Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics

ISBN-13: 9789048163915 / Angielski / Miękka / 2011 / 228 str.

V. I. Shalashilin; E. B. Kuznetsov

Opis

A decade has passed since Problems of Nonlinear Deformation, the first book by E.I. Grigoliuk: and V.I. Shalashilin was published. That work gave a systematic account of the parametric continuation method. Ever since, the understanding of this method has sufficiently broadened. Previously this method was considered as a way to construct solution sets of nonlinear problems with a parameter. Now it is c1ear that one parametric continuation algorithm can efficiently work for building up any parametric set. This fact significantly widens its potential applications. A curve is the simplest example of such a set, and it can be used for solving various problems, inc1uding the Cauchy problem for ordinary differential equations (ODE), interpolation and approximation of curves, etc. Research in this area has led to exciting results. The most interesting of such is the understanding and proof of the fact that the length of the arc calculated along this solution curve is the optimal continuation parameter for this solution. We will refer to the continuation solution with the optimal parameter as the best parametrization and in this book we have applied this method to variable c1asses of problems: in chapter 1 to non-linear problems with a parameter, in chapters 2 and 3 to initial value problems for ODE, in particular to stiff problems, in chapters 4 and 5 to differential-algebraic and functional differential equations.

Spis treści

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. 3. 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.