ABSTRACT
Chapter 1 details the methods of generalized separation of variables that allow one to construct efficiently exact solutions to many nonlinear PDEs. The chapter describes various techniques to find generalized separable solutions following the simple to complex approach. These techniques include a simplified method based on presetting a system of coordinate functions, methods of reduction to a standard bilinear functional equation (followed by solving with the methods of differentiation and splitting), and the method of invariant subspaces. The focus is on nonlinear equations of heat and mass transfer, wave theory, and fluid dynamics, with particular attention to the most challenging equations for analysis that involve one or more arbitrary functions. The application of the methods is illustrated with numerous examples and tables. The chapter also discusses other types of nonlinear equations that admit generalized separable solutions, including delay, integro-differential, fractional derivative, and pseudo-differential equations. The solutions obtained can serve as test problems for numerical methods.
