Exact Methods for Nonlinear PDEs

1. Auflage 2025
ISBN: 978-1-032-71051-8
Verlag: Taylor & Francis Ltd

Buch, Englisch, 384 Seiten, Format (B × H): 156 mm x 234 mm

Reihe: Advances in Applied Mathematics

Exact Methods for Nonlinear PDEs is devoted to the description and practical application of effective analytical methods for finding exact solutions to nonlinear partial differential equations. It covers methods of generalized separation of variables, methods of functional separation of variables, the classical method of symmetry reductions, the direct method of symmetry reductions, the method of weak symmetry reductions, and the method of differential constraints. Furthermore, the book describes several simple methods for finding exact solutions to nonlinear PDEs that do not require specialized knowledge and minimize intermediate calculations. For the first time, the use of non-rigorous reasoning based on heuristic principles such as "from simple to complex" and "structural analogy of solutions" for deriving exact solutions to nonlinear PDEs is discussed. Each section includes numerous examples and exercises to help readers build practical skills in applying the methods. The presentation of the material is illustrated using equations of mass and heat transfer, hydrodynamics, wave theory, nonlinear optics, and other nonlinear equations of mathematical physics.

The key points that distinguish this book from others in the field include:

• Many methods are presented in a simpler and more visual format.

• The material is accessible to a broader range of readers than usual, including those with minimal training and no specialized mathematical education.

• Several simple methods for constructing exact solutions to nonlinear PDEs and delay PDEs, which minimize intermediate calculations are described.

• It emphasizes and details the practical use of non-rigorous reasoning to derive exact solutions for nonlinear PDEs.

The text is intended for a diverse audience including researchers, university professors, engineers, postgraduates, and students specializing in applied mathematics, theoretical physics, and engineering sciences.

Zielgruppe

Postgraduate and Professional Reference

Autoren/Hrsg.

Polyanin, Andrei D.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1.Elementary Invariant Theory: Algebraic Equations and ODEs

2.First-Order Partial Differential Equations

3.Solution Methods for Functional Equations

4.Elementary Invariant Theory: Partial Differential Equations

5.Methods of Generalized Separation of Variables

6.Methods of Functional Separation of Variables

7.DirectMethod of Symmetry Reductions. Weak Symmetries

8.Classical Method of Symmetry Reductions

9.Differential Constraints Method

10.Transformations of Equations of Mathematical Physics

11.Using Simple Solutions to Construct Complex Solutions

12.Constructing Solutions of Complex Equations

Über Autor(innen)

Andrei D. Polyanin, D.Sc., Ph.D., Professor, is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation.