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# Integral equations-a reference text by P.P. Zabreyko (Paperback, 2011)

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## About this product

### Key Features

- Author(s)P.P. Zabreyko
- PublisherSpringer
- Date of Publication12/11/2011
- Language(s)English
- FormatPaperback
- ISBN-109401019118
- ISBN-139789401019118
- GenreMathematics

### Publication Data

- Place of PublicationDordrecht
- Country of PublicationNetherlands
- ImprintSpringer
- Content Notebiography

### Dimensions

- Weight586 g
- Width140 mm
- Height216 mm
- Spine23 mm
- Pagination462

### Editorial Details

- Edition StatementSoftcover reprint of the original 1st ed. 1975

### Description

- Table Of ContentsI General Introduction.- 1 Fredholm and Volterra equations.- 1.1 Fredholm equations.- 1.2 Equations with a weak singularity.- 1.3 Volterra equations.- 2 Other classes of integral equations.- 2.1 Equations with convolution kernels.- 2.2 The Wiener-Hopf equations.- 2.3 Dual equations.- 2.4 Integral transforms.- 2.5 Singular integral equations.- 2.6 Non-linear integral equations.- 3 Some inversion formulas.- 3.1 The inversion of integral transforms.- 3.2 Inversion formulas for equations with convolution kernels.- 3.3 Volterra's equations with one independent variable and a convolution kernel.- 3.4 The Abel equation.- 3.5 Integral equations with kernels defined by hypergeometric functions.- II The Fredholm Theory.- 1 Basic concepts and the Fredholm theorems.- 1.1 Basic concepts.- 1.2 The basic theorems.- 2 The solution of Fredholm equations: The method of successive approximation.- 2.1 The construction of approximations: The Neumann series.- 2.2 The resolvent kernel.- 3 The solution of Fredholm equations: Degenerate equations and the general case.- 3.1 Equations with degenerate kernels.- 3.2 The general case.- 4 The Fredholm resolvent.- 4.1 The Fredholm resolvent.- 4.2 Properties of the resolvent.- 5 The solution of Fredholm equations: The Fredholm series.- 5.1 The Fredholm series. Fredholm determinants and minors.- 5.2 The representation of the eigenfunctions of a kernel in terms of the minors of Fredholm.- 6 Equations with a weak singularity.- 6.1 Boundedness of the integral operator with a weak singularity.- 6.2 Iteration of a kernel with a weak singularity.- 6.3 The method of successive approximation.- 7 Systems of integral equations.- 7.1 The vector form for systems of integral equations.- 7.2 Methods of solution for Fredholm kernels.- 7.3 Methods of solution for kernels with a weak singularity.- 8 The structure of the resolvent in the neighbourhood of a characteristic value.- 8.1 Orthogonal kernels.- 8.2 The principal kernels.- 8.3 The canonical kernels.- 9 The rate of growth of eigenvalues.- III Symmetric Equations.- 1 Basic properties.- 1.1 Symmetric kernels.- 1.2 Basic theorems connected with symmetric kernels.- 1.3 Systems of characteristic values and eigenfunctions.- 1.4 Orthogonalization.- 2 The Hilbert-Schmidt series and its properties.- 2.1 Hilbert-Schmidt theorem.- 2.2 The solution of symmetric integral equations.- 2.3 The resolvent of a symmetric kernel.- 2.4 The bilinear series of a kernel and its iterations.- 3 The classification of symmetric kernels.- 4 Extremal properties of characteristic values and eigenfunctions.- 5 Schmidt kernels and bilinear series for non-symmetric kernels.- 6 The solution of integral equations of the first kind.- 6.1 Symmetric equations.- 6.2 Non-symmetric equations.- IV Integral Equations with Non-Negative Kernels.- 1 Positive eigenvalues.- 1.1 The formulation of the problem.- 1.2 The kernels to be examined.- 1.3 Existence of a positive eigenfunction.- 1.4 The comparison of positive eigenvalues with other eigenvalues.- 1.5 The multiplicity of the positive eigenvalue.- 1.6 Stochastic kernels.- 1.7 Notes.- 2 Positive solutions of the non-homogeneous equation.- 2.1 Existence of a positive solution.- 2.2 Convergence of successive approximations.- 2.3 Note.- 3 Estimates for the spectral radius.- 3.1 Formulation of the problem.- 3.2 Upper estimates.- 3.3 The general method.- 3.4 The block method.- 3.5 Supplementary notes.- 4 Oscillating kernels.- 4.1 The formulation of the problem.- 4.2 Oscillating matrices.- 4.3 Vibrations of an elastic continuum with a discrete distribution of mass.- 4.4 Oscillating kernels.- 4.5 Small vibrations of systems with infinite degrees of freedom.- V Continuous and Compact Linear Operators.- 1 Continuity and compactness for linear integral operators.- 1.1 The formulation of the problem.- 1.2 Linear integral operators with their range in C.- 1.3 General properties of integral operators in the Lp-spaces.- 1.4 L-characteristics of linear integ

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