Exact Methods for Nonlinear PDEs

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.