Book "Separation of Variables and Exact Solutions to Nonlinear PDEs"

Abstract

This book is devoted to describing and applying methods of generalized and functional separation of variables used to find exact solutions of nonlinear partial differential equations (PDEs). It also presents the direct method of symmetry reductions and its more general version. In addition, the authors describe the differential constraint method, which generalizes many other exact methods.

The presentation involves numerous examples of utilizing the methods to find exact solutions to specific nonlinear equations of mathematical physics. The equations of heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion theory, chemical technology, biology, and other disciplines are studied.

Particular attention is paid to nonlinear equations of a reasonably general form that depend on one or several arbitrary functions. Such equations are the most difficult to analyze. Their exact solutions are of significant practical interest, as they are suitable to assess the accuracy of various approximate analytical and numerical methods.

The book contains new material previously unpublished in monographs. It is intended for a broad audience of scientists, engineers, instructors, and students specializing in mathematical physics, applied and computational mathematics, theoretical physics, mechanics, control theory, chemical engineering science, and other disciplines.

Individual sections of the book and examples are suitable for lecture courses on partial differential equations, equations of mathematical physics, and methods of mathematical physics, for delivering special courses and for practical training.

Full text

A. D. Polyanin and A. I. Zhurov, Separation of Variables and Exact
Solutions to Nonlinear PDEs, CRC Press, Boca Raton–London, 2022.
ISBN-978-0367486891
This book is devoted to describing and applying methods of generalized and
functional separation of variables used to find exact solutions of nonlinear partial
differential equations (PDEs). It also presents the direct method of symmetry
reductions and its more general version. In addition, the authors describe the
differential constraint method, which generalizes many other exact methods. The
presentation involves numerous examples of utilizing the methods to find exact
solutions to specific nonlinear equations of mathematical physics. The equations of
heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion
theory, chemical technology, biology, and other disciplines are studied. Particular
attention is paid to nonlinear equations of a reasonably general form that depend on
one or several arbitrary functions. Such equations are the most difficult to analyze.
Their exact solutions are of significant practical interest, as they are suitable to assess
the accuracy of various approximate analytical and numerical methods.
The book contains new material previously unpublished in monographs. It is
intended for a broad audience of scientists, engineers, instructors, and students
specializing in applied and computational mathematics, theoretical physics,
mechanics, control theory, chemical engineering science, and other disciplines.
Individual sections of the book and examples are suitable for lecture courses on
partial differential equations, equations of mathematical physics, and methods of
mathematical physics, for delivering special courses and for practical training.
PDF-file of the entire book is freely available on the author's personal page, see
https://eqworld.ipmnet.ru/Arts_Polyanin/Separation_of_Variables_and_Exact_
Solutions_to_Nonlinear_PDEs_2021.pdf