Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics

A decade has passed since Problems of Nonlinear Deformation, the first book by E.I. Grigoliuk and V.I. Shalashilin was published. This book gave a systematic account of the parametric continuation method. Over the last ten years, the understanding of this method has sufficiently broadened. For example, it is now clear that one parametric continuation algorithm can work efficiently for building up any parametric set. This fact significantly widens its potential applications. In this book the authors refer to the continuation solution with the optimal parameter as the best parametrization. The optimal continuation parameter provides the best conditions in a linearized system of equations at any moment of the continuation process. In this book the authors consider the best parameterization for nonlinear algebraic or transcendental equations, initial value or Cauchy problems for ordinary differential equations (ODEs), including stiff systems, differential-algebraic equations, functional-differential equations, the problems of interpolation and approximation of curves, and for nonlinear boundary-value problems for ODEs with a parameter. They also consider the best parameterization for analyzing the behavior of solutions near singular points. Parametric Continuation and Optimal Parametrization is one of the first books in which the best parametrization is regarded systematically for a wide class of problems. It is of interest to scientists, specialists and postgraduate students working in the field of applied and numerical mathematics and mechanics.