Page 16 - Handbook Of Integral Equations

8.5. Method of Fractional Differentiation
8.5-1. The Definition of Fractional Integrals
8.5-2. The Definition of Fractional Derivatives
8.5-3. Main Properties
8.5-4. The Solution of the Generalized Abel Equation
8.6. Equations With Weakly Singular Kernel
8.6-1. A Method of Transformation of the Kernel
8.6-2. Kernel With Logarithmic Singularity
8.7. Method of Quadratures
8.7-1. Quadrature Formulas
8.7-2. The General Scheme of the Method
8.7-3. An Algorithm Based on the Trapezoidal Rule
8.7-4. An Algorithm for an Equation With Degenerate Kernel
8.8. Equations With Infinite Integration Limit
8.8-1. An Equation of the First Kind With Variable Lower Limit of Integration
8.8-2. Reduction to a Wiener–Hopf Equation of the First Kind
x

9. Methods for Solving Linear Equations of the Form y(x) – K(x, t)y(t) dt = f(x)
a
9.1. Volterra Integral Equations of the Second Kind
9.1-1. Preliminary Remarks. Equations for the Resolvent
9.1-2. A Relationship Between Solutions of Some Integral Equations
9.2. Equations With Degenerate Kernel: K(x, t)= g 1 (x)h 1 (t)+ ·· · + g n (x)h n (t)
9.2-1. Equations With Kernel of the Form K(x, t)= ϕ(x)+ ψ(x)(x – t)
9.2-2. Equations With Kernel of the Form K(x, t)= ϕ(t)+ ψ(t)(t – x)
n m–1

9.2-3. Equations With Kernel of the Form K(x, t)= ϕ m (x)(x – t)
m=1
n m–1
9.2-4. Equations With Kernel of the Form K(x, t)= ϕ m (t)(t – x)
m=1
9.2-5. Equations With Degenerate Kernel of the General Form
9.3. Equations With Difference Kernel: K(x, t)= K(x – t)
9.3-1. A Solution Method Based on the Laplace Transform
9.3-2. A Method Based on the Solution of an Auxiliary Equation
9.3-3. Reduction to Ordinary Differential Equations
9.3-4. Reduction to a Wiener–Hopf Equation of the Second Kind
9.3-5. Method of Fractional Integration for the Generalized Abel Equation
9.3-6. Systems of Volterra Integral Equations
9.4. Operator Methods for Solving Linear Integral Equations
9.4-1. Application of a Solution of a “Truncated” Equation of the First Kind
9.4-2. Application of the Auxiliary Equation of the Second Kind
9.4-3. A Method for Solving “Quadratic” Operator Equations
9.4-4. Solution of Operator Equations of Polynomial Form
9.4-5. A Generalization
9.5. Construction of Solutions of Integral Equations With Special Right-Hand Side
9.5-1. The General Scheme
9.5-2. A Generating Function of Exponential Form
9.5-3. Power-Law Generating Function
9.5-4. Generating Function Containing Sines and Cosines
9.6. The Method of Model Solutions
9.6-1. Preliminary Remarks
9.6-2. Description of the Method
9.6-3. The Model Solution in the Case of an Exponential Right-Hand Side

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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