Interest in nonlinear problems in mechanics has been revived and intensified by the capacity of digital computers. Consequently, a question offundamental importance is the development of solution procedures which can be applied to a large class of problems. Nonlinear problems with a parameter constitute one such class. An important aspect of these problems is, as a rule, a question of the variation of the solution when the parameter is varied. Hence, the method of continuing the solution with respect to a parameter is a natural and, to a certain degree, universal tool for analysis. This book includes details of practical problems and the results of applying this method to a certain class of nonlinear problems in the field of deformable solid mechanics. In the Introduction, two forms of the method are presented, namely continu ous continuation, based on the integration of a Cauchy problem with respect to a parameter using explicit schemes, and discrete continuation, implementing step wise processes with respect to a parameter with the iterative improvement of the solution at each step. Difficulties which arise in continuing the solution in the neighbourhood of singular points are discussed and the problem of choosing the continuation parameter is formulated."

Interest in nonlinear problems in mechanics has been revived and intensified by the capacity of digital computers. Consequently, a question offundamental importance is the development of solution procedures which can be applied to a large class of problems. Nonlinear problems with a parameter constitute one such class. An important aspect of these problems is, as a rule, a question of the variation of the solution when the parameter is varied. Hence, the method of continuing the solution with respect to a parameter is a natural and, to a certain degree, universal tool for analysis. This book includes details of practical problems and the results of applying this method to a certain class of nonlinear problems in the field of deformable solid mechanics. In the Introduction, two forms of the method are presented, namely continu ous continuation, based on the integration of a Cauchy problem with respect to a parameter using explicit schemes, and discrete continuation, implementing step wise processes with respect to a parameter with the iterative improvement of the solution at each step. Difficulties which arise in continuing the solution in the neighbourhood of singular points are discussed and the problem of choosing the continuation parameter is formulated."

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.