Synopses & Reviews
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient.
The book is divided into four chapters, with two useful appendices, an excellent bibliography, and an index. A section of exercises enables the student to check his progress.Contents include Volterra Equations, Fredholm Equations, Symmetric Kernels and Orthogonal Systems of Functions, Types of Singular or Nonlinear Integral Equations, and more.
Professor Tricomi has presented the principal results of the theory with sufficient generality and mathematical rigor to facilitate theoretical applications. On the other hand, the treatment is not so abstract as to be inaccessible to physicists and engineers who need integral equations as a basic mathematical tool. In fact, most of the material in this book falls into an analytical framework whose content and methods are already traditional.
Synopsis
Authoritative, well-written treatment of extremely useful mathematical tool with wide applications. Topics include Volterra Equations, Fredholm Equations, Symmetric Kernels and Orthogonal Systems of Functions, more. Advanced undergraduate to graduate level. Exercises. Bibliography.
Synopsis
Authoritative, well-written treatment of extremely useful mathematical tool with wide applications. Volterra Equations, Fredholm Equations, much more. Advanced undergraduate to graduate level. Exercises. Bibliography.
Table of Contents
1. 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