Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics by V.I. ShalashilinParametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics by V.I. Shalashilin

Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics

byV.I. Shalashilin, Evgenii Kuznetsov

Paperback | September 22, 2011

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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.
Title:Parametric Continuation and Optimal Parametrization in Applied Mathematics and MechanicsFormat:PaperbackDimensions:236 pages, 9.25 × 6.1 × 0 inPublished:September 22, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048163919

ISBN - 13:9789048163915

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Table of Contents

Preface. 1: Nonlinear Equations with a Parameter. 1. Two forms of the method of continuation of the solution with respect to a parameter. 2. The problem of choosing the continuation parameter. Replacement of the parameter. 3. The best continuation parameter. 4. The algorithms using the best continuation parameter and examples of their application. 5. Geometrical visualization of step-by-step processes. 6. The solution continuation in vicinity of essential singularity points. 2: The Cauchy Problem for Ordinary Differential Equations. 1. The Cauchy problem as a problem of solution continuation with respect to a parameter. 2. Certain properties of lambda-transformation. 3. Algorithms, softwares, examples. 3: Stiff Systems of Ordinary Differential Equations. 1. Characteristic features of numerical integration of stiff system of ordinary differential equations. 2. Sinular perturbed equations. 3. Stiff systems. 4. Stiff equations for partial derivatives. 4: Differential-Algebraic Equations. 1. Classification of systems of DAE. 2. The best argument for a system of differential-algebraic equations. 3. Explicit differential-algebraic equations. 4. Implicit ordinary differential equations. 5. Implicit differential-algebraic equations. 5: Functional-Differential Equations. 1. The Cauchy problem for equations with a retarded argument. 2. The Cauchy problem for Volterra's integro-differential equations. 6: The Parametric Approximation. 1. The parametric interpolation. 2. The parametric approximation. The continuous approximation. 7: Nonlinear Boundary Value Problems for Ordinary Differential Equations. 1. The equations of solution continuation for nonlinear one-dimensional boundary value problems. 2. The discrete orthagonal shooting method. 3. The algorithms for continuous and discrete continuation of the solution with respect to a parameter for nonlinear one-dimensional boundary value problems. 4. Example: large deflections of the circle arch.8: Continuation of the Solution Near Singular Points. 1. Classification of singular points. 2. The simplest form of bifurcation equations. 3. The simplest case of branching (rank(J0)=n 1. 4. The case of branching when rank (J0)=n 2. References. Bibliography.