold, I. Green's Functions and Boundary-value Problems. New York, NY: Wiley, 1998. ISBN: 0471610224.

Amazon logo Kanwal, R. P. Linear Integral Equations: Theory and Technique. Boston, MA: Birkhauser, 1996. ISBN: 0817639403.

Amazon logo Drazin, P. G., and R. S. Johnson. Solitons: An Introduction. Cambridge, UK: Cambridge University Press, 1989. ISBN: 052133389X; 0521336554 (paperback).



Green's Functions and Boundary-value Problems

Amazon logo Jereys, H., and B. S. Jereys. Methods of Mathematical Physics. Cambridge, UK: Cambridge University Press, 2000. ISBN: 0521664020.

Coddington, E. A., and N. Levinson. Theory of Ordinary Differential Equations. New York, NY: McGraw-Hill, 1955.

Hellwig, G. Partial Differential Equations. New York, NY: Blaisdell, 1964.

Bergman, S., and M. Schier. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. New York, NY: Academic Press, 1953.

Kellogg, O. D. Foundations of Potential Theory. New York, NY: Dover, 1954.



Elements of Theory of Fredholm Integral Equations

Amazon logo Courant, R., and D. Hilbert. Methods of Mathematical Physics. Vol. I. New York, NY: Wiley, 1989. ISBN: 0471504475.

Feshbach, H., and P. M. Morse. Methods of Theoretical Physics. New York, NY: McGraw-Hill, 1953.

Amazon logo Tricomi, F. G. Integral Equations. New York, NY: Dover, 1985. ISBN: 0486648281

Mikhlin, S. G. Integral Equations and their Applications to Certain Problems in Mechanics, Mathematical Physics, and Technology. New York, NY: Pergamon, 1957.

Smithies, F. Integral Equations. New York, NY: Cambridge University Press, 1958.

Amazon logo Pipkin, A. C. A Course on Integral Equations. New York, NY: Springer-Verlag, 1991. ISBN: 0387975578.

Amazon logo Hochstadt, H. Integral Equations. New York, NY: Wiley, 1973. ISBN: 047140165X, 0471504041.

Amazon logo Weinstein, A., and W. Stenger. Methods of Intermediate Problems for Eigenvalues. New York, NY: Academic Press, 1972. ISBN: 0127424504.

Amazon logo Dunford, N., and J. T. Schwartz. Linear Operators. Vols. I-III. New York, NY: Wiley, 1988. ISBN: 0470226382.



Theory of Functions of a Complex Variable and Fourier Transforms

Amazon logo Carrier, G. F., M. Krook, and C. E. Pearson. Functions of a Complex Variable: Theory and Technique. Ithica, NY: Hod Books, 1983. ISBN: 0070100896.

Amazon logo Whittaker, E. T., and G. N. Watson. A Course of Modern Analysis. Cambridge, UK: Cambridge University Press, 1992. ISBN: 0521588073.

Amazon logo 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

Titchmarsh, E. C. The Theory of Functions. New York, NY: Oxford University Press, 1939.

Amazon logo ———. Introduction to the Theory of Fourier Integrals. New York, NY: Chelsea Pub. Co., 1986. ISBN: 0828403244.

Paley, R., and N. Wiener. Fourier Transforms in the Complex Domain. New York, NY: American Mathematical Society, 1934.

Levinson, N. Gap and Density Theorems. New York, NY: American Mathematical Society, 1940.



Wiener-Hopf Integral Equations

Amazon logo 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.

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.

Krein, M. G. Am Math Transl 22, no. 163 (1962).

Reissner, E., J. Math Phys Mass Inst Tech 20, no. 219 (1941).

Hopf, E. Mathematical Problems of Radiative Equilibrium. London, UK: Cambridge University Press, 1933.

Wiener, N., and E. Hopf. S B Preuss Akad Wiss 696, no. 706 (1931).

Levine, H., and J. Schwinger. Phys Rev 73, 383 (1948).



Wiener-Hopf Sum Equations

Krein, M. G. Am Math Transl 22, no. 163 (1962).

Amazon logo McCoy, B., and T. T. Wu. The Two-dimensional Ising Model. Cambridge, MA: Harvard University Press, 1973. ISBN: 0674914406.



The Hilbert Problem and Singular Integral Equations of the Cauchy Type

Amazon logo 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.

Amazon logo Muskhelishvili, N. I. Singular Integral Equations. New York, NY: Dover, 1992. ISBN: 0486668932.

Amazon logo Gakhov, F. D. Boundary-value Problems. New York, NY: Dover, 1990. ISBN: 0486662756.

Amazon logo Lu, J.-L. Boundary-value Problems for Analytic Functions. River Edge, NJ: World Scientific, 1993. ISBN: 9810210205.



Modern Applications to Engineering

Wiener-Hopf Method in Elastodynamics of Crack Motion

Amazon logo Broberg, K. B. Cracks and Fracture. San Diego, CA: Academic Press, 1999. ISBN: 0121341305.

Amazon logo Freund, L. B. Dynamic Fracture Mechanics. New York, NY: Cambridge University Press, 1990. ISBN: 0521303303.

———. J Mech Phys Solids 20 (1972): 129-140, 141-152.

Willis, J. R., and A. B. Movchan. J Mech Phys Solids 43 (1995): 319-341.

Singular Integral Equations for Dislocations in Solids

Amazon logo Hirth, J. P., and J. Lothe. Theory of Dislocations. New York, NY: Wiley, 1982, esp. chapter 8. ISBN: 0471091251.

Integral Equations in Microwave Engineering

Amazon logo King, R. W. P. The Theory of Linear Antennas. Cambridge, MA: Harvard University Press, 1956. ASIN: B0006AUHPW

Amazon logo Tyras, G. Radiation and Propagation of Electromagnetic Waves. New York, NY: Academic Press, 1969. ISBN: 0127056505.

Amazon logo Collin, R. E. Field Theory of Guided Waves. New York, NY: Wiley-IEEE Press, 1990. ISBN: 0879422378.





Assignments

This section contains documents created from scanned original files and other documents that could not be made accessible to screen reader software. A "#" symbol is used to denote such documents.

The problem sets were due on the lecture dates indicated in the following table. The eighth assignment was more like a project than a problem set, and thus solutions were not given.


LEC #ASSIGNMENTSSOLUTIONS
1Problem Set 1 (PDF)(PDF -1.3 MB)#
2Problem Set 2 (PDF)(PDF)#
3Problem Set 3 (PDF)(PDF)#
4Problem Set 4 (PDF)(PDF)#
5Problem Set 5 (PDF)(PDF)#
6Problem Set 6 (PDF)(PDF)#
7Problem Set 7 (PDF)(PDF)#

Erratum for Problems Set 7 Solutions (PDF)#
8Problem Set 8 (PDF)Solution for Problem 24 (PDF)#




Study Materials

This section contains documents created from scanned original files and other documents that could not be made accessible to screen reader software. A "#" symbol is used to denote such documents.

Practice Set 1 (PDF)

Practice Set 2 (PDF)

Practice Exam 1 (PDF)#

Practice Exam 1 Solutions (PDF)#

A Review of Fourier Series and Fourier Transforms (PDF)#




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