Page 15 - Handbook Of Integral Equations

Part II. Methods for Solving Integral Equations

7 Main Definitions and Formulas. Integral Transforms
7.1. Some Definitions, Remarks, and Formulas
7.1-1. Some Definitions
7.1-2. The Structure of Solutions to Linear Integral Equations
7.1-3. Integral Transforms
7.1-4. Residues. Calculation Formulas
7.1-5. The Jordan Lemma
7.2. The Laplace Transform
7.2-1. Definition. The Inversion Formula
7.2-2. The Inverse Transforms of Rational Functions
7.2-3. The Convolution Theorem for the Laplace Transform
7.2-4. Limit Theorems
7.2-5. Main Properties of the Laplace Transform
7.2-6. The Post–Widder Formula
7.3. The Mellin Transform
7.3-1. Definition. The Inversion Formula
7.3-2. Main Properties of the Mellin Transform
7.3-3. The Relation Among the Mellin, Laplace, and Fourier Transforms
7.4. The Fourier Transform
7.4-1. Definition. The Inversion Formula
7.4-2. An Asymmetric Form of the Transform
7.4-3. The Alternative Fourier Transform
7.4-4. The Convolution Theorem for the Fourier Transform
7.5. The Fourier Sine and Cosine Transforms
7.5-1. The Fourier Cosine Transform
7.5-2. The Fourier Sine Transform
7.6. Other Integral Transforms
7.6-1. The Hankel Transform
7.6-2. The Meijer Transform
7.6-3. The Kontorovich–Lebedev Transform and Other Transforms
x

8. Methods for Solving Linear Equations of the Form K(x, t)y(t) dt = f(x)
a
8.1. Volterra Equations of the First Kind
8.1-1. Equations of the First Kind. Function and Kernel Classes
8.1-2. Existence and Uniqueness of a Solution
8.2. Equations With Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ·· · + g n (x)h n (t)
8.2-1. Equations With Kernel of the Form K(x, t)= g 1 (x)h 1 (t)+ g 2 (x)h 2 (t)
8.2-2. Equations With General Degenerate Kernel
8.3. Reduction of Volterra Equations of the 1st Kind to Volterra Equations of the 2nd Kind
8.3-1. The First Method
8.3-2. The Second Method
8.4. Equations With Difference Kernel: K(x, t)= K(x – t)
8.4-1. A Solution Method Based on the Laplace Transform
8.4-2. The Case in Which the Transform of the Solution is a Rational Function
8.4-3. Convolution Representation of a Solution
8.4-4. Application of an Auxiliary Equation
8.4-5. Reduction to Ordinary Differential Equations
8.4-6. Reduction of a Volterra Equation to a Wiener–Hopf Equation

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page xv