##
**Functional equations and modelling in science and engineering.**
*(English)*
Zbl 0801.39004

This book is an excellent introductory text on the theory and applications of functional equations. With many applications in diverse fields (e.g. mechanics, statistics, economics), the authors have succeeded in the difficult task of bridging the gap between mathematicians on one hand, and engineers and scientists on the other. Engineers and scientists are given a very readable explanation of functional equations techniques which can be incorporated in their work, while mathematicians are provided with illustrations of the usefulness of functional equations in mathematical modelling.

The general structure of this work follows that of the book of J. Aczél [Lectures on functional equations and their applications, (1966; Zbl 0139.093)]. Another book which is rather similar in character is that of W. Eichborn [Functional equations in economics (1978; Zbl 0397.90002)].

Chapter 1 is introductory; it presents basic concepts, definitions and some fundamental solutions methods. Chapter 2 treats the fundamental Cauchy equations and some generalizations, Jensen’s equation, D’Alembert’s equation; several interesting examples of applications (to meteorology, mechanics, geometry, probability and statistics, and others) are given.

Chapter 3 deals with equations for several unknown functions of one variable, such as Pexider equations. In Chapters 4 and 5, equations for functions of several variables are studied; in particular, Chapter 5 contains discussions of the associativity, transitivity, bisymmetry and transformation equations.

Chapter 6 examines connections between functional equations and ordinary or partial differential equations. Chapter 7 treats vector and matrix versions of some equations analyzed in earlier chapters. A variety of examples of applications can be found in each chapter, and the book ends with a collection of miscellaneous applications in Chapter 8. Each chapter concludes with several exercises, and there is an extensive bibliography .

Because of the intended audience, mathematical results are not presented in their most general form, and many proofs have been simplified or omitted. This occasionally leads to mistakes, as in Example 3.5.4 where the authors incorrectly identify the general monotonic solution of Cauchy’s equation as the general invertible solution. Nonetheless this reference/text serves a worthy purpose and would be a valuable addition to any library which includes mathematics or any discipline using mathematical modelling.

The general structure of this work follows that of the book of J. Aczél [Lectures on functional equations and their applications, (1966; Zbl 0139.093)]. Another book which is rather similar in character is that of W. Eichborn [Functional equations in economics (1978; Zbl 0397.90002)].

Chapter 1 is introductory; it presents basic concepts, definitions and some fundamental solutions methods. Chapter 2 treats the fundamental Cauchy equations and some generalizations, Jensen’s equation, D’Alembert’s equation; several interesting examples of applications (to meteorology, mechanics, geometry, probability and statistics, and others) are given.

Chapter 3 deals with equations for several unknown functions of one variable, such as Pexider equations. In Chapters 4 and 5, equations for functions of several variables are studied; in particular, Chapter 5 contains discussions of the associativity, transitivity, bisymmetry and transformation equations.

Chapter 6 examines connections between functional equations and ordinary or partial differential equations. Chapter 7 treats vector and matrix versions of some equations analyzed in earlier chapters. A variety of examples of applications can be found in each chapter, and the book ends with a collection of miscellaneous applications in Chapter 8. Each chapter concludes with several exercises, and there is an extensive bibliography .

Because of the intended audience, mathematical results are not presented in their most general form, and many proofs have been simplified or omitted. This occasionally leads to mistakes, as in Example 3.5.4 where the authors incorrectly identify the general monotonic solution of Cauchy’s equation as the general invertible solution. Nonetheless this reference/text serves a worthy purpose and would be a valuable addition to any library which includes mathematics or any discipline using mathematical modelling.

Reviewer: B.Ebanks (Louisville)

### MSC:

39Bxx | Functional equations and inequalities |

00A71 | General theory of mathematical modeling |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39A10 | Additive difference equations |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

74-XX | Mechanics of deformable solids |

91Bxx | Mathematical economics |