Integral Equations (eBook)

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 Equations2. 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 Problem3. 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 Membrane4. 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 AmplitudeAppendix I. Algebraic Systems of Linear EquationsAppendix II. Hadamard's Theorem Exercises; References; Index