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Integral Equations

F. G. Tricomi
Dover Publications
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a very traditional view of integral equations. It is a proofs book but focuses on practical facts that lead to solutions of integral equations, rather than general properties of solutions. It is traditional partly because it is so old, and partly because the author deliberately concentrates on “well established” methods (p. v). The present book is an unaltered 1985 Dover reprint of the 1957 Prentice-Hall edition.

The book focuses on practical methods but is weak on worked examples. It usually sketches how an applied problem would be set up as an integral equation and indicates the methods for solution, but does not actually solve it. There are 32 exercises in the back that do ask for solutions of integral equations or ask for properties of particular integrals, but there are no hints or solutions for these.

The first two chapters give a lot of specific information on the Volterra and Fredholm equations (including a little bit about non-linear equations). The third chapter is a little more general and deals mostly with eigenfunctions. The fourth chapter is a miscellany and deals with several less-common forms of integral equations. For the most part there’s no discussion of numerical methods, although a few examples are scattered through the book.

An even more practical introduction to integral equations is in Arfken et al., Mathematical Methods for Physicists, Chapter 21. A more modern book (that I have not seen, but that got good reviews) is Kress’s Linear Integral Equations. It uses the operator theory, functional analysis, and Sobolev spaces point of view and covers many additional variants and methods of solution.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.




Volterra Equations
  1.1 A Mechanical Problem Leading to an Integral Equation
  1.2 Integral Equations and Algebraic Systems of Linear Equations
  1.3 Volterra Equations
  1.4 L subscript 2-Kernels and Functions
  1.5 Solution of Volterra Integral Equations of the Second Kind
  1.6 Volterra Equations of the First Kind
  1.7 An Example
  1.8 Volterra Integral Equations and Linear Differential Equations
  1.9 Equations of the Faltung type (Closed Cycle Type)
  1.10 Transverse Oscillations of a Bar
  1.11 Application to the Bessel Functions
  1.12 Some Generalizations of the Theory of Volterra Equations
  1.13 Non-Linear Volterra Equations
2. Fredholm Equations
  2.1 Solution by the Method of Successive Approximations: Neumann's Series
  2.2 An Example
  2.3 Fredholm's Equations with Pincherle-Goursat Kernels
  2.4 The Fredholm Theorem for General Kernels
  2.5 The Formulae of Fredholm
  2.6 Numerical Solution of Integral Equations
  2.7 The Fredholm Solution of the Dirichlet Problem
3. Symmetric Kernels and Orthogonal Systems of Functions
  3.1 Introductory Remarks and a Process of Orthogonalization
  3.2 Approximation and Convergence in the Mean
  3.3 The Riesz-Fischer Theorem
  3.4 Completeness and Closure
  3.5 Completeness of the Trigonometric System and of the Polynomials
  3.6 Approximation of a General L subscript 2-Kernel by Means of PG-Kernels
  3.7 Enskog's Method
  3.8 The Spectrum of a Symmetric Kernel
  3.9 The Bilinear Formula
  3.10 The Hilbert-Schmidt Theorem and Its Applications
  3.11 Extremal Properties and Bounds for Eigenvalues
  3.12 Positive Kernels--Mercer's Theorem
  3.13 Connection with the Theory of Linear Differential Equations
  3.14 Critical Velocities of a Rotating Shaft and Transverse Oscillations of a Beam
  3.15 Symmetric Fredholm Equations of the First Kind
  3.16 Reduction of a Fredholm Equation to a Similar One with a Symmetric Kernel
  3.17 Some Generalizations
  3.18 Vibrations of a Membrane
4. Some Types of Singular or Non-Linear Integral Equations
  4.1 Orientation and Examples
  4.2 Equations with Cauchy's Principal Value of an Integral and Hilbert's Transformation
  4.3 The Finite Hilbert Transformation and the Airfoil Equation
  4.4 Singular Equations of the Carleman Type
  4.5 General Remarks About Non-Linear Integral Equations
  4.6 Non-Linear Equations of the Hammerstein Type
  4.7 Forced Oscillations of Finite Amplitude
Appendix I. Algebraic Systems of Linear Equations
Appendix II. Hadamard's Theorem
  Exercises; References; Index