Abel integral equations. Analysis and applications.

*(English)*Zbl 0717.45002
Lecture Notes in Mathematics, 1461. Berlin etc.: Springer-Verlag. vii, 215 p. DM 37.00 (1991).

The classical Abel integral equation (introduced by Niels Henrik Abel in 1823) and its many generalizations play an important role in the mathematical modelling of various physical phenomena. Moreover, these equations and their underlying integral operators have attracted the interest of numerical and functional analysts.

According to the preface, the aim of the present book is “to stimulate the flow of information between at least three sorts of people. (i) Theoretical mathematicians also interested in applications and application-relevant questions. (ii) Mathematicians working in applications and numerical analysis. (iii) Scientists and engineers working outside of mathematics but applying mathematical methods for modelling and evaluation.”

The first three chapters of the book deal with the basic theory and numerous applications (old and new) of the linear (first-kind) Abel integral equation and its variants. The following two chapters focus on theoretical properties of Abel integral operators and equations: in chapter 4, the authors examine the continuity and smoothness properties of the linear Abel operator, \[ (J^{\alpha}u)(x):=1/\Gamma (\alpha)\int^{x}_{0}(x-t)^{\alpha -1}u(t)dt,\quad 0\leq x\leq a\quad (0<\alpha <1), \] when viewed as an operator acting on \(L^ p(0,a)\), the Hölder space \(C^{\alpha}[0,a]\), or the Sobolev space of fractional order, \(W^{\theta,p}(0,a)\) \((0<\theta <1\), \(p\geq 1)\) Chapter 5 is concerned with existence and uniqueness results for the generalized Abel equation, \[ 1/\Gamma (\alpha)\int^{x}_{0}(x-t)^{\alpha - 1}K(x,t)u(t)dt=f(x), \] and its nonlinear counterpart. Chapter 6 explores some of the relations between the Abel transform \(J^{\alpha}\) and other integral transforms (the Fourier, Mellin, Hankel and (plane) Radon transforms). Chapter 7 brings a digression to linear and nonlinear Abel integral equations of the second kind: here, the two main sections deal with “analysis-motivated investigations” (existence theorems) and “application-motivated investigations” (discussion of a nonlinear Abel equation arising from an initial-boundary-value problem for the heat equation, with a nonlinear radiation condition at one of the boundaries). Brief surveys of related papers and of numerical methods conclude the chapter.

In the final two chapters the authors return to first-kind Abel equations. Chapter 8 deals with the ill-posed nature of such equations (with the discussion being based mainly on specific examples) and with general stability estimates for solutions in the setting of \(L^ p(0,a)\). The numerical treatment of first-kind Abel equations is the topic of the brief Chapter 9 (here, the focus is on methods for equations with noisy data and on a case study involving an equation arising in spectroscopy). An extensive list of references concludes the book.

This clearly written monograph is an important and timely contribution to the literature on integral equations.

According to the preface, the aim of the present book is “to stimulate the flow of information between at least three sorts of people. (i) Theoretical mathematicians also interested in applications and application-relevant questions. (ii) Mathematicians working in applications and numerical analysis. (iii) Scientists and engineers working outside of mathematics but applying mathematical methods for modelling and evaluation.”

The first three chapters of the book deal with the basic theory and numerous applications (old and new) of the linear (first-kind) Abel integral equation and its variants. The following two chapters focus on theoretical properties of Abel integral operators and equations: in chapter 4, the authors examine the continuity and smoothness properties of the linear Abel operator, \[ (J^{\alpha}u)(x):=1/\Gamma (\alpha)\int^{x}_{0}(x-t)^{\alpha -1}u(t)dt,\quad 0\leq x\leq a\quad (0<\alpha <1), \] when viewed as an operator acting on \(L^ p(0,a)\), the Hölder space \(C^{\alpha}[0,a]\), or the Sobolev space of fractional order, \(W^{\theta,p}(0,a)\) \((0<\theta <1\), \(p\geq 1)\) Chapter 5 is concerned with existence and uniqueness results for the generalized Abel equation, \[ 1/\Gamma (\alpha)\int^{x}_{0}(x-t)^{\alpha - 1}K(x,t)u(t)dt=f(x), \] and its nonlinear counterpart. Chapter 6 explores some of the relations between the Abel transform \(J^{\alpha}\) and other integral transforms (the Fourier, Mellin, Hankel and (plane) Radon transforms). Chapter 7 brings a digression to linear and nonlinear Abel integral equations of the second kind: here, the two main sections deal with “analysis-motivated investigations” (existence theorems) and “application-motivated investigations” (discussion of a nonlinear Abel equation arising from an initial-boundary-value problem for the heat equation, with a nonlinear radiation condition at one of the boundaries). Brief surveys of related papers and of numerical methods conclude the chapter.

In the final two chapters the authors return to first-kind Abel equations. Chapter 8 deals with the ill-posed nature of such equations (with the discussion being based mainly on specific examples) and with general stability estimates for solutions in the setting of \(L^ p(0,a)\). The numerical treatment of first-kind Abel equations is the topic of the brief Chapter 9 (here, the focus is on methods for equations with noisy data and on a case study involving an equation arising in spectroscopy). An extensive list of references concludes the book.

This clearly written monograph is an important and timely contribution to the literature on integral equations.

Reviewer: H.Brunner

##### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

45G10 | Other nonlinear integral equations |

45P05 | Integral operators |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |

65R20 | Numerical methods for integral equations |

45D05 | Volterra integral equations |

44A12 | Radon transform |

44A15 | Special integral transforms (Legendre, Hilbert, etc.) |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |