##
**The numerical solution of Volterra equations.**
*(English)*
Zbl 0611.65092

CWI Monographs, 3. Centrum voor Wiskunde en Informatica, Amsterdam. Amsterdam etc.: North-Holland. XVI, 588 p. $ 55.50; Dfl. 150.00 (1986).

The authors which are known for their own contributions to the field have remarkably well succeeded in producing this comprehensive and up-to-date monograph. Its main aim is the presentation of discretization methods for integral equations of the form
\[
\theta y(t)=g(t)+\int^{t}_{0}k(t,s,y(s))ds,\quad t\in I,
\]
where \(I=[0,T]\) or [0,\(\infty)\) and \(\theta =1\) (second kind equation) or \(\theta =0\) (first kind equation). As special cases one has a linear Volterra integral equation (k(t,s)y(s)) instead of k(t,s,y(s)), Volterra integral equations with weakly singular kernels \((t-s)^{-\alpha}k(t,s,y(s))\) with \(0<\alpha <1\) instead of k(t,s,y(s)), and Volterra integral equations with convolution kernels. And there are generalizations: systems of Volterra integral equations, systems of Volterra integro- differential equations. The analogies of analytical theory and numerical methods to those of the initial value problem for an ordinary differential equation is immediately obvious if \(\theta =1\) and g(t)\(\equiv y(0)\) are taken.

In Chapter 1 the basic theory of the types of equations mentioned above is developed (existence, uniqueness, asymptotic behavior) and useful Gronwall inequalities (continuous and discrete) are given. Chapter 2 is a survey on numerical quadrature and on linear multistep methods for ordinary differential equations. These methods are generalized in Chapter 3 to Volterra linear multistep methods. Chapter 4 is devoted to Runge- Kutta-type methods for Volterra equations, and in Chapter 5 collocation methods for Volterra equations with regular kernels are treated, and questions of superconvergence are discussed. Chapter 6 is on Volterra equations with weakly singular kernels (Abel-type integral equations) and contains the theory of fractional convolution quadratures recently (1983, 1985) introduced by Ch. Lubich, descriptions of collocation methods, product integration methods, and a short discussion of the problem of ill-posedness of first-kind Abel integral equations. In Chapter 7, Numerical stability, the various notions of stability of solution of a system of ordinary differential equations \[ y'(t)=f(t,y(t)),\quad t_ 0\leq t<\infty \] are generalized to Volterra integral equations (and systems of such equations) where now the sensitivity to perturbations of the initial function g(t) is important (in contrast to sensitivity to perturbations of an initial value). A numerical method should imitate qualitatively the asymptotic behavior (as \(t\to \infty)\) of the true solution. These problems are thoroughly treated for linear multistep methods and for Runge-Kutta-type methods, and analogously to the case of ordinary differential equations ”test equations” are used, the basic one being \(\theta y(t)=g(t)+\xi \int^{t}_{0}y(s)ds.\) A more complicated one is the ”convolution test equation”. Chapter 8 gives a survey on available software and on test examples. Finally, there are 51 pages of references.

The book is essentially self-contained, and in places where proofs are omitted easily accessible references are given. Results of many illustrative numerical case studies are displayed, and each chapter ends with ”Notes” giving many hints to the original literature where topics have been treated and where more details and special results can be found. The book will be very useful for both the beginner who wants to be introduced and for the researcher who needs to be informed on all the aspects of the subject and on the latest developments.

In Chapter 1 the basic theory of the types of equations mentioned above is developed (existence, uniqueness, asymptotic behavior) and useful Gronwall inequalities (continuous and discrete) are given. Chapter 2 is a survey on numerical quadrature and on linear multistep methods for ordinary differential equations. These methods are generalized in Chapter 3 to Volterra linear multistep methods. Chapter 4 is devoted to Runge- Kutta-type methods for Volterra equations, and in Chapter 5 collocation methods for Volterra equations with regular kernels are treated, and questions of superconvergence are discussed. Chapter 6 is on Volterra equations with weakly singular kernels (Abel-type integral equations) and contains the theory of fractional convolution quadratures recently (1983, 1985) introduced by Ch. Lubich, descriptions of collocation methods, product integration methods, and a short discussion of the problem of ill-posedness of first-kind Abel integral equations. In Chapter 7, Numerical stability, the various notions of stability of solution of a system of ordinary differential equations \[ y'(t)=f(t,y(t)),\quad t_ 0\leq t<\infty \] are generalized to Volterra integral equations (and systems of such equations) where now the sensitivity to perturbations of the initial function g(t) is important (in contrast to sensitivity to perturbations of an initial value). A numerical method should imitate qualitatively the asymptotic behavior (as \(t\to \infty)\) of the true solution. These problems are thoroughly treated for linear multistep methods and for Runge-Kutta-type methods, and analogously to the case of ordinary differential equations ”test equations” are used, the basic one being \(\theta y(t)=g(t)+\xi \int^{t}_{0}y(s)ds.\) A more complicated one is the ”convolution test equation”. Chapter 8 gives a survey on available software and on test examples. Finally, there are 51 pages of references.

The book is essentially self-contained, and in places where proofs are omitted easily accessible references are given. Results of many illustrative numerical case studies are displayed, and each chapter ends with ”Notes” giving many hints to the original literature where topics have been treated and where more details and special results can be found. The book will be very useful for both the beginner who wants to be introduced and for the researcher who needs to be informed on all the aspects of the subject and on the latest developments.

Reviewer: R.Gorenflo

### MSC:

65R20 | Numerical methods for integral equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

45-04 | Software, source code, etc. for problems pertaining to integral equations |

45D05 | Volterra integral equations |

45G10 | Other nonlinear integral equations |

45J05 | Integro-ordinary differential equations |

65D32 | Numerical quadrature and cubature formulas |

41A55 | Approximate quadratures |

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