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**Diffusion-wave fields. Mathematical methods and Green functions.**
*(English)*
Zbl 0976.78001

New York, NY: Springer. xii, 741 p. (2001).

A useful, well balanced, book on the Green function method applied to solutions of numerous problems involving diffusion and wave processes under various physical conditions in Cartesian, cylindrical and spherical geometries. Mathematical procedures are performed in one, two and three dimensions. It is a textbook on the graduate level suitable for students but it can also serve as a reference literature to advanced researchers.

The introduction contains a general mathematical background needed for the Fourier-Laplace and Green function methods in solving inhomogeneous partial differential equations for thermal-wave fields with Dirichlet, Neumann and mixed boundary conditions.

The first eight out of the ten chapters of the book deal with the determination of Green functions and their application in solving equations for thermal-wave fields with given types and distributions of source functions. Thus, Chapters 1 and 2 give Green functions and the resulting thermal-wave field solutions respectively, related to one-dimensional problems in the Cartesian geometry for spatial configurations such as infinite space, semi-infinite space and layers of finite thickness while Chapters 3 and 4 give an extension to two and three dimensional fields. Chapters 5 and 6 consider Green functions and thermal-wave field solutions respectively in the cylindrical geometry for several typical configurations: domains with infinite and finite radii, wedges and edges. Problems on spherical geometry are presented in Chapters 7 and 8 including domains of hollow spheres and spherical cones.

The last two chapters are devoted to physical processes other than thermal diffusion which are also the main field of the authors’ scientific interest. In this sense, Chapter 9 deals with electric density carrier fields in electronic solid and semi-conductor environments. Solutions are obtained for one and three dimensional Cartesian and cylindrical geometries including those of specific composite electronic solids. Chapter 10 is devoted to photon fields and solutions of radiative transfer equations with applications to turbid media and tissue. These two chapters offer an updated foundation for further original scientific research in the domains considered in this investigation.

Each of the ten chapters is followed by a set of illustrative problems and the most relevant list of publication references. Finally, in the appendix, the necessary spatial functions with their basic mathematical properties are presented.

The introduction contains a general mathematical background needed for the Fourier-Laplace and Green function methods in solving inhomogeneous partial differential equations for thermal-wave fields with Dirichlet, Neumann and mixed boundary conditions.

The first eight out of the ten chapters of the book deal with the determination of Green functions and their application in solving equations for thermal-wave fields with given types and distributions of source functions. Thus, Chapters 1 and 2 give Green functions and the resulting thermal-wave field solutions respectively, related to one-dimensional problems in the Cartesian geometry for spatial configurations such as infinite space, semi-infinite space and layers of finite thickness while Chapters 3 and 4 give an extension to two and three dimensional fields. Chapters 5 and 6 consider Green functions and thermal-wave field solutions respectively in the cylindrical geometry for several typical configurations: domains with infinite and finite radii, wedges and edges. Problems on spherical geometry are presented in Chapters 7 and 8 including domains of hollow spheres and spherical cones.

The last two chapters are devoted to physical processes other than thermal diffusion which are also the main field of the authors’ scientific interest. In this sense, Chapter 9 deals with electric density carrier fields in electronic solid and semi-conductor environments. Solutions are obtained for one and three dimensional Cartesian and cylindrical geometries including those of specific composite electronic solids. Chapter 10 is devoted to photon fields and solutions of radiative transfer equations with applications to turbid media and tissue. These two chapters offer an updated foundation for further original scientific research in the domains considered in this investigation.

Each of the ten chapters is followed by a set of illustrative problems and the most relevant list of publication references. Finally, in the appendix, the necessary spatial functions with their basic mathematical properties are presented.

Reviewer: Vladimir Čadež (Bruxelles)

### MSC:

78-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory |

80-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to classical thermodynamics |

35Q60 | PDEs in connection with optics and electromagnetic theory |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

78A02 | Foundations in optics and electromagnetic theory |

80A05 | Foundations of thermodynamics and heat transfer |

78A40 | Waves and radiation in optics and electromagnetic theory |